Statistical Mechanics by Gallavotti

Giovanni Gallavotti

STATISTICAL MECHANICS

Short Treatise

Roma ��

Short treatise of

Statistical Mechanics

Giovanni Gallavotti

Dipartimento di Fisica

Universit�a di Roma La Sapienza�� Roma

Roma ��

i

A Daniela� sempre

Preface

This book is the end result of a long story that started with my involvementas Coordinator of the Statistical Mechanics section of the Italian Encyclo�pedia of Physics�

An Italian edition collecting several papers that I wrote for the Encyclope�dia appeared in September �� with the permission of the Encyclopediaand the sponsorship of Consiglio Nazionale delle Ricerche �CNR�GNFM��

The present work is not a translation of the Italian version but it overlapswith it an important part of it �Ch�I�II�III�VIII� is still based on three arti�cles written as entries for the it Encicopledia della Fisica �namely �Mec�canica Statistica�� Teoria degli Insiemi� and �Moto Browniano�� whichmake up about �� of the present book and� furthermore� it still contains�with little editing and updating� my old review article on phase transitions�Ch�VI� published in La Rivista del Nuovo Cimento�� In translating theideas into English� I introduced many revisions and changes of perspectiveas well as new material �while also suppressing some other material��

The aim was to provide an analysis� intentionally as nontechnical as I wasable to make it� of many fundamental questions of Statistical Mechanics�about two centuries after its birth� Only in a very few places have I en�tered into really technical details� mainly on subjects that I should knowrather well or that I consider particularly important �the convergence of theKirkwood�Salsburg equations� the existence of the thermodynamic limit�the exact soltution of the Ising model� and in part the exact solution of thesix vertex models�� The points of view expressed here were presented ininnumerable lectures and talks mostly to my students in Roma during thelast � years� They are not always �mainstream views�� but I am con�dentthat they are not too far from the conventionally accepted �truth� and I donot consider it appropriate to list the di�erences from other treatments� Ishall consider this book a success if it prompts comments �even if dictatedby strong disagreement or dissatisfaction� on the �few� points that mightbe controversial� This would mean that the work has attained the goal ofbeing noticed and of being worthy of criticism�

I hope that this work might be useful to students by bringing to their at�tention problems which� because of �concreteness necessities� �i�e� becausesuch matters seem useless� or sometimes simply because of lack of time��are usually neglected even in graduate courses�

This does not mean that I intend to encourage students to look at questionsdealing with the foundations of Physics� I rather believe that young studentsshould refrain from such activities� which should� possibly� become a subject

ii

of investigation after gaining an experience that only active and advancedresearch can provide �or at least the attempt at pursuing it over manyyears�� And in any event I hope that the contents and the arguments I haveselected will convey my appreciation for studies on the foundations thatkeep a strong character of concreteness� I hope� in fact� that this book willbe considered concrete and far from speculative�Not that students should not develop their own philosophical beliefs about

the problems of the area of Physics that interests them� Although oneshould be aware that any philosophical belief on the foundations of Physics�and Science�� no matter how clear and irrefutable it might appear to theperson who developed it after long meditations and unending vigils� is veryunlikely to look less than objectionable to any other person who is givena chance to think about it� it is nevertheless necessary� in order to groworiginal ideas or even to just perform work of good technical quality� topossess precise philosophical convictions on the rerum natura� Providedone is always willing to start afresh� avoiding� above all� thinking one has�nally reached the truth� unique� unchangeable and objective �into whoseexistence only vain hope can be laid��I am grateful to the Enciclopedia Italiana for having stimulated the begin�

ning and the realization of this work� by assigning me the task of coordinat�ing the Statistical Mechanics papers� I want to stress that the �nancial andcultural support from the Enciclopedia have been of invaluable aid� Theatmosphere created by the Editors and by my colleagues in the few roomsof their facilities stimulated me deeply� It is important to remark on therather unusual editorial enterprise they led to it was not immediately an�imated by the logic of pro�t that moves the scienti�c book industry whichis very concerned� at the same time� to avoid possible costly risks�I want to thank G� Alippi� G� Altarelli� P� Dominici and V� Cappelletti who

made a �rst version in Italian possible� mainly containing the Encyclopediaarticles� by allowing the collection and reproduction of the texts of which theEncyclopedia retains the rights� I am indebted to V� Cappelletti for grantingpermission to include here the three entries I wrote for the Enciclopedia delleScienze Fisiche �which is now published�� I also thank the Nuovo Cimentofor allowing the use of the �� review paper on the Ising model�I am indebted for critical comments on the various drafts of the work�

in particular� to G� Gentile whose comments have been an essential con�tribution to the revision of the manuscript� I am also indebted to severalcolleagues P� Carta� E� J�arvenp�a�a� N� Nottingham and� furthermore� M�Campanino� V� Mastropietro� H� Spohn whose invaluable comments madethe book more readable than it would otherwise have been�

Giovanni Gallavotti

Roma� January ��

iii

Index

I� Classical Statistical Mechanics � � � � � � � � � � � � � � � � � �

�� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Microscopic Dynamics � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Time Averages and the Ergodic Hypothesis � � � � � � � � � � � � � � � ��

�� Recurrence Times and Macroscopic Observables � � � � � � � � � � � � � � ��

�� Statistical Ensembles or �Monodes and Models of Thermodynamics� Thermodynamics without Dynamics � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Models of Thermodynamics� Microcanonical and Canonical Ensembles and the Ergodic Hypothesis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Critique of the Ergodic Hypothesis � � � � � � � � � � � � � � � � � � � ��

�� Approach to Equilibrium and Boltzmann s Equation� Ergodicity and Irreversibility��

�� A Historical Note� The Etymology of the Word �Ergodic and the Heat Theorems��

Appendix ��A�� Monocyclic Systems� Keplerian Motions and Ergodic Hypothesis � ��

Appendix ��A�� GradBoltzmann Limit and Lorentz s Gas � � � � � � � � � � ��

II� Statistical Ensembles � � � � � � � � � � � � � � � � � � � � � � ��

�� Statistical Ensembles as Models of Thermodynamics� � � � � � � � � � � � ��

�� Canonical and Microcanonical Ensembles� Orthodicity � � � � � � � � � � � ��

�� Equivalence between Canonical and Microcanonical Ensembles � � � � � � � ��

�� Non Equivalence of the Canonical and Microcanonical Ensembles� Phase Transitions�Boltzmann s Constant � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� The Grand Canonical Ensemble and Other Orthodic Ensembles � � � � � � � ��

�� Some Technical Aspects � � � � � � � � � � � � � � � � � � � � � � � � ��

iv

III� Equipartition and Critique � � � � � � � � � � � � � � � � � � ��

�� Equipartition and Other Paradoxes and Applications of Statistical Mechanics ��

�� Classical Statistical Mechanics when Cell Sizes Are Not Negligible � � � � � � ��

�� Introduction to Quantum Statistical Mechanics � � � � � � � � � � � � � ��

�� Philosophical Outlook on the Foundations of Statistical Mechanics � � � � ��

IV� Thermodynamic Limit and Stability � � � � � � � � � � � � ��

�� The Meaning of the Stability Conditions � � � � � � � � � � � � � � � ��

�� Stability Criteria � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Thermodynamic Limit � � � � � � � � � � � � � � � � � � � � � � � � ��

V� Phase Transitions � � � � � � � � � � � � � � � � � � � � � � � ��

�� Virial Theorem� Virial Series and van der Waals Equation � � � � � � � � ��

�� The Modern Interpretation of van der Waals Approximation � � � � � � ��

�� Why a Thermodynamic Formalism� � � � � � � � � � � � � � � � � � ��

�� Phase Space in In�nite Volume and Probability Distributions on it� Gibbs Distributions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Variational Characterization of Translation Invariant Gibbs Distributions � ��

�� Other Characterizations of Gibbs Distributions� The DLR Equations � � � ��

�� Gibbs Distributions and Stochastic Processes � � � � � � � � � � � � � ��

�� Absence of Phase Transitions� d � �� Symmetries� d � � � � � � � � � � ��

�� Absence of Phase Transitions� High Temperature and the KS Equations � � ��

�� Phase Transitions and Models � � � � � � � � � � � � � � � � � � � ��

Appendix ��A�� Absence of Phase Transition in non Nearest Neighbor OneDimensionalSystems � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

V I� Coexistence of Phases � � � � � � � � � � � � � � � � � � � � ��

�� The Ising Model� Inequivalence of Canonical and Grand Canonical Ensembles ��

�� The Model� Grand Canonical and Canonical Ensembles� Their Inequivalence ��

�� Boundary Conditions� Equilibrium States � � � � � � � � � � � � � � � ��

�� The Ising Model in One and Two dimensions and zero �eld � � � � � � � ��

�� Phase Transitions� De�nitions � � � � � � � � � � � � � � � � � � � � ��

�� Geometric Description of the Spin Con�gurations � � � � � � � � � � � ��

�� Phase Transitions� Existence � � � � � � � � � � � � � � � � � � � � � ��

�� Microscopic Description of the Pure Phases � � � � � � � � � � � � � � ��

�� Results on Phase Transitions in a Wider Range of Temperature � � � � � ��

v

�� Separation and Coexistence of Pure Phases� Phenomenological Considerations ��

�� Separation and Coexistence of Phases� Results � � � � � � � � � � � � ��

�� Surface Tension in Two Dimensions� Alternative Description of the SeparationPhenomena � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� The Structure of the Line of Separation� What a Straight Line Really is � ��

�� Phase Separation Phenomena and Boundary Conditions� Further Results � ��

�� Further Results� Some Comments and Some Open Problems � � � � � � ��

V II� Exactly Soluble Models � � � � � � � � � � � � � � � � � � � ��

�� Transfer Matrix in the Ising Model� Results in d � ��

�� Meaning of Exact Solubility and the TwoDimensional Ising Model � � � � ��

�� Vertex Models � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� A Nontrivial Example of Exact Solution� the TwoDimensional Ising Model ��

�� The Six Vertex Model and Bethe s Ansatz � � � � � � � � � � � � � � � ��

V III� Brownian Motion � � � � � � � � � � � � � � � � � � � � � � ��

�� Brownian Motion and Einstein s Theory � � � � � � � � � � � � � � � � ��

�� Smoluchowski s Theory� � � � � � � � � � � � � � � � � � � � � � � � ��

�� The Uhlenbeck�Ornstein Theory � � � � � � � � � � � � � � � � � � � ��

�� Wiener s Theory � � � � � � � � � � � � � � � � � � � � � � � � � � ��

IX� Coarse Graining and Nonequilibrium � � � � � � � � � � � ��

�� Ergodic Hypothesis Revisited � � � � � � � � � � � � � � � � � � � � ��

�� Timed Observations and Discrete Time � � � � � � � � � � � � � � � � ��

�� Chaotic Hypothesis� Anosov Systems � � � � � � � � � � � � � � � � � ��

�� Kinematics of Chaotic Motions� Anosov Systems � � � � � � � � � � � � ��

�� Symbolic Dynamics and Chaos � � � � � � � � � � � � � � � � � � � � ��

�� Statistics of Chaotic Attractors� SRB Distributions � � � � � � � � � � � ��

�� Entropy Generation� Time Reversibility and Fluctuation Theorem� ExperimentalTests of the Chaotic Hypothesis � � � � � � � � � � � � � � � � � � � � � ��

�� Fluctuation Patterns � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Conditional Reversibility and �Fluctuation Theorems � � � � � � � � ��

�� Onsager Reciprocity and GreenKubo s Formula � � � � � � � � � � � � ��

�� Reversible Versus Irreversible Dissipation� Nonequilibrium Ensembles� � � ��

Appendix ��A�� M�ecanique statistique hors �equilibre� l h�eritage de Boltzmann � ��

Appendix ��A�� Heuristic Derivation of the SRB Distribution � � � � � � � � ��

vi

Appendix ��A�� Aperiodic Motions Can be Begarded as Periodic with In�nite Period��

Appendix ��A�� Gauss Least Constraint Principle � � � � � � � � � � � � � ��

Biblography � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Names index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Analytic index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Citations index � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�

Chapter I�

Classical Statistical Mechanics

I� Classical Statistical Mechanics �

x�� IntroductionStatistical mechanics poses the problem of deducing macroscopic properties

of matter from the atomic hypothesis� According to the hypothesis matterconsists of atoms or molecules that move subject to the laws of classicalmechanics or of quantum mechanics�Matter is therefore thought of as consisting of a very large number N

of particles� essentially point masses� interacting via simple conservativeforces��

A microscopic state is described by specifying� at a given instant� the valueof positions and momenta �or� equivalently� velocities� of each of the Nparticles� Hence one has to specify �N � �N coordinates that determine apoint in phase space� in the sense of mechanics�

It does not seem that in the original viewpoint Boltzmann particles werereally thought of as susceptible of assuming a �N dimensional continuumof states� ��Bo�� p� ��

Therefore if we wish to get a picture of the continuum in words� we �rsthave to imagine a large� but �nite number of particles with certain propertiesand investigate the behavior of the ensemble of such particles� Certain prop�erties of the ensemble may approach a de�nite limit as we allow the numberof particles ever more to increase and their size ever more to decrease� Ofthese properties one can then assert that they apply to a continuum� andin my opinion this is the only non�contradictory de�nition of a continuumwith certain properties

and likewise the phase space itself is really thought of as divided into a �nitenumber of very small cells of essentially equal dimensions� each of whichdetermines the position and momentum of each particle with a maximumprecision�

This should mean the maximum precision that the most perfect measure�ment apparatus can possibly provide� And a matter of principle arises canwe suppose that every lack of precision can be improved by improving theinstruments we use�If we believe this possibility then phase space cells� representing microscopic

states with maximal precision� must be points and they must be conceivedof as a �N dimensional continuum� But since atoms and molecules are notdirectly observable one is legitimized in his doubts about being allowed toassume perfect measurability of momentum and position coordinates�

In fact in �recent� times the foundations of classical mechanics have been

� N � �� particles per mole � �Avogadro s number� this implies� for instance�that � cm� of Hydrogen� or of any other �perfect� gas� at normal conditions �� atm at� �C� contains about �� molecules�

� I� Classical Statistical Mechanics

subject to intense critique and the indetermination principle postulates thetheoretical impossibility of the simultaneous measurement of a componentp of a particle momentum and of the corresponding component q of theposition with respective precisions �p and �q without the constraint

�p�q � h ��

where h � �� erg � sec is the Planck�s constant�Without attempting a discussion of the conceptual problems that the above

brief and super�cial comments raise it is better to proceed by imagining thatthe microscopic states of a N particles system are represented by phase spacecells consisting in the points of R�N with coordinates� �e�g� �Bo��

p�� p� � p� � p�� p� q�� q� � q� � q�� q�

� � �� N ��

if p�� p�� p� are the momentum coordinates of the �rst particle� p�� p�� p�of the second� etc� and q�� q�� q� are the position coordinates of the �rstparticle� q�� q�� q� of the second� etc�� The coordinate p�� and q�� are usedto identify the center of the cell� hence the cell itself�The cell size will be supposed to be such that

�p�q � h ��

where h is an a priori arbitrary constant� which it is convenient not to �xbecause it is interesting �for the reasons just given� to see how the theorydepends upon it� Here the meaning of h is that of a limitation to the preci�sion that is assumed to be possible when measuring a pair of correspondingposition and momentum coordinates�Therefore the space of the microscopic states is the collection of the cubic

cells �� with volume h�N into which we imagine that the phase space isdivided� By assumption it has no meaning to pose the problem of attemptingto determine the microscopic state with a greater precision�The optimistic viewpoint of orthodox statistical mechanics �which admits

perfect simultaneous measurements of positions and momenta as possible�will be obtained by considering� in the more general theory with h � �� thelimit as h � �� which will mean �p � �p�� q � �q�� with p�� q� �xed and�� Even if we wish to ignore �one should not�� the development of quan�

tum mechanics� the real possibility of the situation in which h � � cannotbe directly checked because of the practical impossibility of observing anindividual atom with in�nite precision �or just with �great� precision��

x�� Microscopic Dynamics

The atomic hypothesis� apart from supposing the existence of atoms andmolecules� assumes also that their motions are governed by a deterministiclaw of motion�


This hypothesis can be imposed by thinking that there is a map S

S� � ��

transforming the phase space cells into each other and describing the systemdynamics�If at time t the state of the system is microscopically determined by the

phase space cell �� then at a later time t � � it will be determined by thecell �� Here � is a time step extremely small compared to the macroscopictime intervals over which the system evolution is followed by an observerit is� nevertheless� a time interval directly accessible to observation� at leastin principle�The evolution law S is not arbitrary it must satisfy some fundamental

properties� namely it must agree with the laws of mechanics in order toproperly enact the deterministic principle which is basic to the atomic hy�pothesis�This means� in essence� that one can associate with each phase space cell

three fundamental dynamical quantities the kinetic energy� the potentialenergy and the total energy� respectively denoted by K�� E��For simplicity assume the system to consist of N identical particles with

mass m� pairwise interacting via a conservative force with potential energy� If � is the phase space cell determined by �see �� p�� q�� then theabove basic quantities are de�ned respectively by

K�p�� K��

NXi�

�p�i�� m p�

i� �p��i�� p

��i�� p

��i�

��q��

��NXi�j

�q�i� q�

j� q�

i� �q��i�� q

��i�� q

��i�

E�p�� q�� E�� K�p�� q��

��

where p�i

� �p��i�� p

��i�� p

��i�� q

�i

� �q��i�� q

��i�� q

��i� are the momentum

and position of the i�th particle� i � �� N � in the microscopic statecorresponding to the center �p�� q�� of ��Replacing p�� q�� i�e� the center of �� by another point �p� q� in � one

obtains values K�p��q�� E�p� q� for the kinetic� potential and total energiesdi�erent from K��E�� however such a di�erence has to be nonobservable otherwise the cells � would not be the smallest ones to beobservable� as supposed above�If � is a �xed time interval and we consider the solutions of Hamilton s

equations of motion

!q �E

p�p� q�� !p � �E

q�p� q� ��

with initial data �p�� q�� at time � the point �p�� q�� will evolve in time �into a point S� �p�� q�� p�� q�� S� �p�� q�� One then de�nes S so that


S� � �� if �� is the cell containing �p�� q�� The evolution �� may senda few particles outside the volume V � cubic for simplicity� that we imagine tocontain the particles one has therefore to supplement �� by boundaryconditions that will tell us the physical nature of the walls of V �

They could be re�ecting� if the collisions with the walls are elastic� orperiodic if the opposite faces of the region V are identi�ed �a very convenient�mathematical �ction�� useful to test various models and to minimize the��nite size� e�ects� i�e� dependence of observations on system size��

One cannot� however� escape some questions of principle on the structureof the map S that it is convenient not to ignore� although their deep under�standing may become a necessity only on a second reading�

First we shall neglect the possibility that �p�� q�� is on the boundary of a cell�a case in which �� is not uniquely determined� but which can be avoidedby imagining that the cells walls are slightly deformed��

More important� in fact crucial� is the question of whether S�� S��

implies �� the latter is a property which is certainly true in thecase of point cells �h � �� because of the uniqueness of the solutions ofdi�erential equations� It has an obvious intuitive meaning and an interestdue to its relation with reversibility of motion�

In the following analysis a key role is played by Liouville s theorem whichtells us that the transformation mapping a generic initial datum �p� q� intothe con�guration �p�� q�� S�p� q� is a volume preserving transformation�

This means that the set of initial data �p� q� in � evolves in the time � into

a set "� with volume equal to that of �� Although having the same volumeof � it will no longer have the same form of a square parallelepiped withdimensions �p or �q� For h small it will be a rather small parallelepiped ob�tained from � via a linear transformation that expands in certain directionswhile contracting in others�

It is also clear that in order that the representation of the microscopicstates of the system be consistent it is necessary to impose some non trivialconditions on the time interval� so far unspeci�ed� that elapses betweensuccessive �thought� observations of the motions� Such conditions can beunderstood via the following reasoning�

Suppose that h is very small �actually by this we mean� here and below�that both �p and �q are small� so that the region "� can be regarded asobtained by translating � and possibly by deforming it via a linear dilata�tion in some directions or a linear contraction in others �contraction anddilatation balance each other because� as remarked� the volume remainsconstant�� This is easily realized if h is small enough since the solutions toordinary di�erential equations can always be thought of� locally� as lineartransformations close to the identity� for small evolution times � � Then

i� If S dilates and contracts in various directions� even by a small amount�there must necessarily exist pairs of distinct cells �� for which S�� S�� an example is provided by the map of the plane transforming �x� y�into S�x� y� � �� x� �� y�� and its action on the lattice of


the integers� Assuming that one decides that the cell �� into which a givencell � evolves is� among those which intersect its image S�� the one whichhad the largest intersection with it� then indetermination arises for a set ofcells spaced by about ��It is therefore necessary that � be small� say

� � ��

with such that the map S �associated with S� � see �� hence closeto the identity� produces contractions and expansions of � that can beneglected for the large majority of the cells �� Only in this way will it bepossible that S�� S�� with �� for just a small fraction of the cellsand� hence� one can hope that this possibility is negligible�

It should be remarked explicitly that the above point of view is systemat�ically taken by physicists performing numerical experiments� Phase spaceis represented in computers as a �nite� but very large� set of points whosepositions are changed by the time evolution �how many depends on theprecision of the representation of the reals�� Even if the system studied ismodeled by a nice di�erential equation with global uniqueness and existenceof solutions� the computer program� while trying to generate a permutationof phase space points� will commit errors� i�e� two distinct points will besent to the same point �we do not talk here of round�o� errors� which arenot really errors as they are a priori known� in principle� one thus hopesthat such errors are rare enough to be negligible� This seems inevitableexcept in some remarkable cases� the only nontrivial one I know of being in�LV��

ii� But � cannot be too small either� if one wishes to maintain coherentlythe point of view that microscopic states are described by phase space cells�In fact to a cell � is associated a natural time scale �� which can bede�ned as the minimum time in order that � becomes distinguishable fromthe cell into which it evolves in time �� And � must be necessarilylarger than the latter minimum time scale

��

�otherwise we have Zeno s paradox and nothing moves��

Summarizing we can say that in order to be able to de�ne the dynamicevolution as a map permuting the phase space cells it must be that � bechosen so that

� � ��max�

��

where the quotes mean that the maximum has to be taken as � varieswithin the �majority� of the cells� where one can suppose that ��

implies S�� S��

# I� Classical Statistical Mechanics

One should realize that if is a �reasonable� molecular potential �a typicalmodel for is� for instance� the Lennard�Jones potential with intensity �and range r� given by �r� � �� r�r �� r�r �� it will generically be that

limh��

� � � ��

while for small h the right hand side of �� which has a purely kine�matical nature� becomes h independent�Hence it will be possible� at least in the limit in which h tends to �� to de�ne

a � so that �� hold� i�e� it will be possible to ful�ll the aboveconsistency criteria for the describing microscopic states of the system via�nite cells�On the other hand if h � �� and a posteriori one should think that h �

�� erg sec� the question we are discussing becomes quite delicatewere it not because we do not know what we should understand when wesay the �large majority� of the phase space cells�In fact� on the basis of the results of the theory it will become possible to

evaluate the in$uence on the results themselves of the existence of pairs ofcells �� with S�� S��Logically at this point the analysis of the question should be postponed

until the consequences of the hypotheses that we are assuming allow us toreexamine it� It is nevertheless useful� in order to better grasp the delicatenature of the problem and the orders of magnitude involved� to anticipatesome of the basic results and to provide estimates of � readers preferringto think in purely classical terms� by imagining that h � � on the basis ofa dogmatic interpretation of the �classical� atomic hypothesis� can skip thediscussion and proceed by systematically taking the limit as h � � of thetheory that follows�It is however worth stressing that setting h � � is an illusory simpli�cation

avoiding posing a problem that is today well known to be deep� Assumingthat� at least in principle� it should be possible to measure exactly positionsand momenta of a very large number of molecules �or even of a single one�means supposing it is possible to perform a physical operation that no onewould be able to perform� It was the obvious di%culty� one should recall� ofsuch an operation that in the last century made it hard for some to acceptthe atomic hypothesis�Coming to the problem of providing an idea of the orders of magnitude of � one can interpret �max� in �� as evaluated by considering as typicalcells those for which the momenta and the reciprocal distances of the parti�cles take values �close� to their �average values�� The theory of statisticalensembles �see below� will lead to a natural probability distribution givingthe probability of each cell in phase space� when the system is in macro�scopic equilibrium� Therefore we shall be able to compute� by using thisprobability distribution� the average values of various quantities in terms ofmacroscopic quantities like the absolute temperature T � the particle massm� the particle number N � and the volume V available to the system�


The main property of the probability distributions of the microscopic statesobserved in a situation in which the macroscopic state of the system is inequilibrium is that the average velocity and average momentum v� p will berelated to the temperature by

p � mv �p

�mkBT � mv� � �kBT �� #��

where kB � ��#� �� erg �K�� is the universal Boltzmann s constant�Other relevant quantities are the characteristic parameters of the interac�

tion� i�e� the strength � with the dimension of energy and the range r� withthe dimension of length� It follows from the developments of the theoryof equilibrium statistical mechanics� independently of the particular form of�r� �as long as it is �reasonable�� like for instance the above mentionedLennard�Jones potential�� that � � kBT

�c where T �

c is the critical liquefac�tion temperature and r� is of the order of the molecular diameter �between �� cm and �� cm in the simplest gases like H�� He�O�� CO�� seeChap�V��We estimate �rst �the time scale over which expansion and contraction

of a phase space cell become sensible� looking at a typical cell where onecan assume that the particles evolve in time without undergoing multiplecollisions� In such a situation the relative variation of a linear dimension of� in the time � will be� for small � � proportional to � and it may depend on��m� r�� v the pure numbers related to � and to the phase space dilatations�i�e� to the derivatives of the forces appearing in the equations of motion�

that one can form with the above quantities are �� mr��

�� and ��mv�

mr��

Hence the phase space changes in volume will be negligible� recalling thatmv� � �kBT � from �� #�� and setting � � kBT

�c � provided

� � min

�� mr��kBT �

c

��mr��kBT

��

The condition �p��mr�� means that� even during a microscopic collision

taking place while the time � elapses� there is no sensible expansion whilethe second condition � � r�� v means that the time � is short with respectto the total collision duration �which therefore takes several units of � tobe completed��To estimate � �the time scale over which a cell evolves enough to be

distinguishable from itself� note that� given �� the coordinates p�� q� of thephase space points in the cell � change obeying the Hamiltonian equationsof motion� in the time � � by

j�q�j � j� Ep�

�p� q�j � �� E

�p

j�p�j � j � �E

q��p� q�j � �

�� E

�q

��

�� I� Classical Statistical Mechanics

where �� E� �

�� E are the variations of the energy E in the cell � when the

coordinates p� or� respectively� q� vary by the amount �p or �q� i�e� theyvary by a quantity equaling the linear dimensions of the cell �� while theothers stay constant �so that the variations in �� are related to thepartial derivatives of the energy function E��De�ning therefore the energy indetermination� which we denote by �E��

in the cell � as�E�� max

�� E� �� E� ��

we see that the minimum time �� that one has to wait in order to seethat the cell evolves into a cell which is distinguishable from � itself is

�� max�� E�p � �q� or �� max�

�� E�q � �p ��

in fact� �p and �q being the linear dimensions of �� Eq� �� just saysthat at least one of the sides of � has moved away by a quantity of theorder of its length �thus becoming distinguishable from itself��Since we set �p �q � h one deduces from ��

��E�� h ��

and � � �t can be chosen so that� introducing the notation

�E � ��min�� E��

we have�t�E � h � ��

��

We can therefore see� on the basis of �� whether or not aninterval � �� admissible for � exists� We can in fact imagine that �p p�hence �E � p �p�m � &p��m � �kBT and

� � h�kBT � ��

Equation �� gives a remarkable interpretation of the time scale h�kBT it is the time necessary so that a phase space cell� typical among thosedescribing the microscopic equilibrium states at temperature T � becomesdistinguishable from itself�One can say� di�erently� that � is determined by the size of !p� !q� i�e� by

the size of the �rst derivatives of the Hamiltonian� while is related to thephase space expansion� i�e� to the second derivatives of the Hamiltonian�With some algebra one derives� from ��

� � � �mr��kBT�c �h

�� min�T�T �c � �T�T

�c ��

Therefore it is clear that the relation � � � �� necessary for a consistentdescription of the microscopic states in terms of phase space cells� will be

I� Classical Statistical Mechanics ��

satis�ed for large T � say T � T�� but not for small T �unless one takesh � �� And from the expression just derived for the ratio � � one gets � � � � if T � T� with

T��T�c � max

� h

�mr��kBT�c ��

�h�

��mr��kBT�c ��

�� #�

��

Table �� below gives an idea of the orders of magnitude it is elaboratedhaving chosen h � �� erg sec� i�e� Planck s constant� A similartable would be derived if� ignoring Planck�s constant suggested as a naturalaction unit from quantum mechanics� we took �p p� �q �

pV�N � as

Boltzmann himself did when performing various conceptual calculations��Bo�� Bo�� in his attempts to explain why his physical theory did notcontradict mathematical logic�In fact with the latter choice the action unit �p �q � p �

pV�N would be�

in �reasonable cases� � � cm� of hydrogen� m � �� g� T � �� K�N � �� kB � ��#�� ergoK�� of the same order of magnitudeas Planck s constant� namely it would be �p �q � �� erg � sec�The corresponding order of magnitude of � is � � �� sec�The sizes of the estimates for T��T

�c in the table show that the question

of logical consistency of the microscopic states representation in terms ofphase space cells permuted by the dynamics� if taken literally� depends ina very sensitive way on the value of h and� in any event� it is doomed toinconsistency if T � � and � �� hence � � �� and �

pmr��

��

Table � Orders of magnitude �NA denotes Avogadro s number�

r� � � kBT�c T��T

�c T true

c A B m

��cm ��erg �K �� cgs cgs �� gH� �� CO� �� He �� # �� N� �� #�� #O� ��

The columns A�B give empirical data� directly accessible from experiments and expressedin cgs units �i�e� A in erg � cm� and B in cm�� of the van der Waals equation of state�If n � N�NA � number of moles� R � kBNA� see x�� for �� below� then theequation of state is�

�P �An��V ��V � nB� � nRT ��

which is supposed here in order to derive values for �� r� via the relations�

�B�NA� � ��

�

�r�

�

�� v� A�N�

A ��

��v� ��

which lead to the expressions �see x�� r� � ��B��NA�� A��BNA �

�� kBT�c �� T

truec � experimental value of the critical temperature � T �

c �


In this book I will choose the attitude of not attempting to discuss whichwould be the structure of a statistical mechanics theory of phenomenabelow T� if a strict� �axiomatic�� classical viewpoint was taken assuming�p � �� q � � the theory would be extremely complicated as discoveredin the famous simulation �FPU�� and it is still not well understood eventhough it is full of very interesting phenomena� see �GS� �� Be�� Be��and Chap�III� x��

x�� Time Averages and the Ergodic Hypothesis

We are led� therefore� to describe a mechanical system of N identical massm particles �at least at not too low temperatures� T � T�� see �� #��in terms of �a� an energy function ��Hamiltonian�� de�ned on the �N �dimensional phase space and �b� a subdivision of such a space into cells� of equal volume h�N � whose size is related to the highest precision withwhich we presume to be able to measure positions and momenta or timesand energies�Time evolution is studied on time intervals multiples of a unit � large

compared to the time scale �t associated with the cell decomposition ofphase space by �� and small compared with the collisiontime scale �� see �Bo�� p� �� In this situation time evolutioncan be regarded as a permutation of the cells with given energy we neglect�in fact� on the basis of the analysis in x�� the possibility that there may bea small fraction of di�erent cells evolving into the same cell�In this context we ask what will be the qualitative behavior of the system

with an energy ��xed� macroscopically� i�e� in an interval between E�DEand E� if its observations are timed at intervals � and the quantity DE ismacroscopically small but DE � �E � h��t� see �� Boltzmann assumed� very boldly� that in the interesting cases the ergodichypothesis held� according to which ��Bo�� Bo#�� Ma��

Ergodic hypothesis the action of the evolution transformation S� as a cellpermutation of the phase space cells on the surface of constant energy� is aone cycle permutation of the N phase space cells with the given energy

S�k � �k� k � �� N ��

if the cells are suitably enumerated �and �N� � ��

In other words as time evolves every cell evolves� visiting successively allother cells with equal energy� The action of S is the simplest thinkablepermutation�Even if not strictly true this should hold at least for the purpose of com�puting the time averages of the observables relevant for the macroscopicproperties of the system�The basis for such a celebrated �and much criticized� hypothesis rests on

its conceptual simplicity it says that in the system under analysis all cellswith the same energy are equivalent�


There are cases �already well known to Boltzmann� �Bo#�� in which thehypothesis is manifestly false for instance if the system is enclosed in aperfect spherical container then the evolution keeps the angular momentumM��

PNi� q

�i p�

iconstant� Hence cells with a di�erent total angular

momentum cannot evolve into each other�

This extends to the cases in which the system admits other constants of mo�tion� besides the energy� because the evolving cells must keep the constantsof motion equal to their initial values� And this means that the existence ofother constants of motion besides the energy is� essentially� the most generalcase in which the ergodic hypothesis fails in fact when the evolution is nota single cycle permutation of the phase space cells with given energy� thenone can decompose it into cycles� One can correspondingly de�ne a functionA by associating with each cell of the same cycle the very same �arbitrarilychosen� value of A� di�erent from that of cells of any other cycle�

Obviously the function A so de�ned is a constant of motion that can playthe same role as the angular momentum in the previous example�

Thus� if the ergodic hypothesis failed to be veri�ed� then the system wouldbe subject to other conservation laws� besides that of the energy� In suchcases it would be natural to imagine that all the conserved quantities were�xed and to ask oneself which are the qualitative properties of the motionswith energy E� when all the other constants of motion are also �xed� Clearlyin this situation the motion will be by construction a simple cyclic permuta�tion of all the cells compatible with the pre�xed energy and other constantsof motion values�

Hence it is convenient to de�ne formally the notion of ergodic probabilitydistribution on phase space

De�nition a set of phase space cells is ergodic if S maps it into itself andif S acting on the set of cells is a one�cycle permutation of them�

Therefore� in some sense� the ergodic hypothesis would not be restrictiveand it would simply become the statement that one studies the motionafter having a priori �xed all the values of the constants of motion�

The latter remark� as Boltzmann himself realized� does not make less inter�esting the concrete question of determining whether a system is ergodic inthe strict sense of the ergodic hypothesis �i�e� no other constants of motionbesides the energy�� On the contrary it serves well to put in evidence somesubtle and deep aspects of the problem�

In fact the decomposition of S into cycles �ergodic decomposition of S�might turn out to be so involved and intricated to render its constructionpractically impossible� i�e� useless for practical purposes� This would hap�pen if the regions of phase space corresponding to the various cycles were�at least in some directions� of microscopic size or of size much smaller thana macroscopic size� or if they were very irregular on a microscopic scale aquite di�erent a situation if compared to the above simple example of theconservation of angular momentum�


It is not at all inconceivable that in interesting systems there could bevery complicated constants of motion� without a direct macroscopic physicalmeaning important examples are discussed in �Za#��Therefore the ergodic problem� i�e� the problem of verifying the validity

of the ergodic hypothesis for speci�c systems� in cases in which no partic�ular symmetry properties can be invoked to imply the existence of otherconstants of motion� is a problem that remains to be understood on a case�by�case analysis� A satisfactory solution would be the proof of strict validityof the ergodic hypothesis or the possibility of identifying the cycles of S vialevel surfaces of simple functions admitting a macroscopic physical meaning�e�g� simple constants of motion associated with macroscopic �conservationlaws�� as in the case of the angular momentum illustrated above��It is useful to stress that one should not think that there are no other simple

and interesting cases in which the ergodic hypothesis is manifestly false� Themost classical example is the chain of harmonic oscillators described by

T �

NXi�

p�i � m� � �

NXi�

m�qi� � qi��

where� for simplicity� qN� � q� �periodic boundary condition��In this case there exist a large number of constants of motion� namely N

Ak � �p � �k�� k��q � �

k�� k � �� N ��

where ��

��

Nare N suitable orthonormal vectors �normal modes�

and ��k� are the �intrinsic pulsations� of the chain

��k�� cos �k�N� � ��

The constants of motion in �� can be arranged into an N �vector A�� A�� A�� AN �� The phase space cells � and �� for which thevectors A�� and A�� do not coincide cannot belong to the same cycleso that the system is not ergodic�Nevertheless Boltzmann thought that circumstances like this should be

considered exceptional� Hence it will be convenient not to go immediatelyinto a deeper analysis of the ergodic problem not only because of its di%�culty but mainly because it is more urgent to see how one can proceed inthe foundations of classical statistical mechanics�Given a mechanical system of N identical �for the sake of simplicity� par�

ticles consider the problem of studying a �xed observable f�p� q� de�ned onphase space�The �rst important quantity that one can study� and often the only one

that it is necessary to study� is the average value of f

f�� limT��

�

T

T��Xk�

f�Sk��


where f�� f�p� q� if �p� q� is a point determining the cell �� If �� N is the cycle to which the cell � belongs� then

f��

NNXk�

f��k� ��

and in the ergodic case the cycle consists of the set of all cells with the sameenergy as ��If the system energy is determined up to a macroscopic error DE� macro�

scopically negligible �but large with respect to the microscopic indetermi�nation of energy �E� �� the cells with energy between E �DE andE will be divided into cycles with �slightly� di�erent energies� On each ofthe cycles the function f can be supposed to have the �continuity� propertyof having the same average value �i�e� energy independent up to negligiblevariations��Hence� denoting by the symbol JE the domain of the variables �p� q� whereE �DE � E�p� q� � E holds� one �nds

f��

ZJE

f�p� q� dp dq�Z

JE

dp dq � ��

Recalling in fact that the cells all have the same volume� �� follows im�mediately from �� and from the assumed negligibility of the dependenceof f�� from E�� provided h is so small that the sum over the cells canbe replaced by an integral�The above relation� which Boltzmann conjectured ��Bo��b�� Bo#�� to be

always valid �discarding exceptional cases� �like the harmonic oscillatorchain just described� and wrote in the suggestive form� �Bo��b�� and p� �in �EE��

limT��

dt

T�

dpdqRJE

dpdq��#�

��

is read �the time average of an observable equals its average on the surfaceof constant energy�� As we shall see� �x�� #� provides a heuristicbasis of the microcanonical model for classical thermodynamics�Note that if ��#� holds� i�e� if �� holds� the average value of an

observable will depend only upon E and not on the particular phase spacecell � in which the system is found initially� The latter property is certainlya prerequisite that any theory aiming at deducing macroscopic propertiesof matter from the atomic hypothesis must possess�It is� in fact� obvious that such properties cannot depend on the detailed

microscopic properties of the con�guration � in which the system happensto be at the initial time of our observations�It is also relevant to note that in �� the microscopic dynamics has

disappeared it is in fact implicit in the phase space cell enumeration� madeso that �� are the cells into which � successively evolves at time


intervals � � But it is clear that in �� the order of such enumeration isnot important and the same result would follow if the phase space cells withthe same energy were enumerated di�erently�Hence we can appreciate the fascination that the ergodic hypothesis exer�

cises in apparently freeing us from the necessity of knowing the details ofthe microscopic dynamics� at least for the purposes of computing the ob�servables averages� That this turns out to be an illusion� already clear toBoltzmann� see for instance p� �� in �Bo�� will emerge from the analysiscarried out in the following sections�

x�� Recurrence Times and Macroscopic Observables

In applications it has always been of great importance to be able to estimatethe rapidity at which the limit f is reached in order that �� be usefulit is necessary that the limit in �� be attained within a time intervalt which might be long compared to the microscopic � but which shouldstill be very short compared to the time intervals relevant for macroscopicobservations that one wants to make on the system� It is� in fact� only onscales of the order of the macroscopic times t that the observable f mayappear as constant and equal to its average value�It is perfectly possible to conceive of a situation in which the system is er�

godic� but the value f�Sk�� is ever changing� along the trajectories� so thatthe average value of f is reached on time scales of the order of magnitude ofthe time necessary to visit the entire surface of constant energy� The latteris necessarily enormous�For instance� referring to the orders of magnitude discussed at the end

of x�� see the values of �p� �E preceding �� and �� itself� wecan estimate this time by computing the number of cells with volume h�N

contained in the region between E and E � �E and then multiplying theresult by the characteristic time h�kBT in �� Bo�� Bo��

If the surface of the d�dimensional unit sphere is written p�d'�d� ��

�with ' Euler s Gamma function� then the volume of the mentioned region�if h is very small� can be computed by using polar coordinates in momentum

space� The cells are those such that P �q

(iP�i varies between P �

p mE

and P � �P �p

m�E � �E�� hence we introduce� see x�� thequantities

P �p

mE� �p � p �p

�mkBT

�E ��kBT � p�p�m�E

N�

�kBT

�P �p�p

P�

ppN� �q �

�VN

��

where kB is Boltzmann s constant� kB � ��# � �� erg�K�� T is theabsolute temperature� V is the volume occupied by the gas and N is the


particle number� One �nds that the volume we are trying to estimate is�setting h � �p�q and using Stirling s formula to evaluate '�N� �

w � V Np

mE�N��

�Pp��N

'��N� ��

� �N�q��N �pN�p��N��P

p��N

'��N� ��

� ��p�q��NN��N��p��N '��N� ��

� h�NN

��N��

N��N� �

�

� �e

�

��N�� p�� h�NNN� �

�

� �e

�

��N��

The number of cells N would then be w�h�N � But we shall assume� onthe grounds of particle indistinguishability� that cells di�ering because in�dividual particles are permuted are in fact identical� Then �� has tobe divided by N � � NNe�N

p �N and therefore the recurrence time� if the

system did move ergodically on the surface of energy E� would be

Trecurrence � N � � N h

kBT� N�� h

kBT

� �e��

�

��N��

� ��

As discussed in x�� the order of magnitude of � � h�kBT is� if T � ��K�of about �� sec� For our present purposes it makes no di�erence whetherwe use the expression h � �p�q with �p� �q given in �� with V � � cm��N � �� m � �� g � hydrogen molecule mass� or whetherwe use Planck s constant �see comment after �� #��

Hence� �� e

�� being � �� the recurrence time in �� is unimaginably

longer than the age of the Universe as soon as N reaches a few decades�still very small compared to Avogadro s number�� If T is chosen to be ��Cfor � cm� of hydrogen at ��C� � atm one has N �� and Trecurrence ��

��

sec� while the age of the Universe is only �� sec�Boltzmann s idea to reconcile ergodicity with the observed rapidity of the

approach to equilibrium was that the interesting observables� the macro�scopic observables� had an essentially constant value on the surface of givenenergy with the exception of an extremely small fraction � of the cells� �Bo��p� �� See x�� below for further comments�Hence the time necessary to attain the asymptotic average value will not

be of the order of magnitude of the hyperastronomic recurrence time� butrather of the order of T � � �Trecurrence� And one should think that �� asthe number of particles grows and that T � is very many orders of magnitudesmaller than T so that it becomes observable on �human� time scales� seex��# for a quantitative discussion �actually T � sets� essentially by de�nition�the size of the human time scale��Examples of important macroscopic observables are

�� the ratio between the number of particles located in a small cube Qand the volume of Q this is an observable that will be denoted ��Q� andits average value has the interpretation of density in Q�

�# I� Classical Statistical Mechanics

�� the sum of the kinetic energies of the particles K�� (ip�i� m�

�� the total potential energy of the system ��q� � (i�j�qi� q

j��

�� the number of particles in a small cube Q adherent to the containerwalls� and having a negative component of velocity along the inner normalwith value in ��v��v � dv�� v � �� This number divided by the volumeof Q is the �density� n�Q� v�dv of particles with normal velocity �v thatare about to collide with the external walls of Q� Such particles will cede amomentum mv normally to the wall at the moment of their collision �astheir momentum will change from �mv to mv� with the wall� Consider theobservable de�ned by the sum over the values of v and over the cubes Qadjacent to the boundary of the container V

XQ

Zv�

d v n�Q� v�� mv�vs

S� P ��

with s � area of a face of Q and S � area of container surface this is themomentum transferred� per unit time and surface area� to the wall �notethat the number of collisions with the wall per unit time on the face s ofQ adjacent to the walls and with normal velocity v is n�Q� v�vs dv�� Thequantity �� is an observable �i�e� a function on phase space� whoseaverage value has the interpretation of macroscopic pressure� therefore itcan be called the �microscopic pressure� in the phase space point �p� q��

�� the product ��Q��Q�� is also interesting and its average value is calledthe density pair correlation function between the cubes Q�Q�� Its averagevalue provides information on the joint probability of �nding simultaneouslya particle in Q and one in Q��

x�� Statistical Ensembles or �Monodes and Models of Ther�

modynamics� Thermodynamics without Dynamics

From a more general viewpoint and without assuming the ergodic hypoth�esis it is clear that the average value of an observable will always exist andit will be equal to its average over the cycle containing the initial datum�see ��For a more quantitative formulation of this remark we introduce the notion

of stationary distribution it is a function associating with each phase spacecell a number �� probability or measure of �� so that

�� X�

�� S��

if S is the time evolution map which permutes the cells� see x�� and x��One usually says that � is an invariant probability distribution or a sta�tionary probability distribution on phase space �or� better� on phase spacecells�� The following de�nition will be convenient �see x��


De�nition Let � be an invariant probability distribution on phase spacecells� If the dynamics map S acts as a one�cycle permutation of the set ofcells � for which �� then � is called ergodic�

If one imagines covering phase space with a $uid so that the $uid massin � is �� and if the phase space point are moved by the permutationS associated with the dynamics then the $uid looks immobile� i�e� its dis�tribution on phase space remains invariant �or stationary� as time goes bythis gives motivation for the name used for ��It is clear that �� must have the same value on all cells belonging to

the same cycle C� of the permutation S �here � is a label distinguishingthe various cycles of S�� If N �C�� is the number of cells in the cycle C� itmust� therefore� be that �� p��N �C�� with p� � � and

P� p� � ��

for � � C��It is useful to de�ne� for each cycle C� of S� a �ergodic� stationary distri�

bution �� by setting

�� n

��N �C�� if � � C�� otherwise

��

and this allows us to think that any invariant probability distribution is alinear combination of the �� associated with the various cycles of S

�� X�

p��

where p� � � are suitable coe%cients withP

� p� � �� which can be calledthe �probabilities of the cycles� in the distribution �� Note that� by def�inition� each of the distributions �� is ergodic because it gives a positiveprobability only to cells that are part of the same cycle �namely C��The decomposition �� of the most general S�invariant distribution � as

a sum of S�ergodic distributions is naturally called the ergodic decompositionof � �with respect to the dynamics S��In the deep paper �Bo#�� Boltzmann formulated the hypothesis that sta�

tionary distributions � could be interpreted as macroscopic equilibriumstates so that the set of macroscopic equilibrium states could be identi��ed with a subset E of the stationary distributions on phase space cells�The current terminology refers to this concept as an ensemble� after Gibbswhile Boltzmann used the word monode� We shall call it an ensemble or astatistical ensemble�Identi�cation between an individual stationary probability distribution �

on phase space and a corresponding macroscopic equilibrium state takesplace by identifying �� with the probability of �nding the system in thecell �i�e� in the microscopic state� � if one performed� at a randomly chosentime� the observation of the microscopic state�Therefore the average value in time� in the macroscopic equilibrium state

described by �� of a generic observable f would be

f �X�

��f��


This relation correctly gives� in principle� the average value of f in time� ifthe initial data are chosen randomly with a distribution � which is ergodic�But in general even if � is ergodic one should not think that �� isdirectly related to the physical properties of �� This was already becomingclear in x�� and x�� when we referred to the length of the recurrence timesand hence to the necessity of further assumptions to derive �� #��We shall come back to �� and to the ergodic hypothesis in x�� Return�

ing to Boltzmann s statistical ensembles he raised the following questioninin the paper �Bo#�� letting aside the ergodic hypothesis or any other at�tempt at a dynamical justi�cation of �� consider all possible statisticalensembles E of stationary distributions on phase space� Fix E and� for each� � E � de�ne

�� X�

�� average potential energy��

T �� X�

��K�� average kinetic energy��

U�� T �� average total energy��

P �� X�

��P �� pressure�� see ��

�� N�V � � � ��v � �density��

V �

Zdq � �volume��

where V is the volume assigned to the system �i�e� the volume of the con�tainer� and N is the particle number�

Question ��orthodicity problem�� which statistical ensembles� or monodes�E have the property that as � changes in�nitesimally within E the corre�sponding in�nitesimal variations dU � dV of U � U�� and V � see �� are related to the pressure p � P �� and to the average kinetic energy perparticle T � T ��N� so that

dU � P dV

T� exact di�erential ��

at least in the thermodynamic limit in which the volume V � � and alsoN�U � � so that the densities N�V� U�V remain constant �assuming forsimplicity that the container keeps cubic shape��

Ensembles �or monodes� satisfying the property �� were called byBoltzmann orthodes they are� in other words� the statistical ensemblesE in which it is possible to interpret the average kinetic energy per particle�T � as proportional to the absolute temperature T �via a proportionality con�

stant� to be determined empirically and conventionally denoted � ��kB��

so that T � ��kB

T ��N �� and furthermore it is possible to de�ne via �� a


function S�� on E so that the observables U � �� T � V � P � S satisfy the rela�tions that the classical thermodynamics quantities with the correspondingname satisfy� at least in the thermodynamic limit�In this identi�cation the function S�� would become� naturally� the en�tropy and the validity of �� would be called the second law�In other words Boltzmann posed the question of when it would be possible

to interpret the elements � of a statistical ensemble E of stationary distri�butions on phase space as macroscopic states of a system governed by thelaws of classical thermodynamics�The ergodic hypothesis combined with the other assumptions used in x��

to deduce �� #� leads us to think that the statistical ensemble Econsisting of the distributions � on phase space de�ned by

�� N �U� V � if E�� U �DE�U�

�� otherwise��

where U� V are pre�xed parameters corresponding to the total energy andvolume of the system� should necessarily be a statistical ensemble apt atdescribing the macroscopic equilibrium states� Here N �U� V � is a normal�ization constant to be identi�ed as proportional to the integral

Rdpdq over

the region JE of p� q in which E�p� q� � �U �DE�U�� and the parameterDE is �arbitrary� as discussed before ��However the orthodicity or nonorthodicity of a statistical ensemble E whoseelements are parameterized by U� V as in �� is �only� the question ofwhether �� second law� holds or not and this problem is not� in itself�logically or mathematically related to any microscopic dynamics property�The relation between orthodicity of a statistical ensemble and the hy�

potheses on microscopic dynamics �like the ergodic hypothesis� that woulda priori guarantee the physical validity of the ensuing model of thermody�namics will be reexamined in more detail at the end of x��If there were several orthodic statistical ensembles then each of them would

provide us with a mechanical microscopic model of thermodynamics ofcourse if there were several possible models of thermodynamics �i�e� severalorthodic statistical ensembles� it should also happen that they give equiva�lent descriptions� i�e� that they give the same expression to the entropy S asa function of the other thermodynamic quantities� so that thermodynamicswould be described in mechanical terms in a nonambiguous way� This checkis therefore one of the main tasks of the statistical ensembles theory�It appears that in attempting to abandon the �hard� fundamental aim at

founding thermodynamics on microscopic dynamics one shall neverthelessnot avoid having to attack di%cult questions like that of the nonambiguityof the thermodynamics that corresponds to a given system� The latter is aproblem that has been studied and solved in various important cases� but weare far from being sure that such cases �the microcanonical or the canonicalor the grand canonical ensembles to be discussed below� and others� exhaustall possible ones� Hence a �complete� understanding of this question could

I� Classical Statistical Mechanics

reveal itself equivalent to the dynamical foundation of thermodynamics thevery problem that one is hoping to circumvent by deciding to �only� builda mechanical model of thermodynamics� i�e� an orthodic ensemble�

x�� Models of Thermodynamics� Microcanonical � Canonical

Ensembles and the Ergodic Hypothesis�

The problem of the existence of statistical ensembles �i�e� a family of sta�tionary probability distributions on phase space� that provides mechanicalmodels of thermodynamics� was solved by Boltzmann in the same paperquoted above� �Bo#�� following earlier basic papers on the canonical en�semble �Bo��a�� Bo��b� where the notion of ensemble seems to appear forthe �rst time��Here Boltzmann showed that the statistical ensembles described below and

called� after Gibbs� the microcanonical and the canonical ensemble are or�thodic� i�e� they de�ne a microscopic model of thermodynamics in which theaverage kinetic energy per particle is proportional to absolute temperature�see below and x��

�� The microcanonical ensemble

It was named in this way by Gibbs while Boltzmann referred to it by thestill famous� but never used� name of ergode� The microcanonical ensembleconsists in the collection E of stationary distributions � parameterized bytwo parameters U� total energy and V � system volume so that� see ��

�� N �U� V � if U �DE � E�� U

�� otherwise��

��

with

N �U� V � �X

U�DE�E��U� � fnumber of cells � with

energy E�� U �DE�U�g��

��

where the quantity DE has to be a quantity� possibly V �dependent� �macro�scopically negligible� compared to U � such that one may think that all cellswith energy between U �DE and U have the �same energy� from a macro�scopic point of view�The importance of the microcanonical ensemble in the relation between

classical thermodynamics and the atomic hypothesis is illustrated by theargument leading to ��#� which proposes it as the natural candidate foran example of an orthodic ensemble�

� At least in the thermodynamic limit� see �� in which the volume becomes in�nitebut the average density and energy per particle stay �xed�


However� as discussed in x�� the argument leading to �� #�cannot possibly be regarded as a �proof on physical grounds� of orthodicityof the microcanonical ensemble��

Following the general de�nition in x�� of orthodic statistical ensemble�i�e� of an ensemble generating a model of thermodynamics� we can de�ne the�absolute temperature� and the �entropy� of every element � ��macroscopicstate�� so that the temperature T is proportional to the average kineticenergy� Boltzmann showed that such functions T and S are given by thecelebrated relations

T �

�kB

T ��

N� S�� kB logN �U� V � ��

where kB � �Boltzmann s constant�� is a universal constant to be empiricallydetermined by comparison between theory and experiment�� The factor �

� isconventional and its choice simpli�es some of the following formulae� besidesthe second of ��

The statement that �� provide us with a microscopic model ofthermodynamics in the thermodynamic limit V � �� U � �� N � � sothat u � U�N � v � V�N remain constant has to be interpreted as follows�One evaluates� starting from �� see also ��

u � U�N � �speci�c energy� � v � V�N � �speci�c volume��

T � T ��kBN � �temperature� �

s � S��N � �entropy�� p � P �� pressure��

Since the quantities u� v determine � � E it will be possible to express T� p� sin terms of u� v via functions T �u� v�� P �u� v�� s�u� v� that we shall supposeto admit a limit value in the thermodynamic limit �i�e� V �� with �xedu� v��Then to say that �� give a model of thermodynamics means

�see also x�� that such functions satisfy the same relations that link thequantities with the same name in classical thermodynamics� namely

du � T ds� P dv � ��

Equation �� is read as follows if the state � de�ned by �� is subject to a small variation by changing the parameters U� V that de�neit� then the corresponding variations of u� s� v verify �� i�e� the secondprinciple of thermodynamics see Chap�II for a discussion and a proof of�� The proof of a statement like �� for the ensemble E was called� by

Boltzmann� a proof of the heat theorem�

� Which it is worth stressing once more does not depend on the microscopic dynamics�� As already said and as it will be discussed later� one �nds kB � �� erg�K�� Mainly it simpli�es the relation between T and � in the �rst of �� below�


� � The canonical ensemble�

The name was introduced by Gibbs� while Boltzmann referred to it with thename of holode� It consists in the collection E of stationary distributions� parameterized by two parameters � and v � V�N � via the de�nition

�� exp��E��Z�� V � ��

withZ�� V � �

X�

exp��E��

Boltzmann proved the proportionality between T �� and �� as well as theorthodicity of this statistical ensemble by showing that temperature andentropy can be de�ned by

T � T ��kBN � ��kB� S � �kB��U � logZ�� V �� #��

where kB is a universal constant to be empirically determined�The statement that �� #� provide us with a model of thermody�

namics� in the thermodynamic limit V � �� V�N � v� � � constant�has the same meaning discussed in the previous case of the microcanonicalensemble� See Chap�II for the analysis of the orthodicity of the canonicalensemble� i�e� for a proof of the heat theorem for the canonical ensemble�

The relations �� hold� as already pointed out� for both ensembles con�sidered� hence each of them gives a microscopic �mechanical� model of clas�sical thermodynamics�Since entropy� pressure� temperature� etc� are in both cases explicitly ex�

pressible in terms of two independent parameters �u� v or �� v� it will bepossible to compute the equation of state �i�e� the relation between p� v andT � in terms of the microscopic properties of the system� at least in principlethis is enormous progress with respect to classical thermodynamics wherethe equation of state always has a phenomenological character� i�e� it is arelation that can only be deduced by means of experiments�It is clear� however� that the models of thermodynamics described above

must respond� to be acceptable as physical theories� to the basic prerequisiteof de�ning not only a possible thermodynamics� but also of de�ning thethermodynamics of the system� which is experimentally accessible� One cancall the check of the two prerequisites a check of theoretical and experimentalconsistency� respectively�For this it is necessary� �rst� that the two models of thermodynamics co�

incide �i�e� lead to the same relations between the basic thermodynamicquantities u� v� T � P � s� but it is also necessary that the two models agreewith the experimental observations�

� I�e a thermodynamics that does not come into con�ict with the basic principles� expressed by ��


But a priori there are no reasons that imply that the above two prerequi�sites hold�Here it is worthwhile to get more deeply into the questions we raised in

connection with �� and to attempt a justi�cation of the validity of themicrocanonical ensemble as a model for thermodynamics with physicallyacceptable consequences and predictions� This leads us once more to discussthe ergodic hypothesis that is sometimes invoked at this point to guaranteea priori or to explain a posteriori the success of theoretical and experimentalconsistency checks� whose necessity has been just pointed out�In x�� we have seen how the microcanonical distribution could be justi�

�ed as describing macroscopic equilibrium states on the basis of the ergodichypothesis and of a continuity property of the averages of the relevant ob�servables �see the lines preceding �� in that analysis� leading to ��we have not taken into account the time scales involved� Their utmost im�portance has been stated in x�� if �� held but the average value overtime of the observable f � given by the right�hand side of �� was at�tained in a hyperastronomic time� comparable to the one given by ��then �� would� obviously� have little practical interest and value�

x�� Critique of the Ergodic HypothesisSummarizing to deduce �� hence for an a priori justi�cation of the

connection between the microcanonical ensemble and the set of states ofmacroscopic thermodynamic equilibrium� one meets three main di%culties�

� The �rst is a veri�cation of the ergodic hypothesis� x�� as a mathemat�ical problem�

� The second is that even accepting the ergodic hypothesis for the cyclicityof the dynamics on the surface with constant energy �i�e� with energy �xedwithin microscopic uncertainty �E� one has to solve the di%culty that� inspite of the ergodicity� the elements of the microcanonical ensemble arenot ergodic because the �trivial� non ergodicity is due to the fact that inthe microcanonical ensemble the energy varies by a small but macroscopicquantity DE � �E�

� The third is that� in any event� it would seem that enormous times areneeded before the $uctuations of the time averages over �nite times stabilizearound the equilibrium limit value �times enormously longer than the ageof the Universe��

The three di%culties would be solved if one supposed that� simultaneously

�i� the phase space cells with �xed energy �microscopically �xed� are partof a single cycle of the dynamics S this is the ergodic hypothesis� see x��

�ii� the values of the �relevant� macroscopic observables are essentiallythe same on cells corresponding to a given macroscopic value of the energy


E� with the possible exception of a small fraction of the number of cells�negligible for large systems i�e� in the thermodynamic limit�

�iii� the common average value that the relevant observables assume ontrajectories of cells of energy E changes only slightly as the total energychanges between the values U and U�DE� if U and DE are two macroscopicvalues with U � DE �but DE � �E�� This can be called a continuityassumption�

The hypotheses �i� and �iii�� see x�� show that the average values ofthe macroscopic observables can be computed by using� equivalently� anyergodic component of a given microcanonical distribution ��

Hypothesis �ii� allows us to say that the time necessary in order that anaverage value of an observable be attained� if computed on the evolutionof a particular microscopic state �� is by far shorter than the recurrencetime �too long to be interesting or relevant�� The region of phase spacewhere macroscopic observables take the equilibrium value sometimes hasbeen pictorially called the �Boltzmann s sea� �see �Bo�� p� �� and �Ul�#��p� �� g� ��

Accepting �i�� ii� and �iii� implies �by the physical meaning that u� p� vacquire� that the microcanonical ensemble must provide a model for ther�modynamics in the sense that dU � p dV must admit an integrating factor�to be identi�ed with the absolute temperature�� The fact that this factorturns out to be proportional to the average kinetic energy is� from this view�point �and only in the case of classical statistical mechanics as one shouldalways keep in mind�� a consequence �as we shall show in Chap�II��

One can remark that assumptions �ii� and �iii� are assumptions that donot involve explicitly the dynamical properties� at least on a qualitativelevel one says that they are equilibrium properties of the system� And itis quite reasonable to think that they are satis�ed for the vast majority ofsystems encountered in applications� because in many cases it is possibleto really verify them� sometimes even with complete mathematical rigor��Fi�� Ru��

Hence the deeper assumption is in �i�� and it is for this reason that some�times� quite improperly� it is claimed that the ergodic hypothesis is �thetheoretical foundation for using the microcanonical ensemble as a model forthe equilibrium states of a system��

The improper nature of the above locution lies in the fact that �i� can begreatly weakened without leading to a modi�cation of the inferences on themicrocanonical ensemble�

For instance one could simply require that only the time average of fewmacroscopically interesting observables should have the same value on everycycle �or on the great majority of cycles� of the dynamics with a �xed energy�

This can be done while accepting the possibility of many di�erent cycles�on which non macroscopically interesting observables would take di�er�ent average values�� An essentially exhaustive list of the �few� interesting


observables for monoatomic gases is given by ��

Furthermore the above�mentioned locution is improper also because� evenif one accepts it� it cannot release us from checking �ii�� iii� which� inparticular� require a quantitative veri�cation evidently one cannot be sat�is�ed with a simple qualitative veri�cation since the orders of magnitudeinvolved are very di�erent� One could� in fact� raise doubts that the time�for reaching equilibrium� could really come down from the recurrence times�superastronomical� to the times experimentally recorded �usually of a fewmicroseconds��For what concerns the canonical ensemble its use could be justi�ed simply

by proving that it leads to the same results that one obtains by using themicrocanonical ensemble� at least in the thermodynamic limit and for thefew interesting observables �see above for a list��

But� as already mentioned� the ergodic hypothesis �with or without theextra two assumptions �ii�� iii� above� is technically too di%cult to studyand for this reason an attempt has been made to construct models of ther�modynamics while avoiding solving� even if partially� the ergodic problem�The proposal is simply to prove that all the orthodic ensembles �at least

the reasonable ones�� generate the same macroscopic thermodynamics �forinstance the same equation of state�� This property� by itself very notableand remarkable� should then be considered su%cient to postulate� by the�principle of su%cient reason�� that the equations of state of a system canbe calculated from the microscopic properties �i�e� from the Hamiltonian ofthe system� by evaluating the average values of the basic observables �see�� via the distributions of the microcanonical or canonical ensembles�or more generally of any orthodic ensemble�

The latter is the point of view usually attributed to Gibbs virtually allthe treatises on statistical mechanics are based on it�It is well understandable why such a point of view appeared unsatisfac�

tory to Boltzmann who had the ambition of reducing thermodynamics tomechanics without introducing any new postulate on the other hand� thepragmatic approach of Gibbs is also very understandable if one keeps in

� One should not think that it is di�cult to devise ensembles which are orthodic and whichmay seem �not reasonable �for a thermodynamic interpretation�� in fact Boltzmann spaper� �Bo�� on the ensembles starts with such an example involving the motion ofone of Saturn s rings regarded as a massive line �in a parallel paper the example was theMoon� whose orbit was replaced by an ellipse of mass such that each arc contained anamount of mass proportional to the time spent on it by the Moon�� This may have beenone of the reasons this fundamental paper has been overlooked for so many years� Such�unphysical examples come from Helmoltz� �He��a�� He��b�� and played an importantrole for Boltzmann �who was considering them in a less systematic way even muchearlier� �Bo�� In fact if one can de�ne the mechanical analogue of thermodynamicsfor any system� small or large� then it is natural to think that in large systems theaverage quantities will also satisfy the second law� And the idea �of Boltzmann� thatthe macroscopic observables have the same value on most of the energyy surface makesthe law easily observable in large systems� while this may not be the case in very smallsystems� In other words the onedegreeoffreedom examples are not at all unphysical�rather the contrary holds� see Appendix ��A� �to Chap�I� for Helmoltz s theory andChap�IX for a recent application of the same viewpoint�

# I� Classical Statistical Mechanics

mind the necessity of deducing all the applicative consequences stemmingfrom the marvelous discovery of the possibility of unambiguously derivingvalues of thermodynamical quantities in terms of mechanical properties ofthe atomic model of matter�For the past few decades� about a century after the birth of the above

theories� we seem to feel again the necessity of a uni�ed derivation of ther�modynamics from mechanics without the arti�cial a priori postulate thatthermodynamics is described by the orthodic statistical ensembles� a pos�tulate made possible� i�e� consistent� by the mentioned independence �dis�cussed in the following Chap�II� of the results as functions of the statisticalensemble used�The ergodic problem and the statistical dynamics are therefore again at

the center of research� and are stimulating new interesting ideas and results�Boltzmann tried to justify the microcanonical and canonical ensembles

also following a path rather di�erent from the one of studying the ergodicproblem and the hypotheses �i��ii��iii� above� �UF�� And his attemptled him� �Bo� �� to deduce the Boltzmann�s equation which revealed itselfessential even for technical applications� although it presented and presentsvarious conceptually unsatisfactory aspects� see x��# for a �rst analysis ofthis equation�

x�� Approach to Equilibrium and Boltzmann�s Equation� Er�

godicity and Irreversibility

As discussed in the previous sections macroscopic equilibrium states can beidenti�ed with elements of the orthodic statistical ensembles �microcanoni�cal� canonical� grand canonical� � � �� It is not quantitatively clear� however�through which mechanism a mechanical system initially in non equilibriumcan reach equilibrium�We have argued that the ergodic hypothesis� by itself� is not su%cient to

explain why a system reaches equilibrium within times usually relativelyshort�Boltzmann developed a model� �Bo� �� for describing approach to equilib�

rium which was strongly criticized since its formulation� much as his otherintuitions� and which is considered by some �perhaps incorrectly� his great�est contribution to Science�The validity of the model is limited to systems with so low a density that

they can be considered rare�ed gases and this shows how it can� in concretecases� happen that assumptions �i�� ii�� iii� of x�� could be� for practicalpurposes� veri�ed in such systems and how it could be possible that theinteresting observables reach their average values over time scales accessibleto our senses rather than on the absurdly long recurrence time scales�One imagines the system to consist of N identical particles �for simplicity��

each of which is described by momentum p and by position q� They moveas if free� except that from time to time they collide�Assuming that such particles are rigid spheres with radius R �again only


for simplicity� and that they have an average speed v� the low�density as�sumption is that the density � � N�V is such that

�R� � � ��#��

which means that it is very unlikely that there are two particles at a distanceof the order R� i�e� �colliding��At the same time one requires that the number of collisions that each

particle undergoes per unit time does not vanish� Evidently this numberhas order of magnitude

�R� v � ��#� ��

Hence the limit situation in which the gas is very rare�ed but� nevertheless�the number of collisions that each particle undergoes per unit time is notnegligible� is described by

R� �� so that

�R� � �� R� v � w � �xed quantity ��#��

The quantity � � ��w is the time of �ight between two collisions while themean free path is � v � ��R��The limit situation that is obtained by letting R � � and � � � as in

��#�� is called Grad�s limit� In the situation envisaged by Boltzmann onesupposes that we are �close� to this limit� i�e� one supposes that we areclose to and �R� v � w � ��It is of some interest to compute �R�� and � v for a Hydrogen sample at

atmospheric pressure and room temperature �p � � atm� T � ��oK� one�nds �R� � ��#� �� sec� v � �� m�sec�Let then f�p� q� dpdq be the number of particles that can be found in the

cell Q � dpdq of the phase space describing the states of a single particle�not to be confused with the phase space which we have been using so far�which describes the states of N particles��Boltzmann remarks that f can change in time either by virtue of collisions

or because particles move in space� If � is a pre�xed time interval� thenumber of particles that at a certain instant are in the cell Q is

f�p� q� t� dpdq � f�p� q � �p�m� t� �� dp dq�

�XQ��Q��

�number of particles in Q� that collide per unit of time with ��#��particles in Q�� producing particles in Q��Q� with Q� � Q�

�XQ��Q��

�number of particles in Q� � Q that collide per unit of time

with particles in Q� producing particles in Q��Q��

If we consider the collision that transforms two particles in Q�� Q�� intotwo others in Q�� Q� we must have� by the conservation of momentum andenergy in the collision�

p� � p�� p�

� p�� p�� p�� p�

�� p�

��#��


and the number of collisions leading from p�� p�� to p�� p

�can be expressed

in terms of the notion of collision cross�section � � ��p�� p�� p� p��

The latter is de�ned to be the fraction of particles of a stream withmomentum in dp and spatial density n�p�dp streaming around one par�ticle with momentum p

�that collides with it in time dt� experienc�

ing a collision that trasnforms p� p�

into p�� p�� This number is writ�

ten as n�p�dpjp�p

�j

m ��p�� p�� p� p�� dt and � has the dimension of a sur�

face� Hence the total number of such collision per unit volume will be

n�p��dp

�n�p�dp �

jp�p�j

m ��p�� p�� p� p�� dt� i�e� the the number of particles

with momentum in dp that experience a collision with one p��particle in

time dt is the number of particles with momentum in dp and contained

in a volume of size �jp�p

�j

m ��p�� p�� p� p�� dt� It is therefore natural to call

collision volume �per unit time� the quantity in square brackets becauseit gives� after multiplication by the density of particles with momentum indp� the number of collisions per unit time and volume that particles withmomentum p would undergo against a momentum p

�particle if there was

only one such particle�

Introducing

f�p�� q� dp� dq � number of particles with momentum p��within dp� in the cube dq �

�number of collision centers

f�p�� q� dp�� density of particles with momentum p��within dp�� in q �

��density of particles thatcan undergo collision

��p�� p�� p� p�� di erential crosssection per unit solid

angle for the considered collision�

��#��

Note that the collision volume associated with a single collision center is�since the relative velocity at collision is jp� � p��j�m � jp� p

�j�m n� also

�jp� � p��j�m��p�� p�� p� p�� #��

Hence the total number of collisions from Q�� Q�� to Q�� Q� is� per unittime

f�p�� q� dp��jp� � p��j

m��p�� p�� p� p

��f�p�� q� dp� dq � ��#�#��

clearly symmetric in p�� p�� although derived by treating p� and p�� asym�metrically�By a similar argument the number of �inverse� collisions is

f�p� q�f�p�� q� dpdp

�dqjp� p

�j

m��p� p

�� p�� p�� #��


One then remarks that ��#�� imply

dp� dp�� dp dp�

��Liouville s theorem� �

jp� � p��j � jp� p�j ��conservation of momentum��

��#��

moreover the cross�section� as is in general true in collisions governed bycentral forces� depends exclusively on the de�ection angle between �p�� p��and �p� p

�� and on the relative speed jp�� p��j�m� and it is proportional to

the �normalized� solid angle d� into which �p� p�� points with respect to

�p� � p��Note in this respect that the collision �nal data� i�e� p� p

�� do not determine

p�� p�� via ��#�� but they leave the direction d� of p� � p�� arbitrary�We shall then set ��p�� p�� p� p

�� jp� � p��j� d� � �� d� where

the last relation is only valid when the interaction between the spheres isassumed a rigid sphere interaction� and from the scattering theory it followsthat in this case �� is independent of � �� R��Hence ��#�� allow us to rewrite ��#�#�� #�� as

f�p�� q�f�p�� q� dp dp�

dq d� �jp� � p��j�m��

f�p� q�f�p�� q� dp dp

�dq d� �jp� p

�j�m��

��#��

where� given p� p�� the vectors p�� p�� are computed from ��#�� and from

the information that the solid angle between p� p�

and p� � p�� is ��Introducing ��#�� in ��#�� and dividing by � one �nds the Boltzmannequation

f

t�p� q� �

p

m� fq

�p� q� �

Z jp� p�j

m�� d� dp

��

� �f�p�� q�f�p�� q�� f�p� q�f�p�� q�� #��

In ��#�� one supposes that q varies over the whole space but the mostinteresting cases concern systems �one should say �rare�ed gases� becauseof the conditions under which ��#�� has been derived� con�ned in a givenvolume V � In such cases ��#�� must be complemented by suitable bound�ary conditions that depend on the microscopic nature of the collisions ofthe particles against the walls�Since the discussion of the boundary conditions is delicate we shall avoid

it and in the case of con�ned systems we shall suppose� for simplicity� thatperiodic boundary conditions hold� This means that we imagine the volumeV as a cube with opposite faces identi�ed i�e� a particle that collides withone of the cube faces reemerges� after the collision� from the opposite faceand with the same velocity� For a deeper analysis of the problem of theboundary conditions �and in general of Boltzmann s equation� see �Ce��It should be clear that ��#�� is an approximation because we neglected

�i� the possibility of multiple collisions�


�ii� the possibility that particles in the same volume element do not behaveindependently� as implicitly assumed in deriving ��#�� and instead� astime goes by� correlations between positions and velocities build up� andmake certain collisions more probable than others� or multiple collisionsrelatively more probable with respect to binary ones� This approximationis sometimes called molecular chaos�

Such e�ects should disappear in the Grad�Boltzmann limit ��#�� providedthey are absent at the initial time� This �conjecture� is known as Grad�sconjecture on the validity of the �Stosszahlansatz�� a word that� for tradi�tional reasons� just denotes the lack of correlation between the motions ofdi�erent particles at various instants of time�In Appendix ��A we show how an analysis of the corresponding conjecture

can be easily performed in a much simpler case� in which a gas of particlesmoves in a space occupied by randomly placed spherical scatterers a modelcalled Lorentz�s model� The particles collide with the scatterers but do notinteract with each other� so that Boltzmann s equation for this model turnsout to be linear�Returning to Boltzmann s equation and postponing the analysis of the

fundamental assumptions �i� and �ii� discussed above� the irreversibility ofthe approach to equilibrium that it implies can be demonstrated on thebasis of the following remarks�Multiplying both sides of ��#�� by �� p� �

�m p� or by log f�p� q� and in�tegrating over p and q �under the assumption that f�p� q� � � rapidly as�p� q� �� or� when q is restricted to a �xed container� that f satis�es suit�able boundary conditions on the q coordinate� one �nds that the quantities

N �

Zf�p� q� dpdq� P �

Zpf�p� q� dpdq

T �

Z�p�� m�f�p� q� dp dq � H � �

Zf�p� q� log f�p� q� dp dq

��#��satisfy the relations

dN

dt�

dP

dt�

dT

dt� � ��#��

dH

dt�

�

�

Z jp� p�j

m�� d��f�p�� q�f�p�� q�� f�p� q�f�p

�� q��

� � log f�p�� q�f�p�� q�� log f�p� q�f�p�� q��

dpdp�

dq � �

as can be checked by a simple calculation in which an essential role is playedby the symmetry of the right�hand side of ��#�� between p� p

�and p�� p��

and dp dp�

� dp� dp�� i�e� Liouville s theorem� see ��#�� and momentumconservation� see ��#�� is used together with the relations logx� log y �logxy and �x� y��logx� log y� � ��Therefore while the �rst three relations in ��#�� imply �ve conserva�tion laws �of the particle number� of momentum and of �kinetic� energy��


the fourth manifestly implies irreversibility and is called Boltzmann�s H�theorem�Furthermore ��#�� shows that the only possible equilibrium distributionsf�p� q� can be those for which

f�p�� q�f�p�� q� � f�p� q�f�p�� q� ��#��

where p� p�� p�� p�� satisfy ��#��

Equation ��#�� and the arbitrariness of p� p�� p�� p�� imply� via a simple

argument that we leave out�

f�p� q� ��q�e��q��p�p��q��

��m

� ��q��m��#��

where ��q�� p��q�� q� are arbitrary functions and the factor in the denomi�

nator of the right�hand side has been introduced for the purposes of a simplenormalization� so that ��q� could be interpreted as the density at the pointq since it is then ��q� �

Rf�p� q� dp�

This means that the equilibrium distributions f�p� q� necessarily have theform ��#�� Considering subsequently the simple case of a system in acubic container with periodic boundary conditions it is easy to show that�if f satis�es ��#�� #�� and f�t � � �i�e� it is stationary� then itmust necessarily be that ��q�� q�� p

��q� are q�independent�

In fact if f has the form ��#�� the right�hand side of ��#�� vanishes

and� therefore� f�t � � implies p � f�q � �� Hence denoting by )f�p� k�

the Fourier transform of f with respect to q this implies that p�k )f�p� k� � ��

so that if )f�p� k� is continuous in p it must be that )f�p� k� � � for k �� This means that f is q�independent and that ��q�� q�� p

��q� are constants�

We see that the H�theorem not only shows that the system evolves irre�versibly� but it also shows that the one�particle distribution f�p� q� evolvestowards the free Maxwell�Boltzmann distribution which� one should not failto note� is just a typical property of an element � of the canonical ensemblein a system in which the interaction energy between the particles is so small�when their hard cores do not overlap� that the total energy of the systemcan be identi�ed with the kinetic energy� The parameters �� p

�of this dis�

tribution are uniquely determined by the initial data via the conservationlaws in ��#��It is natural to think that the H�theorem is� for rare�ed gases� the micro�

scopic version of the second law of thermodynamics which states that inisolated systems entropy increases �while equilibrium is approached� en�tropy should be identi�ed as proportional to H �It is therefore important to stress that the H�theorem is manifestly incontrast with the reversibility properties of Newtons�s equations and� conse�quently� it cannot be a mathematical consequence of the latter� as alreadyremarked� at least not in the literal� i�e� naive� sense of the word�


Just for this reason it becomes essential to understand whether this con�trast between macroscopic irreversibility and microscopic reversibility canbe overcome�

The alleged incompatibility between the two con$icting properties was thecause of violent critiques to Boltzmann who created the ergodic hypoth�esis �thus laying the foundations of the modern ergodic theory� in one ofhis attempts to answer his critics on a theoretical basis� solidly resting onmechanics rather than on the admittedly obscure Stosszahlansatz�

To investigate the question one can take two viewpoints� which appearsomewhat to overlap in Boltzmann s brilliant and misunderstood �by hiscontemporaries� attempt at defending his theory and his H�theorem�

The �rst point of view is that the ergodic hypothesis holds� in the sense of�i� in x�� reinforced by �ii� and �iii�� and therefore dH

dt � � could be onlyapproximately true in the sense that it should hold �most of the time� whenthe cell Sk� that represents the microscopic state at the instant k� runsthrough a great part of the ergodic cycle of given energy �i�e� the part inwhich the interesting macroscopic observables� see �� are also constantfor practical purposes�� The relation dH� dt � � would then become falsewhen Sk� exits such region�

The latter circumstance however can only happen� in really macroscopicsystems as well as in systems with few decades of particles� with a temporalfrequency longer� by far� than the longest astronomical scales� see x��

Therefore the system would for all practical purposes evolve irreversibly�and the evolution irreversibility would be symmetrical in time�� Reversibil�ity could manifest itself over time scales beyond eternity� i�e� of many ordersof magnitude greater than the age of the Universe� already for systems likea gas at normal conditions in a container of the size of a room� or of a verysmall box� Or� alternatively� for an extremely short time around the initialtime enough to �forget� the peculiarity of the preparation of the initialstate�

A system set up initially in an �atypical� condition� e�g� occupying onlyhalf of the container� would expand to occupy the whole container and thenit would continue to evolve without �ever� returning to occupy the initialhalf�

Of course if a daemon acting a few seconds after the initial time invertsall the velocities of all the particles of the system� then the system wouldretrace its previous evolution coming back to the initial state �but just fora very short time� and then it would evolve by again occupying the wholecontainer proceeding towards equilibrium exactly as it would have doneif its velocities had not been inverted �and� furthermore� according to anevolution law described approximately by Boltzmann s equation��

This inversion of the motion with production of an atypical situation aftera short time �i�e� a non astronomically long time� from the initial instantrequires the exact inversion of all velocities if they were inverted with anerror� even very small �provided not �astronomically small�� the system


would not go back and� instead� it would probably continue to evolve asif �nothing had happened� from a macroscopic point of view� �LV�� Thee�ort of the daemonic heavenly creature to intervene on earthly a�airs� afterleaving the realm of metaphysics� would therefore be in vain�The second point of view has a more mathematical character and attempts

at making quantitative the argument just described by relating it to Boltz�mann s equation�One imagines an initial datum in which particles� hard spheres with radiusR� are independently distributed in phase space� we suppose that the densitywith which each of them is distributed is �f��p� q� where f� is normalized

to �Rf��p� q� dpdq � ��

This system is evolved with Hamilton s equations �i�e� with elastic collisionrules� and at time t one supposes that it is described by a distribution�ft�p� q�� without however assuming that the particles are independentlydistributed� this means that the one�particle distribution �ft�p� q� providesonly the information on the number of particles in dp dq but no longer theircorrelations �as was the case at time �� by construction� which will be nontrivial� just because the �Stosszahlansatz� will not hold�We now imagine� keeping t �xed� that �� R� � so that �R� � � but�R� � � � �xed quantity i�e� we consider the Grad�Boltzmann limit� see��#�� If the above qualitative discussion is correct and if one remarks thatin this limit the gas becomes a perfect gas �because the particles becomepoint masses� in which equilibrium is attained by virtue of collisions be�tween pairs of particles without two particles ever colliding more than once�because R � � implies just this� as it is easy to estimate the probabilityof recollision� i�e� of the event in the Fig� ��#��

A

B

C

Fig� �� Trajectory of C collides twice with A �A� B imagined �xed to simplify

� Note that if one throws randomly and independently hard spheres in a box then someof them may overlap� It is convenient not to exclude such a possibility provided onedisregards completely the interaction of the overlapping spheres as long as they overlapand starts considering it only after they separate because of the motion� this is clearlya trick that introduces some minor simpli�cation of the discussion while not a ectingthe macroscopic properties of the �rare�ed� gas�


drawing��

as proportional to �R� � �� per unit time�� then recalling that we denote�ft the evolving one particle density �to keep the integral of ft normalizedto �� one has to conclude that the evolution of the limit lim

R��ft � f t should

be rigorously described by the Boltzmann equation which for f t is

f tt

� p

m� f tq

� ��R��

Z jp� � p��jm

��

R�dp

�d��

� � f t�p�� q� f t�p�� q�� ft�p� q�f�p�� q��

��#��

One should note that this is ��R�independent because �R� and ��R�

are independent of R �recall that we are considering the case of hard spheresystems��

Hence the Boltzmann s equation should describe correctly the evolutionof a rare�ed gas for arbitrary times t in fact we expect that in the Gradlimit the recurrence times grow in�nitely large while the collisions make thesystem evolve on a time scale �xed by the �ight time ��R�� v�� This canalso be seen from ��#�� in which the time scale is �xed by �R�jp��p��j�mwhich� in the average� is just �R� v�

Equation ��#�� has been proved with complete mathematical rigor onlyrecently and for times t � ��R� v for systems of hard spheres and forinteresting classes of initial data f� this is the content of Lanford�s theoremon the Grad conjecture� �La��This is an important con�rmation� mathematically rigorous� of Boltzman�

n s point of view according to which reversibility� and the correspondingrecurrence times� is not in contradiction with the experimental observationof irreversibility� Because the time scale over which reversibility manifestsitself is not observable while that in which irreversibilty can be observedis related to the time of free $ight ��R� v�� Furthermore we see thatirreversibility is not incompatible with the ergodic hypothesis� and Boltz�mann s equation provides us with a model of the development of irreversiblemotions in situations in which the recurrence times are �in�nitely� longer�even on astronomical scales� than the average time needed for a moleculeto travel the free path �i�e� the $ight time��

Thus Lanford s theorem� although it presents moderate interest for theapplications due to the shortness of the interval of validity� t� ��R� v��has an enormous conceptual importance �apparently not yet fully appreci�ated by many� because it shows in a mathematically precise and rigorousfashion that there is no incompatibility between irreversible evolutions likethe one described by the Boltzmann equation and the completely reversibleHamilton equations that describe the details of the microscopic motions� Infact mathematical rigor is particularly welcome here in consideration of theenormous amount of speculation on the theme and of pretended proofs ofinconsistency between macroscopic irreversibility and mechanics� It has to


be hoped that� with time� Lanford s theorem will be appreciated as a basicadvancement of statistical mechanics�

There is already a vast literature that developed following the spirit ofLanford s work �which was heralded by various works� and here I cannotdiscuss the matter further for a proof developed with attention to the laterdevelopments and for the developments themselves the reader is referred tothe recent treatise by Spohn� �Sp�� pp� �#��

This concludes our general introduction to statistical mechanics� We haveseen that classical statistical mechanics holds only under certain conditions�see x�� for instance� at least as formulated here� It remains to analyze itsconsequences to deduce some of its applications and a better understandingof its validity and limitations�

Such an understanding is based� as already remarked� on the very con�sequences of the theory and it cannot be derived a priori as shown� forinstance� by the fact that the basic condition in x�� namely � � � �is compatible with very reasonable values of the temperature for �everydayphysics� only because the intensity � of the molecular interaction energyhas order of magnitude of �� erg and the radius of the molecules hassize �� cm� If this experimentally determined data had been verydi�erent the condition � � � � could be impossible to satisfy at tem�peratures of importance for the observations usually carried out by classicalthermodynamics� See Chap�II�III for a discussion of the latter points�

x�� A Historical Note� The Etymology of the Word �Ergodic

and the Heat Theorems

This section and Appendix ��A� are written in a way to be independent ofthe preceding sections therefore there are here and there a few repetitionsof subjects already analyzed in x��x��#� Few references to the previoussections are meant for readers familiar with them� but they are not essentialfor reading this section and Appendix ��A��

What follows is an expanded and revised version of various of my writingson Boltzmann s work� �Ga#�� Ga#��

�� The etymology of the word �ergodic� and the heat theorems�

Trying to �nd the meaning of the word ergodic one is led to a paper byBoltzmann� �Bo#�� see the footnote of S� Brush in his edition� �Bo�� ofthe Lectures on Gas Theory� on p� �� x�� here Boltzmann s paper isquoted as the �rst place where the word is introduced� Brush acutely warnsthe reader that Ehrenfests paper misrepresents the opinions and even theterminology of Boltzmann and Maxwell and dates �in agreement with Gibbs�

Readers might be interested in the referee report to one of my papers� �Ga��a�� as itshows� in my opinion� how blind to evidence an historian of Science can be at times�The contents of the paper in question are reproduced here� the referee report and thecorresponding unamended original version can be found in �Ga��b� �in English��

�# I� Classical Statistical Mechanics

see p� vi of the introduction of �Gi#�� the �rst appearance of the concept to�#�� Bo��b�� For instance the etymology that one �nds in the Ehrenfests paper is incorrect on this point see �EE�� note *�� p�#�� where also the�rst appearance of the word is incorrectly dated and quoted��

In fact the basic idea of ergodicity can perhaps be traced to even earlierworks� namely to the �rst work of Boltzmann on the theory of heat� �Bo��on p� �� one �nds that �� this explanation is nothing else but the mathe�matical formulaton of the theorem according to which the paths that do notclose themselves in any �nite time can be regarded as closed in an in�nitetime� �in this paper one also �nds a general derivation of the necessity of theidenti�cation between average kinetic energy and absolute temperature��

The �Bo#�� paper by Boltzmann is seldom quoted� I found only Brush sreference in �Bo�� and a partial account in �Br�� p� � and p� ��#�before my own etymological discussion appeared in print in �Ga#�� Ga#��Ga��a�� More recently the paper has been appropriately quoted by �VP� ��the paper was discussed also by �Ma##�� However no English translation of�Bo#�� is available yet� Nevertheless I think that this is one of the mostinteresting papers of Boltzmann it is a precursor of the work of Gibbs��Gi#�� on ensembles� containing it almost entirely �if one recalls that theequivalence of the canonical and microcanonical ensembles was already es�tablished �elsewhere� by Boltzmann himself� �Bo�#�� Bo�� and I will tryto motivate this statement�

The paper stems from the important� not too well known� work ofHelmholtz� �He��a�� He��b�� who considered what we call today a systemwhose phase space contains only periodic orbits� or cycles of distinct ener�gies i�e� essentially a one�dimensional conservative system� He called suchsystems monocyclic systems and noted that they could be used to providemodels of thermodynamics in a sense that Boltzmann undertakes to extendto a major generalization�

After an introduction� whose relative obscurity has been probably respon�sible for the little attention this paper has received� Boltzmann gives thenotion of stationary probability distribution on the phase space of N in�teracting particles enclosed in a vessel with volume V � He calls a familyE of such probabilities a monode� generalizing an �analogous� concept onmonocyclic systems� In fact Boltzmann �rst calls a monode just a singlestationary distribution regarded as an ensemble� But sometimes later heimplicitly� or explicitly� thinks of a monode as a collection of stationarydistributions parameterized by some parameters the distinction is alwaysclear from the context� Therefore� for simplicity� I here take the liberty ofcalling monode a collection of stationary distributions� and the individualelements of the collection will be called �elements of the monode��

The etymology that follows� however� is more appropriate for the elementsof the monodes� as they are thought to consist of many copies of the samesystem in di�erent con�gurations� By reading Boltzmann s analysis onecan get the impression� see p� �� of �Bo#�� that the word �monode� hadalready been introduced by Maxwell� in �Ma�� however the reference to


Maxwell is probably meant to refer to the notion of stationarity rather thanto the word monode which does not seem to appear in �Ma��In fact the orbits of a monocyclic system can be regarded as endowed with

a probability distribution giving an arc length a probability proportional tothe time spent on it by the motion hence their family forms a family ofstationary probability distributions�Etymologically� from the context of �Bo#�� this appears to mean a family of

stationary distributions with a �unique nature�� each consisting of systemswith a �unique nature�� di�ering only by the initial conditions�� from m�nosand e�dos � with a probable reference to Plato and Leibnitz� The conceptappears� in fact� in some of Plato s dialogues� see the entry monoeid�s ��onein kind�� in �LS��Then the following question is posed� Given an element � of a monodeE we can compute the average values of various observables� e�g� averagekinetic energy� average total energy� average momentum transfer per unittime and unit surface in the collisions with the vessel walls� average volumeoccupied and density� denoted� respectively

T ��

NhKi�� U � hK � �i�� p� V� � �

N

V��

where � denotes the potential interaction energy and K the total kineticenergy� We then imagine varying � in the monode E � by an in�nitesimalamount �this means changing any of the parameters which determine theelement�� Question is it true that the corresponding variations dU and dVare such that

dU � p dV

Tis an exact di�erential dS � ��

In other words is it true that the above quantities� de�ned in purely me�chanical terms� satisfy the same relation that would hold between them if�for some thermodynamic system� they were the thermodynamic quantitiesbearing the same name� with the further identi�cation of the average kineticenergy with the absolute temperature� �x��That the temperature should be identi�ed with the average kinetic energy

per particle was quite well established �for free gases� since the paper byClausius� �Cl�� and the paper on the equipartition of kinetic energy byBoltzmann� �Bo�� Bo�#� �in the interacting cases�� see the discussion ofit in Maxwell s last scienti�c work� �Ma�� The latter paper is also veryinteresting as Maxwell asks there whether there are other stationary distri�butions on the energy surface� and tries to answer the question by puttingforward the ergodic hypothesis� If so the monode would provide a �me�chanical model of thermodynamics� extending� by far� the early examplesof Helmholtz on monocyclic systems�Thus Boltzmann is led to the following de�nition� see x��

De�nition a monode E is called an orthode if the property described by�� holds�


By reading �Bo#�� the etymology of �orthode� is composed by �rj�s ande�dos� i�e� �right nature� or �correct nature��The above deep de�nition has not been taken up by the subsequent liter�

ature� This is surprising� even more so as Boltzmann� in the same paper�proceeds to discuss �examples� of mechanical models of thermodynamics�i�e� examples of orthodic monodes� The above orthodicity concept is stillattributed to Gibbs� see �Br�� p� � �The examples of orthodes discussed by Boltzmann in his paper are theholode and the ergode which are two ensembles whose elements are param�eterized with two parameters ��N or U�N � respectively� Their elementsare

��N�dpdq� �dp

�� dp

ndq

�� dq

n

conste��K��

and

�U�N �dpdq� �dp

�� dp

ndq

�� dq

n

const��K�p� � ��q�� U� ��

Boltzmann proves that the above two ensembles are both orthodes� thusestablishing that the canonical and the microcanonical ensembles �usingour modern terminology� are equilibrium ensembles and provide mechanicalmodels of thermodynamics� see Chap�II for a discussion of similar proofs�Boltzmann s simple proof makes use of the auxiliary �with respect to the

above de�nition� notion of heat transfer� In the canonical case it yieldsexactly the desired result� in the microcanonical it is also very simple butsomehow based on a di�erent notion of heat transfer� An analysis of thematter easily shows� see x�� in Chap�II� that a de�nition of heat transfer forthe microcanonical ensemble consistent with that of the canonical ensemblegives the result �� but only up to corrections expected to be of orderO�N�� As we have alluded to in x�� there is a problem only if one insistsin de�ning in the same way the notion of heat transfer in the two casesBoltzmann does not even mention this� possibly because he saw as obviousthat the two notions would become equivalent in the thermodynamic limit�Again from the context of �Bo#�� one sees that the word �holode� has the

etymological origin of �los and e�dos while �ergode� is a shorthand for�ergomonode� and it has the etymological root of �rgon and e�dos � meaninga �monode with given energy�� Ga#�� Ga��a��

The word �ergode� appears for the �rst time on p� �� of �Bo#�� but thismust be a curious misprint� as the concept is really introduced on p� ��On p� �� the author probably meant to say �holode� instead� this has beencorrectly remarked by �VP� �� The above etymology was probably proposedfor the �rst time by myself in various lectures in Roma� and it was includedin the �rst section of �Ga#�� It has also been proposed in �Ja#�� Ma##��The word �holode� is probably a shorthand for �holomonode�� meaning a�global monode� �perhaps a monode involving states with arbitrary energy�i�e� spread over the whole phase space��


This is not what is usually believed to be the etymology of �ergode� theusual belief� comes from the Ehrenfests statement that the etymology is�rgon and �d�s � with the meaning of �unique path on the surface of con�stant energy� see note *�� in �EE�� The latter etymology has been takenup universally and has been attached to the subject of �ergodic theory��which is a theory dealing with time evolution properties�

� � The ergodic hypothesis� continuous and discrete phase space

The etymological error of the Ehrenfests could be just an amusing factbut it had a rather deep negative in$uence in the development of the ��th century physics� They present their etymology in connection with thediscussion �amounting to a de facto rejection� of the ergodic hypothesis ofBoltzmann� In fact Boltzmann had come to the ergodic hypothesis in hisattempts to justify a priori that the ergode� as a model of thermodynamics�had to produce the thermodynamics of a system with the given Hamiltonianfunction �and not just a model��Boltzmann had argued that the trajectory of any initial datum evolves on

the surface of constant energy� visiting all phase space points and spendingequal fractions of time in regions of equal Liouville measure� See x��The Ehrenfests criticize such a viewpoint on surprisingly abstract math�

ematical grounds basically they say that one can attach to each di�erenttrajectory a di�erent label� say a real number� thus constructing a functionon phase space which is constant on trajectories� Such a function would ofcourse have to have the same value on points on the same trajectory �i�e� itwould be a constant of motion�� This is stated in the note *�� p� #�where the number of di�erent paths is even �counted�� and referred to inthe note *�� p� #�� Therefore� they conclude� it is impossible that there isa single path on the surface of constant energy� i�e� the ergodic hypothesisis inconsistent �except for monocyclic systems� for which it trivially holds��The abstract mathematical nature of this argument� see also below for a

critique� was apparently remarked on only by a mathematician� see �VP� �p� #�� i�e� by Borel� �� but it escaped many physicists� It is worryingto note how literally so many took the Ehrenfests version of the ergodichypothesis and how easily they disposed of it� taking for granted that theirformulation was the original one by Boltzmann and Maxwell� see �Br�� p��#��Having disposed of the ergodic hypothesis of Boltzmann� the Ehrenfests

proceeded to formulate a new hypothesis� the rather obscure �and somewhatvague as no mention is made of the frequency of visits to regions in phasespace� quasi�ergodic hypothesis see notes *�# and *�� p� �� in �EE��it led physicists away from the subject and it inspired mathematicians to

� It is important� in this respect� to be aware that Boltzmann had studied the Greeklanguage and� by his own account� quite well� see �Bo�� p� �� to the point of havingknown at least small parts of Homer by heart� Hence there should be no doubt that hedid distinguish the meanings of e�dos and �d�s which are among the most common

words�


�nd the appropriate de�nition giving birth to ergodic theory and to its �rstnontrivial results�

The modern notion of ergodicity is not the quasi�ergodicity of the Ehren�fests� It is simply based on the remark that the Ehrenfests had de�ned anontrivial constant of motion very abstractly� by using the axiom of choice�In fact from the de�nition� consisting in attaching a di�erent number� oreven �N� di�erent numbers� to each distinct trajectory� there is in princi�ple no way of constructing a table of the values of the function so de�ned inorder to distinguish the di�erent trajectories� In a system which is ergodicin the modern sense the Ehrenfests construction would lead to a nonmea�surable function� and to a physicist endowed with common sense such afunction� which in principle cannot be tabulated� should appear as not ex�istent� or as not interesting� Thus motion on the energy surface is calledergodic if there are no measurable constants of motion here measurable is amathematical notion which essentially states the possibility of a tabulationof the function�

It is surprising that a generation of physicists could be in$uenced �in be�lieving that the ergodic hypothesis of Boltzmann had to be abandoned as atoo naive viewpoint� by an argument of such an exquisitely abstract nature�resting on the properties of a function that could not be tabulated �and noteven de�ned if one did not accept the sinister axiom of choice�� What isremarkable is the coincidence that the recognition and the development ofthe axiom of choice was due essentially to the same Zermelo who was oneof the strongest opponents of Boltzmann s ideas on irreversibility� see also�Sc#��

Therefore it is worth� perhaps� trying to understand what Boltzmann mayhave meant when he formulated the ergodic hypothesis� Here one cannotfully rely on published work� as the question was never really directly ad�dressed by Boltzmann in a critical fashion �he might have thought� rightly�that what he was saying was clear enough�� The following analysis is anelaboration of �Ga#�� Ga��a� in some respects it gets quite close to �VP� ��It should be noted that �VP� � has a somewhat di�erent point of viewon several key issues� although we seem to share the main thesis that the�EE�� paper is responsible for most of the still persisting misunderstand�ings on Boltzmann s work� including the exclusive attribution to Gibbs ofBoltzmann s ideas on ensembles� so clearly elaborated in �Bo#�� This isso even though� by reading the literature carefully� it is possible to realizethat many were aware of the connection of Gibbs work with Boltzmann s�see for instance �Br�� p� � � �rst of all Gibbs� see p� vi of �Gi#�� wherehe quotes the �rst section �Bo��c� of �Bo��b��

My point of view� adopted in the preceding sections� is that of those whobelieve that Boltzmann always conceived of the phase space and time asdiscrete spaces� divided into small cells� see �Bo� �� p� �� He alwaysstressed that the continuum must be understood as a limit� see x�� seealso �Br�� p� �� and �Kl� �� Kl� �� Kl�� Du�� The book by Dugas��Du�� is particularly illuminating �also� on this respect �see for instance


Chap�I and the quotations of Boltzmann presented there� where he appearsto identify the discrete viewpoint with the atomistic conceptions�� In hiswritings Boltzmann very often makes this point see for instance p� � ��note � on p� �� discrete time�� p� �� p� ��#� p� �� p� �� discretetime�� p� � + �� in �Bo��

Although Boltzmann seems to have sometimes been quite apologetic aboutsuch a viewpoint �even calling it a �mathematical �ction�� Ba�� p��#� from�Bo� �� see also �VP� �� p� �� he took advantage of it to the point thatone can say that most of his arguments are based on a discrete conceptionof phase space� followed at the end by a passage to the continuum limit� seex�� It should be understood however that the discretization that Boltz�mann had in mind is by no means to be identi�ed with the later concept ofcoarse graining� see Chap�IX where a modern version of Boltzmann s dis�cretization is considered and where a distinction has to be made betweencells and volume elements� see also �VP� � and �Ga��a��

It is easier for us� by now used to numerical simulations� to grasp themeaning of a �cell� in the numerical simulations a cell is simply an ele�ment of the discrete set of points in phase space� each represented withincomputer precision �which is �nite�� One should always discuss how muchthe apparently harmless discreteness of phase space a�ects results� This is�however� almost never attempted� see �Ga��a� for an attempt� A �volumeelement� in phase space has� instead� a size much larger than the machineresolution� so that it looks like a continuum �for some purposes�� In theprevious sections we have been careful to keep the discrete treatment ofphase space always quite explicit� so that later we shall be easily able to seewhich are the consequences of a verbatim interpretation of the phase spacediscreteness�

Hence one can say that an essential characteristic of Boltzmann s thought isto have regarded a system of N atoms� or molecules� as described by a cell ofdimension �q and �p in each position and momentum coordinates� He alwaysproceeded by regarding such quantities as very small� avoiding entering intothe analysis of their size� but every time this had some importance he seemsto have regarded them as positive quantities�

A proof of this is when he refutes Zermelo s paradoxes by counting thenumber of cells of the energy surface of � cm� of normal air� �Bo�� a featthat can only be achieved if one considers phase space as discrete� Hiscalculation has been discussed in x��

In particular this point of view must have been taken when he formulatedthe ergodic hypothesis in fact conceiving the energy surface as discretemakes it possible to assume that the motion on it is �ergodic�� i�e� it visitsall the phase space points identi�ed with cells� compatible with the givenenergy �and possibly with other �trivial� constants of motion�� thus behav�ing as in a monocyclic system �as all the motions are necessarily periodic��This is in fact the de�nition in x��

The passage to the continuum limit� which seems to have never been madeby Boltzmann� of such an assumption is of course extremely delicate� and


it does not lead necessarily to the interpretation given by the Ehrenfests� Itcan easily lead to other interpretations� among which the modern notion ofergodicity� but it should not be attempted here� as Boltzmann himself didnot attempt it�In general one can hardly conceive that studying the continuum problem

could lead to really new information� cannot be obtained by taking a discreteviewpoint� Of course some problems might still be easier if studied in thecontinuum� and the few results on ergodicity of physical systems do in factrely explicitly on continuum models� �Si�� However I rather interpret suchresults as illustrations of the complex nature of the discrete model forinstance the ergodicity theory of a system like billiards is very enlighteningas it allows us to get some ideas on the question of whether there existother ergodic distributions on the energy surface �in the sense of ergodictheory the answer is a%rmative�� and what is their meaning� The theoryof the continuum models has been essential in providing new insights in thedescription of nonequilibrium phenomena� �RT�� Ru�#�� CELS��Finally the fruitfulness of the discrete models can be even more appreciated

if one notes that they have been the origin of the quantum theory of radi�ation it has even been maintained that Boltzmann had already obtainedthe Bose�Einstein statistics� �Ba��The latter is a somewhat strong intepretation of the �#�� paper� �Bo��

The most attentive readers of Boltzmann have� in fact� noted that in his dis�cretizations he uses� eventually� the continuum limit as a device to expeditethe computations� manifestly not remarking that sticking to the discreteviewpoint would lead to important di�erences in some extreme cases� Infact he does not discuss the two main �errors�� see Chap�III� that one com�mits in regarding a continuum formulation as an approximation �based onreplacing integrals with sums�� they were exploited for the �rst time byPlanck� much later� The latter errors amount� in modern language� seeChap�III� to the identi�cation of the Maxwell�Boltzmann statistics and theBose�Einstein statistics� and to neglecting the variation of physically rele�vant quantities over the cells see the lucid analysis in �Ku#�� p�� for atechnical discussion see Chap�III�The above �oversight� might simply be a proof that Boltzmann never took

the discretization viewpoint to its extreme consequences� among which thereis that the equilibrium ensembles are no longer orthodic in the sense ofBoltzmann� see Chap�III� �although they still provide a model for thermo�dynamics provided the temperature is no longer identi�ed with the averagekinetic energy�� a remark that very likely was not made by Boltzmann in


spite of his consideration and interest on the possibility of �nding otherintegrating factors for the heat transfer dQ� see the footnote on p� �� in�Bo#��

The necessity of an understanding of this �oversight� has been in particularclearly advocated by Kuhn referring to Boltzmann s �little studied viewsabout the relation between the continuum and the discrete�� Ku#�� forinstance�There are many directions into which the analysis of the foundations of

classical statistical mechanics can be developed� A somewhat di�erent view�point for instance can be found in �Kr�� this work of Krylov� and particu�larly part III has been very in$uential on Russian theoretical physics� In it�besides a very detailed critique of the foundations and of Boltzmann s andGibbs work� the foundations of the theory of the ergodicity of hard spheresystems is laid down it was pursued later by Sinai� It also provided groundsfor subsequent work on coarse graining �see Chap�IX� of Sinai and� in Sinai sinterpretation� �Si�� also inspiration for the later theory of chaotic systems��Si� �� quite close to Ruelle s proposal� see Chap�IX and �Ru�#c��

Appendix ��A�� Monocyclic systems� Keplerian Motions and Er�

godic Hypothesis

Consider a one�dimensional system with potential �x� such that j��x�j �� for jxj � �� and �x��x�� All motions are periodic sothat the system is monocyclic� We suppose that the potential �x� dependson a parameter V �One de�nes a state a motion with given energy E and given V � And

U � total energy of the system � K � T � time average of the kinetic energy KV � the parameter on which is supposed to dependp � � average of V �

A state is parameterized by U� V and if such parameters change by dU� dV �respectively� we de�ne

dL � �pdV� dQ � dU � pdV � ��A��A��

Then

Theorem �Helmoltz� The di�erential �dU � pdV ��T is exact�

In fact let x��U� V � be the extremes of the oscillations of the motion withgiven U� V and de�ne S as

�� In checking my understanding of the original paper as partially discussed in �Ga�� Ihave pro�ted from an English translation that Dr� J� Renn kindly provided me withlater� �� He noticed this footnote in �Bo�� while performing his translation� �unfortunately still unpublished��


S � log

Z x�U�V �

x��U�V �

pK�x�U� V �dx � log

Z x�U�V �

x��U�V �

pU � �x�dx

��A�� A��so that

dS �

R �dU � V �x�dV

�dxpKR

K dxpK

��A��A��

and� noting that dxpK

�q

�mdt we see that the time averages are given by

integrating with respect to dxpK

and dividing by the integral of �pK

� We �nd

therefore

dS �dU � pdV

T��A��A��

The above analysis admits an extension to keplerian motions such sys�tems are not monocyclic in the sense of Helmoltz� but if one considers onlymotions with a �xed eccentricity they have the same properties�It is convenient to study motions in polar coordinates �� so that ifA � �� ! �E � �

�m !x� � g m � m being the mass and g the strength of the

attraction due to gravity �g � kM if k is the gravitational constant and Mis the central mass� then

E ��

m !��

mA�

�� mg

�� gm

��A��A��

and

!��

m�E � mA�

��mg

��def� A��

�

��

��

�

��

��

�

�� E

mA��

�

��

�

��

g

A��

� � ��

def� a �

mg

� E��A��A��

p��

def� a

p�� e��

p�� e� �

Apag

�

Furthermore if a motion with parameters �E�A� g� is periodic �hence E � ��and if h�i denotes a time average over a period then

E � �mg

a� hi � �mg

a� h �

��i �

�

a�p

�� e�

hKi �mg

a� �E� T �

mg

a� hKi� Tecc �

��

p�� e�

�T

��A��A��Hence if S is de�ned by

S � log

Z

�

s

m�E � mA�

��mg

�� d� ��A��#�

��A��


its di�erential is

dS �dE � mA�

�a�p��e�

dA�

A� � ma dg�

��p�� e��T

��A��A��

This means that

�� If �E�A� g� are regarded as parameters then Tecc is an integrating factorof

dQ � dE � pAdA� � pgdg� pA �

�m a�

p�� e�

� pg �m

a��A��A��

� � Suppose that e is kept constant� so that the states as characterized by�E� g�� Then using

p�� e� �

p� E�mAg�� and �E � T � i�e�

dA�

A��d��E�

�E � dg

g� � ��A��A��

one can eliminate dA��A� from dS and �nd �after some simple algebra�

dS �dE � �� E�g��dg

T� d log

g�

�E ��A�� A��

so that T is the integrating factor of dQ � dE � pdV if V � g and p ��Eg � m

a � �Boltzmann�� Note that the equations pg � T and E � �T canbe interpreted as� respectively� analogues of the �equation of state� and the�ideal speci�c heat� laws �with the �volume� being g� the �gas constant�being R � and the �speci�c heat� CV � ��

�� If g is kept constant and �E�A�� determine the states the integratingfactor of dQ � dE�pAdA

�� with pA � � m�a�

p��e� is not the average kinetic

energy T but the eccentric temperature Tecc�

To check � � note that by ��A�� A��

dS �dE �� mA�

�a�p��e�

��E � � dg �ma � �mA�

�a�p��e�

�g ��

��p�� e��T

�

��dE � m

a dg�� A�

agp��e� ��

��p�� e��T

�dE � m

a dg

T�

�dE � ��E

g dg

�E � d logg�

� E�

��A��A��

This concludes the discussion of Boltzman s version of Helmoltz s theory�

In general one can call a system monocyclic when it has the propertythat there is a curve � � x�� parameterized by its curvilinear abscissa ��varying in an interval � � � � L�E�� closed and such that x�� covers allthe positions compatible with the given energy E�

�# Boltzmann s equation �Lorentz gas�

Let x � x�� be the parametric equations so that the conservation of energycan be written

�

m �

�dx

d�

��

�� x�� E � ��A��A��

Then if we suppose that the potential energy depends on a parameter Vand if T is the average kinetic energy� p � �hV i it follows

dS �dE � p dV

T� p � �hV i� T � hKi � ��A��A��

A typical case to which the above can be applied is the case in which thewhole space of con�gurations is covered by the projection of a single periodicmotion and the whole energy surface consists of just one periodic orbit� orat least only the phase space points that are on such an orbit are observable�Such systems provide natural models of thermodynamic behavior�Noting that a chaotic system like a gas in a container of volume V will

satisfy �for practical purposes� the above property we see that we shouldbe able to �nd a quantity p such that dE � pdV admits the inverse of theaverage kinetic energy as an integrating factor�On the other hand the distribution generated on the surface of constant

energy by the time averages over the trajectory should be an invariant dis�tribution and therefore a natural candidate for it is the uniform distribution�Liouville distribution� on the surface of constant energy�It follows that if � is the Liouville distribution and T is the average kinetic

energy with respect to � then there should exist a function p such that T��

is the integrating factor of dE � p dV �Boltzmann showed that this is the case and� in fact� p is the average mo�

mentum transfer to the walls per unit time and unit surface� i�e� it is thephysical pressure�Clearly this is not a proof that the equilibria are described by the micro�

canonical ensemble� However it shows that for most systems� independentlyof the number of degrees of freedom� one can de�ne a mechanical model ofthermodynamics� The reason we observe approach to equilibrium over timescales far shorter than the recurrence times is due �as discussed in the pre�vious sections� to the property that on most of the energy surface the actualvalues of the observables whose averages yield the pressure and tempera�ture assume the same value� This implies that this value coincides with theaverage and therefore satis�es the heat theorem� as Boltzmann called thestatement that �dE � p dV ��T is an exact di�erential if p is the pressure�de�ned as the average momentum transfer to the walls per unit time andunit surface� and T is proportional to the average kinetic energy�

Appendix ��A� Grad�Boltzmann Limit and Lorentz�s Gas

It is interesting to see how to derive Boltzmann s equation in simple modelsin which it becomes a linear equation� The models are well known since

Boltzmann s equation �Lorentz gas� ��

Lorentz introduced them in his attempt to establish more �rmly Drude stheory of electric conduction in metals�In the models there are two types of particles the W �particles �wind�

particles� and the T �particles �tree�particles��The W �particles move through space interacting only with the T �particles

which� however� are supposed to be in�nitely heavy compared to the W �particle and are supposed to be at rest and randomly distributed in space�Each model is completely described by the W � T interaction and by theT �particle distribution� From now on we shall focus our interest on the casein which the T �particles are distributed as the space distribution of a perfectgas �Poisson s distribution� with density n� We shall also assume that theT �particles are� with respect to the W �particles� hard spheres of radius a�re$ecting the W �particles on their surface�The assumed tree distribution is such that the probability for �nding inside

a given region ,� with volume V �,�� exactly N tree particles� and for �ndingthem in the in�nitesimal cubes dc�� dcN around c�� cN � is

f��c�� cN �dc�� dcN

N �� e�nV ��n

N

N �dc�� dcN ��A ��A��

where the parameter n has the interpretation of density of the tree particles�Note that� since the T �particles are hard spheres only with respect to theW �particles but not with respect to the each other� there are con�gurationsc�� cN of trees in which the hard spheres overlap� �for some commentson this point see x��If x � �p� q� is a W �particle phase space coordinate �p� velocity� q� posi�

tion� the symbolSc��cNt x ��A � ��A��

will denote the W �particle coordinate x� � �p�� q�� into which x evolves intime t in the presence of N tree�particles located at c�� cN � The symbol��p� will denote the direction of p and )x will denote the pair ��p�� q� ifx � �p� q��

Since the velocity jpj is conserved it is clear that Sc��c

Nt x depends only

on the trees located within a distance �jpjt � a� from q� The symbols

�Sc��c

Nt x

��Sc��c

Nt x

�� Sc��c

Nt x

��A ��A��

will� respectively� denote the velocity� position and momentum direction of��A � �� and we also set

Sc��c

Nt )x �

��Sc��c

Nt x

��Sc��c

Nt x

��

��A ��A��

Similarly we can give a natural meaning to the evolution of m W �particles

Sc��c

Nt �x�� xm� �

�Sc��c

Nt x�� S

c��c

Nt xm

�� A ��A��

which takes into account the fact that there are no W �W interactions�

�� Boltzmann s equation �Lorentz gas�

It is easy to derive Boltzmann s equation for W �particles in the case of theLorentz gas described above� see ��A �� below� One realizes that the

assumptions to be made in order to derive the Boltzmann s equation areessentially the same as conditions �� of Sect� #� They are

�i� a W �particle never hits twice the same particle�

�ii� molecular chaos is assumed�

�iii� the size of the T �particles is negligible�

Here by a �chaotic� W �particle state we again mean a state such that theW �particle correlation functions are a product of one W �particle distribu�tion which are independent of the T �particle distribution� More preciselya chaotic state is such that the probability distribution for �nding a certaincon�guration C of T �particles and a set of W �particles in x�� xm hasthe form p�C�

Qmi� f��xi�� where p�C� denotes the distribution ��A ��

and this is interpreted as � if any wind particle is inside the hard cores ofC��

Clearly assumptions �i�� ii� and �iii� can be only approximately true�Let us formulate Grad s limit conjecture for the Lorentz gas� Assume that

the initial W �particles state is such that the probability density for �ndingW �particle in dx� � � � dxm is m�� times

f�x�� xm� ��

ZC comp �x��xm�

p�C�

mYi�

f��xi� ��A ��A��

where f��x� is a given function of x and the �integral� is the �sum� overall the T �particle con�gurations compatible with x�� xm �i�e� over theC s such that no W �particle is located inside the hard core of a T �particle��The compatibility between �x�� xm� and C is expressed by the notation�x�� xm�compC�Note that ��A �� is not a product state for the W �particles this is so

because here we have hard core interactions between them and the T par�ticles�Consider the state obtained by evolving the initial state ��A ��

f�x�� xm� t� �

ZC comp �x��xm�

p�C�mYi�

f��SC�txj� ��A ��

��A��

�� Explicitly this means the following� Let p be the probability of �nding the W particles in a in�nitesimal cube dx� � � � dxm around the con�guration X � �x�� xm�in the box !�� and a tree con�guration in the in�nitesimal cube dc� � � � dcM aroundC � �c�� cM � in the box !� assuming it wider by an amount a than !�� atleast� Here xi � �p

i� qi�� Then p is the product of the probability in ��A�� times

m��Qm

i��f��xi� dxi� e

�RCf��d�

� where � � �p� q� andRCd� means integration over

p and over the q � !� which are outside the hard spheres centered on C � �c�� cN ��

In other words the W particles also have a Poisson distribution� in the region outsidethe T particles� with a density function f��


and then let the T �particle density n tend to in�nity and the hard core W �Tradius tend to zero in such a way that na� � � but na� � l �� Weshall imagine that the solid angle integration is normalized so that

Rd� � ��

Grad s conjecture can then be formulated as

If t � � and under �mild assumptions� on f�� the following limit exists

limna��

na�� const ��

f�x�� xm� t� � "f�x�� xm� t� ��A �#��A��

and

"f�x�� xm� t� �

mYi�

"f�xi� t� ��A ��A��

and "f�x� t� satis�es the Boltzmann equation

"f

t�x� t� � p �

"f

q�x� t� � ��jpj

Z� "f�x�� t�� "f�x� t�� d� ��A ��A��

where x � �p� q�� x� � �p�� q� and p� is a vector with the same length as p but

forming an angle � with it� a�� a� is the scattering cross�section of ahard sphere with radius a and �� na��A similar conjecture can be formulated in a two�dimensional model� here thesolid angle � has to be replaced by the de�ection angle � �see Fig� �A��and �� by ��

� sin �� and �� an� Of course the Boltzmann limit

will be� in this case� na� � �� na� ��

It is easy to construct a proof of the above conjecture in the two�dimensional case� The three�dimensional case could be treated along thesame lines as will become apparent from the proofs� We shall assume� forsimplicity� the spatial dimension to be two� The direction ��p� will be inthis case the angle between p and a �xed axis�We �rst specify the �mild assumptions� on f�� The function f��x� will be

thought as f��jpj� ��p�� q�� if x � �p� q�� and we can write

f��jpj� ��p�� q� �

Zd q� d��f��jpj� �� q��q � q��p�� A ��A��

we shall abbreviate �� q�� to �� dq�d�� to d� and ��q � q��p� � �� to��x � �� Hence� by using de�nition ��A �� Eq� ��A �� becomes� form � �

f�x� t� �

Zd �f��jpj� ��

ZC comp x

��SC�t)x� ��p�C� � ��A �� A��

It is therefore useful to consider the function

g��x� t� � ena�

ZC comp x

p�C��SC�t)x� �� A ��A��

� Boltzmann s equation �Lorentz gas�

where the factor ena�

has been introduced for normalization purposes �notethat it tends to �� in the Boltzmann limit�� It is easily checked that

g��x� �� )x � ��

Zg��x� t�d� � �

f�x� t� � e�na�

Zd � f��jpj� �� g��x� t� �

��A ��A��

We shall show that as na� � �� na � �� the function g��x� t�will tend to a limit "g��x� t� which satis�es the two dimensional analogue ofrelation ��A �� with initial condition "g��x� �� )x � �� and jpj �xed�The linearity of ��A �� and of the third ��A �� will imply� undersuitable assumptions on f�� that also "f�x� t� satis�es ��A ��We will not insist in discussing in which sense g��x� t� converges to

"g��x� t�� It will appear from the proofs below that� if x�t� � �q �

pt� p�� at least g��x� t�def� g��x� t� � e��jpjt��)x � )x�t�� converges to

"g��x� t�def� "g��x� t� � e��jpjt��)x � )x�t�� pointwise for t �� and in the

sense of the distributions for all t � �� However a close examination of theproof will provide evidence against any uniformity of the convergence in t�unless t is restricted to a bounded interval �for further remarks on this pointsee below��Under the above convergence conditions� �mild assumptions� could� for

instance� be continuity and boundedness of f�� The proof is based on asimple change of variables in ��A ��

Let x � �p� q� and let R�x� t� be the sphere with center q and radius�jpjt � a�� then Sc�tx depends only on the T �particles in c contained inR�x� t�� Hence the integral ��A �� can be explicitly written as

g��x� t� � ��A ��A��

� ena�

�XM�

ZR�x�t�M

e�nV �R�x�t��nM

M ��Sc��c

M�t )x� ��dc� � � � dcM

where V �R�x� t�� area of R�x� t� and where use has been made of theassumed Poisson distribution of the T �particles ��A ��Note that� in general� not all the T �particles c�� cM in ��A �� will

be hit by the trajectory Sc��cM�� x� � � � � t� Let Ax�t�N denote the set

of con�gurations c�� cN of N T �particles such that a W �particle withinitial coordinate x hits� in the time t� all the N particles in c�� cN atleast once� We deduce from ��A �� see Fig� ��A ��


Fig� ��A�� The set P �t� c�� cN � is the dashed region� The circles represent trees

c�� cN � �N � �� and the length of the trajectory in the dashed region is jpjt�

g��x� t� � ena�

�XN�

ZAx�t�N

nNdc

��dc

N

N � �c��c

N�x� � � �Sc��cN�t )x� �

� �� XMN

RR

dc��dc�

M�N

�M�N�� nM�Ne�nV �R�x�t��

��A ��A��

R � set of points in R�x� t�M�N such thatc�� c�M�N �� P �t� c�� cN �

where �c��c

N�x� is � if x is compatible with the hard cores of c�� cN

and � otherwise the region P �t� c�� cN� is the tube like region �see Fig��A �� swept by an ideal T �particle when its center is moved along the pathSc��c

N�� x� � � � � t�The sum within square brackets in ��A �� can be performed �since theintegrals are trivial� and yields

e�nV �P �t�c��c

N�� A ��A��

so that g�� x� t� is

g��x� t� �ena�

�XN�

ZAx�t�N

nNe�nV �P �t�c��c

N�� c

��c

N�x��

� � �Sc��cN�t )x� �� dc� � � � dcN

N �

��A ��#��A��

The reader should note the very simple probabilistic meaning of this equa�tion which makes it almost self�evident the T �particles in Ax�t�N can be hitmore than once in the time t� Divide Ax�t�N as A�

x�t�N �A�x�t�N where A�x�t�N

is the set of T �con�gurations in Ax�t�N such that all their T �particles are hitjust once by the trajectory S

c��c

N�� x� � � � � t�


To this decomposition of A�x�t�N there corresponds a decomposition g��x� t�

� g��x� t� � g��x� t� with

g��x� t� �ena�

�XN�

ZA�x�t�N

nNdc�� dcN

N ��c

��c

N�x��

��Sc��c

N�t )x� ��e�nV �P �t�c

��c

N��

��A ��A��

We now perform the change of variables� illustrated in Fig� ��A � from the N variables c�� cN to the new N�� variables l�� lN�� N �we get

dc�� dcNN �

� aN�

�N�Xi�

li � jpjt��

N�Yi�

dlj

�NYj�

�d�j

sin�j

��A � ��A��

represented as

q

l�

��

c�

l�

c��

lNcN

�N

lN�

� �� q��

Fig� ��A �

Hence the N �th order contribution to ��A �� is given by �if ��p� � and x � �p� q� � �jpj� � q�� jpj� �� q��

� ena�

� na�NZ �

�

N�Yi�

dli

Z �

�

NYi�

�sin�i

d�i�

��

N�Xi�

li � jpjt��

� ��N�Xi�

li � �q� � q��

NXi�

�i � � � � ��p��e�nV �P �t�c��c

N��A � ��A��

where li are the vectors represented by arrows in Fig� ��A � �jl�j � li�� the -in ��A � �� means that there is an extra condition on the integration region�It is the condition that none of the spheres of radius a around c�� cNhas intersection with the straight segments of the broken line representingthe trajectory in Fig� ��A � �i�e� this is the condition that c�� cN really

belongs to A�x�t�N�� Of course in ��A � ��

�PNi�� i � � � � ��p��

�meansP�

h�� P

i �i � � � � ��p�� h��


In the limit na� � �� na � �� the restrictions indicated bythe - in ��A � �� become unimportant and nV �P �t� c�� cN �� simpli�esenormously

nV �P �t� c�� cN �� na

N�Xj�

lj � ��jpjt ��A � ��A��

Hence the limit "g��x� t� as na� � � and na� �� of g��x� t� is

�XN�

��NZ �

�

Z �

�

� NYi�

sin�j

d�i dlj�

��

N�Xi�

li � jpjt��

� ��N�Xi�

li � �q� � q�

��

NXi�

�i � � � � ��p��

�� e��jpjt

��A � ��A��

In the derivation of ��A � �� we have systematically disregarded conver�gence problems connected with the summation over N � M � etc�� since theyare trivial as a consequence of the presence of the factorials and of theboundedness of the integration regions� The limit ��A � �� is pointwise fort �� and it could be checked that it holds also in the sense of distributionsfor t � ��Furthermore it could be checked that for t � � the function g�� x� t� �g�� x� t� is bounded above by a L��d�� function once the delta functioncontribution coming from the collisionless paths is subtracted to both terms�hence the limit ��A � �� holds also in the L��d�� sense� Finally� by directcomputation� it follows from ��A � �� thatZ

"g��x� t�d� � � ��A � ��A��

and this fact� together with the above convergence properties and ��A ��

implies the validity of the limit relation lim na��na��

g��x� t�� e��jpjt��)x�)x�t�� "g��x� t�e��jpjt��)x � )x�t�� in L��d�� for t � �� furthermore it couldbe proved that this limit holds� for t � �� in the sense of the distributions�That ��A � �� is a solution of the Boltzmann equation can be checked

directly by substituting "g into ��A �� with initial condition "g��x� �� )x�� and jpj �xed� see for instance �Ga�� or check directly �recalling that��d� �

� sin ��d�� as� with our conventions� d� � d�

� if � is the �solidangle� in the direction ��To complete the proof of Grad s limit conjecture it remains to deal with

the m�particle distributions� However we skip this point since it involvesstraightforward calculations based on changes of variable of the type illus�trated in Fig� ��A � �

We have thus described a proof of the Boltzmann limit conjecture in thecase of a two�dimensional Lorentz gas with hard core W �T interactions


and free gas distribution of the T �particles� The generalization to threedimensions would be trivial� See also �LS# � where a more detailed andcareful study of the mathematical aspects of the above analysis is performedwith further insights and applications�A less trivial generalization would be obtained by keeping the hard coreW �T interaction but assuming that the T �particles are spatially distributedas if they were a gas of hard spheres with hard core size being proportionalto the W �T radius� Other generalizations are conceivable in the directionof allowing soft W � T particle interactions and more general T �particledistributions�Much more di%cult and interesting would be the treatment of Knud�sen�s model� in which the T �particles are allowed to move without su�eringchanges in their momentum in the collisions with the W �particles�Had we done the calculations associated with the proof of ��A �� we

would have also found evidence of a lack of uniformity of the Boltzmannlimit in the number m of W �particles even at �xed t the larger m is� thecloser one has to get near the Boltzmann limit in order to see factorizationof the W �particle correlations�We also wish to remark that even when the Boltzmann limit conjecture

is true� one cannot expect that the solution � f�r� v� t� to the Boltzmann sequation �see x��# and ��A �#� above� is such that f�r� v� t� is a goodapproximation to the actual distribution f�r� v� t� for large t in fact oneintuitively expects that for times of the order of tm�f�p��na

� some nontrivialcorrelations will start building up thus destroying the molecular chaos andspoiling the validity of the Boltzmann equation�This last remark is quite deceiving since it tells us that we cannot use�without further assumptions� the Boltzmann equation to investigate thelong time behavior and� in particular� to compute the transport coe%cients�From a rigorous point of view we cannot even be sure that the lowest orderin na of the transport coe%cients is correctly given by the value obtained inthe Boltzmann limit� However it seems reasonable that this is� indeed� thecase at least if the dimension of the space is larger than two �in one dimen�sion a simple counterexample can be found by using soluble models �LP��in this case� however� the Boltzmann equation is a priori not expected tobe a good approximation��For further reading on the Lorentz gas see �WL�� LS# ��The idea of the Boltzmann limit is clearly stated in �Gr�#�� see p� �� the

present proof in the case of the Lorentz gas is done in �Ga�� for the caseof "g��x� t� only� and was inspired by discussions and suggestions from J�L�Lebowitz�

��

Chapter II�

Statistical Ensembles

�#

�

II� Statistical Ensembles ��

x�� Statistical Ensembles as Models of Thermodynamics

Given a mechanical system its microscopic states are described by themicroscopic con�gurations of N �identical for simplicity� particles with massm wandering in a given volume V such con�gurations are represented byphase space cells of equal phase space volume h�N �The cells have dimensions �p and �q in momentum and position coordi�

nates and they represent the maximal resolution with which we supposethat the microscopic states can possibly be observed since we suppose theparticles to be identical the phase space cells di�ering by a permutationof the particles must be regarded as identical� The parameter h � �p�qempirically represents the precision with which the microscopic states canbe determined� see Chap�I� x�� and x�� Time evolution transforms cells into cells in a small time � so that cell �

is transformed into �� S� by a transformation S de�ned in terms of thetotal energy or Hamiltonian function E�� the sum of the kinetic energyK�p� and the total potential energy ��q�

E�� E�p� q� � K�p� � ��q� �NXi�

p�� m �Xi�j

�qi� q

j� � ��

E�p� q� � Umin � minE�p� q� � ��where p � �p

�� p

N�� q � �q

�� q

N� are the momentum and position

coordinates of the N particles and is interaction potential between par�ticles� see x�� The second of � �� is a stability constraint that we shallassume to hold for all N �with Umin dependent on N� without it many ofthe integrals that we shall consider would be divergent�In fact we shall see that the properly signi�cant physical condition is thatUmin can be taken � �BN for some B� see � � �� below�We have then considered the stationary probability distributions � that

associate with every cell� i�e� with every microscopic state� its probability�� so that �� S��Families E of stationary distributions can be identi�ed with families of

macroscopic equilibrium states in which a generic observable f � i�e� a genericfunction de�ned on the phase space cells� takes an average value in the state� � E

f �X�

��f��

Given a family E of stationary distributions on the space of microscopicstates one can consider the averages that the most physically relevant ob�servables take in a state � � E

U�� X�

��E�� energy�

�� II� Statistical Ensembles

V �� V �� V �volume�

K�� X�

��K�� kinetic energy� � ��

P �� X�

��P �� pressure�

where P �� is the momentum variation per unit time and per unit surfacearea undergone by the particles in the microscopic state � in the collisionswith the container walls� i�e� P �� is the force per unit surface area exertedover the walls� see x��Therefore� given a family E of stationary distributions on the space of

microscopic states� we shall call its elements a statistical ensemble or simplyensemble� after Gibbs� or monode� after Boltzmann� see x�� If E is such afamily we can associate with every �macroscopic� state � � E the quantitiesU� V�K� p �energy� volume� average kinetic energy and average pressure�and we can ask whether the statistical ensemble E de�nes a �model ofthermodynamics� in which the absolute temperature T can be identi�edwith the average kinetic energy per particle up to a proportionality factorthat� to simplify various expressions� is written as ��kB

T �

�kB

K��

N� ��

where kB is a constant to be determined empirically�The precise meaning of the locution �de�nes a model of thermodynamics�

has been discussed in Chap�I� �see �� it means that by varying � in Eand following the variations of U� V� T� p the relation

� dU � p dV ��T � exact di�erential � ��

holds� Hence it will be possible� by integrating � �� to de�ne a functionS�� on E so that the quantities U� V� S� T� p satisfy the relations of classicalthermodynamics in which S has the interpretation of �entropy�

� dU � p dV ��T � dS� � ��

see ��It is possible� in this way� to associate with each macroscopic state � � E

the quantities U� T� S� p� V and de�ne a �model of thermodynamics� thestatistical ensembles E that enjoy the latter property � �� were brie$ycalled by Boltzmann �orthodes�� see x�� and x�� and therefore we shallrefer to � �� by calling it the orthodicity property of the ensemble E �The existence of important classes of orthodic ensembles was demonstrated

by Boltzmann who also provided some a priori reasons to expect that hisexamples should not only give mechanical �models of thermodynamics� butprecisely the thermodynamics of the system� given to us by the experimental


observations in this attempt he founded ergodic theory and the Boltzmannequation� see x��Therefore the �theory of ensembles� poses three questions

�� existence and description of orthodic ensembles

� � equivalence of the thermodynamics that they describe

�� comparison of the equations of state computed from the ensembles andthe corresponding ones obtained experimentally�

In this chapter we shall consider the two basic ensembles studied by Boltz�mann and we shall show their orthodicity� following the lines of Boltzmann��Bo#��The canonical ensemble �see x�� consists of the probability distributions� on the space of the microscopic states � which describe particles roamingin a volume V that� for simplicity� we shall suppose cubic and with perfectlyre$ecting walls� The probability of a cell is� by de�nition

�� e��E��

Z�� V ��

with E�� E�p� q�� p� q� � �� being the energy of the microscopicalcon�guration �� and

Z�� V � �X�

e��E�� #��

is a normalization factor Z which will be called canonical partition function�the elements � of the canonical ensemble are therefore parameterized by thevolume V and the quantity ��The microcanonical ensemble consists of the probability distributions �

parameterized by the parameters U and V de�ned by

�� n

��N �U� V � if U �DE � E�� U� otherwise

� ��

where N �U� V �� called the microcanonical partition function� is

N �U� V � �X

E�D��U�DE�U ��

�number of cells � ofenergy E�� U �DE�U �

� ��

where DE is a macroscopic energy� albeit very small compared to U ��

� Or better compared with U �BN if the energy is bounded below by a stability boundUmin � �BN � our conventions give energy � to con�gurations in which the N particlesare in�nitely far apart and with zero speed� hence� if Umin is the minimum �potential�energy then the energy above the minimum energy con�guration� the �ground state isU � U� � U � BN and �BN is a lower bound for U��

� II� Statistical Ensembles

In other words in the microcanonical ensemble one attributes equal proba�bility to all cells with macroscopic energy U and � probability to the others�while in the canonical ensemble one attributes relative probability �alsocalled weight� e��U to all cells with microscopic energy U which� however�can take all possible values �i�e� all values between the minimum of thepotential energy and ��Proving �orthodicity� of the above ensembles means

�a� expressing U�K� p in terms of two parameters �� v�� with v � V�N� � �� in the case of the canonical ensemble� or �u� v� with u � U�N� v � V�N �in the case of the microcanonical ensemble� and

�b� showing that� de�ning T equal to ��kB

times the average kinetic energyper particle� then

� du � p dv��T � exact di�erential � ��

as �� v� or �u� v� vary� respectively�

We shall see that while the canonical ensemble is already orthodic in �nitevolume� the microcanonical ensemble is orthodic �only� in the �thermody�namic limit� N � �� U � �� V � � so that U�N � u� V�N � v stayconstant �or tend to a constant��This will be the physically interesting limiting situation� if one keeps in

mind the size of N � in real physical systems�

x�� Canonical and Microcanonical Ensembles� Orthodicity�

There are many other examples of ensembles which are orthodic at least inthe thermodynamic limit� However before proceeding to the discussion ofother ensembles and of their equivalence �i�e� of their identity as models ofthermodynamics� it is convenient to describe how one can check orthodicityof the canonical and microcanonical ensembles� This check is a key tothe understanding of Boltzmann s ideas and to the understanding of themathematical mechanisms that make tractable a problem that at �rst sightmight look formidable�Consider �rst the canonical ensemble case � �� #��The partition sum Z�� V � can be computed� if the cell size h � �p �q is

small� as

Z�� V � �

ZR�N

ZqV N

e��K�p�e��q�dp dq

N �h�N� � ��

where the factor N � takes into account that the N particles are strictlyidentical and� therefore� indistinguishable as a matter of principle� so thatby permuting the N particles one obtains microscopic states described byphase space cells that must be regarded as identical�We can identify a con�guration � �i�e� a phase space cell� as the two no�

tions coincide having adopted a discrete viewpoint �see x�� of the system


by giving the �occupation numbers� n� of particles in a small cube C� ofdimension ��p�q�� h�� n� in the cube C�� etc �the cubes should not beconfused with the phase space cells they are �.dimensional boxes in thephase space of a single particle�� Therefore in � � �� we replace the sumcorresponding to � ��#� with an integral in this way a twofold error iscommitted

�i� an approximation error due to the fact that E�p� q� � E�� only at thecenter of the cell � we shall call this error �i�e� the act of confusing theaverage value in a cell with the actual value at the center� an analytic error�

�ii� an error due to the fact that a microscopic con�guration � � R�N

described by giving the numbers n�� n�� is counted in the integral � � ��N ��n��n�� times instead of N � times� We call this a combinatorial error�

Both errors are� obviously� in�nitesimal as h � � �if one means that bothdimensions �p and �q tend to � as h � �� as we always assume�� Theywere neglected by Boltzmann in his analysis since he had no reason to thinkthat the cell size would play any role in his classical world� besides that ofallowing one to speak of the �number of con�gurations� of given energy�We shall neglect them here as well� postponing their analysis until it will

become possible to discuss them a posteriori on the basis of theoreticalconsequences of the theory drawn by neglecting them hence the theory�once developed� will allow us to evaluate under which physical conditionsnegligibility of the above errors becomes reasonable�Anticipating the results of the analysis �see also x�� details will be pro�

vided in x �� these errors will become negligible at �high temperature�and� in the example of a perfect gas ��q� � �� at least for

T � Tq � �mkBh��

��

where � � N�V � kB � ��#� �� erg�K��

The relation � � � � is obtained �summarizing part of the discussion in x�� by remarking that the representation of the microscopic states by cells canbe consistent only if �p and �q are smaller than the average values of themomentum and of the intermolecular distances��

Since by � �� the absolute temperature is such that �kBT� is the av�erage value of the kinetic energy per particle� i�e� the average of p�

�� m�

it is clear that the average momentum will have order of magnitudep �

pmkBT � while the average interparticle distance will be q � �

pV�N �

� The inequality �� in the case of hydrogen at normal density and pressure� m �� g� N � �� particles in V � � cm�� and choosing h � Planck sconstant � �� erg��K� gives Tq � ��K� very di erent values for Tq areobtained for other gases� see x�� where other necessary conditions for the validity ofthe approximations are also taken into account�

� This is a condition less stringent than the one examined in x�� where T � T�also imposes the compatibility of the description in terms of cells with the classicalmicroscopic dynamics as a cell permutation�


�� and therefore the condition h � �p �q p q �pmkBT�

�� follows�necessary but not su%cient� yielding � � � �� See x �� below for a moredetailed analysis�It is important� however� to keep in mind that when � � � � is not valid�

hence the cells sizes cannot be neglected� the very consistency of a cellrepresentation of the microscopic states fails� and the whole theory shouldbe reexamined from scratch� It will appear that in such circumstancesquantum mechanics becomes important and classical statistical mechanicsmay lose validity in a fundamental sense�Making the assumption that � � �� is correct without the necessary ana�

lytic and combinatorial corrections and performing the orthodicity analysisis equivalent to setting h � �� i�e� to admitting the possibility of in�niteprecision �simultaneous� measurements of position and momenta of �all�particles�We can evaluate� following Boltzmann� �Bo#�� the thermodynamic quan�

tities in the state described by the canonical distribution � with parameters�� V �To simplify notation we shall identify the region V occupied by the system

with the measure V of its volume �which we always think of as cubic��We shall use the fact that in our approximations the probability of �nding

the system in the microscopic state dp dq is e��E�p�q� dp dq�N �h�NZ�� V ��so that � �� become

K � K��

Z �NXi�

p�i

m

�e��K�p��q� dp dq

h�NN �Z�� V �

v � V�N

U � U��

logZ�� V � � � ��

p � P �� XQ

N

Z�� V ��

�Zv�

e��K�p��q�� mv�s

S

dq�� dq

Ndp

�� dp

N

h�NN �

where the sum is over the small cubes Q adjacent to the boundary of the boxV by a side with area s while S �

PQ s is the total area of the container

surface and q�

is the center of Q �note that S � �V �� see x�� It is not di%cult to transform the last of � � �� into a more useful form

p � ��

VlogZ�� V � � � � ��

the calculation is illustrated in detail in x �� below where we also collectother more technical deductions�At this point we only need a simple direct check� Let

F � �� logZ�� V �� S � �U � F ��T �� F � U � TS � � ��


and use � � �� to obtain

T �

�kB

K��

N�

�

kB�

dT

T� �d�

��

because the integral over q in � � �� factorizes and the one over p is elemen�tary �i�e� gaussian�

dF � �� logZ�� V � � ��U� d� � ��

VlogZ�� V � dV �

� �F � U� dT�T � p dV � �S dT � p dV � � ��

hence

T dS � d�F � TS� � p dV � dU � p dV � � �#��

which coincides with � ��We also read from the above relations the physical interpretation of the

partition function Z�� V � in fact the function F � �� logZ�� V � isthe free energy of thermodynamics�Equation � � �#� shows the orthodicity of the canonical ensemble�Note that � � �#� has been derived without any necessity to consider the

thermodynamic limit N � �� V � �� V�N � v� as long as one acceptsthe approximations leading to � � �� i�e� if the cells size h can be taken as� or� more physically� as negligible�� This �unconditional� validity� for allN and V � should be regarded as a coincidence� as the following discussionshows� In the other ensemble cases consideration of the thermodynamiclimit is necessary to establish the correct thermodynamic relations betweenU� T� S� p� �� V � In fact� to prove orthodicity of ensembles other than thecanonical it is necessary to impose some physically important conditionson the interaction potential energy ��q� the �stability and temperednessconditions�� see below�In particular the situation is somewhat more involved in the microcanonical

ensemble case because in this case it becomes really necessary to considerthe thermodynamic limit�The microcanonical partition function is de�ned in � �� and� up to the

errors already pointed out in the case of the canonical ensembles� it can bewritten as

N �U� V � �

ZJE

dp dq

h�NN ��

where JE is the phase space set in which �U �DE � E�p� q� � U��The thermodynamic quantities are de�ned by � �� and the pressure can

be written just as in � � �� withN �U� V �� replacing Z�� V �� e��K�p��q�

respectively� and with the integral extended to the domain U � DE �K�p� �F�q� � U � In this case U is a parameter de�ning� together with V �the elements of the ensemble� The temperature is de�ned as ��NkB timesthe average kinetic energy per particle� See x��


Also in this case orthodicity is derived by a direct check� Let

S � kB logN �U� V � � � ��

and let T be � ��kB� times the average kinetic energy per particle� one�nds

dS � kB

��

N �U� V �

NU

�U� V � dU ��

N �U� V �

NV

�U� V � dV

��

and we ask whether the right�hand side of � � �� can be written as � dU �p dV ��T with p� V� T de�ned in � �� i�e� by relations analogous to � � ��The derivatives ofN can be studied as in the case of the canonical ensemble�

and one �nds that � � �� can be rewritten� see x �� as

dS � kB�N

K�p��

��

�N� dU �

p dV

hK�p�ihK�p��i�

� � ��

where �if JE is the domain U �DE � E�p� q� � U� we have set for � real

hK�p��i �

RJE

K�p�� dp dq�h�NN �RJE

dp dq�h�NN �� real � � ��

hK�p��i �

RJE�q

� dV K�p�� dp dq�h�NN �R

JE�q� dV dp dq�h�NN �

� real � � ��

dV being an in�nitesimal region �with volume also denoted dV � around Vobtained by displacing by a distance �� along the external normal to V � thesurface elements of V �In other words hK�p��i is the average value of the ��th power of K�p�

with respect to the considered microcanonical distribution� while hK�p�iis the average value of K�p� with respect to a distribution � obtained byimposing the condition that one among the N particles is constrained to bein the region dV around the surface of V � If the relations

hK�p��i � hK�p��i � K�� N � � � ��

were valid� with N ��N�� and with K�� equal to the average kineticenergy in the microcanonical ensemble� then one could deduce that � � �� becomes� after dividing both sides by N and letting N � � with V

N �UN

constantsds � �du � p dv��T � � � ��

In the microcanonical case one sees from � � �� that the parti�tion sum directly has the physical meaning of entropy S � kB logN �U� V ��Since N �U� V � is the number of microscopic states with energy U and al�lowed volume V �see also x�� this is the well�known Boltzmann s relation


expressing entropy as proportional to the logarithm of the number of pos�sible microscopic states with given energy and volume� see �Bo�� and x��To complete the analysis of the microcanonical ensemble orthodicity it

remains to check � � �� as already mentioned one needs for this purposesuitable assumptions on the potential energy ��q��Such assumptions� which will have an important physical meaning� are

�a� Stability this means that there is a constant B such that for everycon�guration �q

�� q

N� � q

��q� �Xi�j

�qi� q

j� � �BN � � ��

This property not only says that the potential energy is bounded below �asusual in many mechanical systems� but it also says that its minimum cannotbe too small as N grows�

�b� Temperedness there are three constants C � �� x � �� R � � forwhich

j�q � q��j � Cjq � q�j��x for jq � q�j � R � � ��#��

This is essentially a condition that says that �far� particles have �small�interaction by this hypothesis the interaction energy between a particleand a uniformly �lled half�space approaches � as the distance between thetwo tends to �� In a large system the macroscopic subsystems have �small�interaction energy �i�e� much smaller than the product of the volumes oc�cupied by each�� This can be considered as a property of the �short range�of the forces�

Relations � � �� #� are not satis�ed in the special but very impor�tant case of systems of charged particles interacting via the Coulomb forcequalitatively the problem really comes only from condition �b� because �a�is satis�ed as one thinks that in realistic cases particles have hard cores�however� in spite of this� we shall see that even �a� poses a problem of aquantitative nature as the �obvious� hard cores are often of nuclear sizewhich turns out to be too small for compatibility with the observations��statistical mechanics of systems interacting via Coulomb forces is thereforemore delicate than that of systems interacting via phenomenological pairforces with short range �like Lennard�Jones potentials� which mean e�ectivehard cores of atomic size �rather than nuclear size��Even more delicate is the statistical mechanics of gravitationally interacting

particles� We shall see that while systems of charged hard core particles withthe property of a neutral total charge do obey �normal thermodynamics�the same is not true for gravitationally interacting particles �so that weshould not expect that a Star obeys the same thermodynamics as a pot ofgas� just in case this idea occurred to you��

�# II� Statistical Ensembles

Equations � � �� are related to the law of large numbers they say thatthe variables �

NK�p�� regarded as random variables with a distribution givenby an element � of the microcanonical ensemble or of the corresponding ��see � � �� are variables with a �dispersion that approaches �� in thelimit N � �� because the ratio K�p��K�� is such that h�K�p��K��i�h�K�p��K��i � � for all �� or the $uctuations of K�p�� with respectto its average value hK�p��i K�� do not have the order of magnitudeof hK�p��i itself� but are much smaller�

The kinetic energy K�p� is however a sum of N �almost independent�

variablesp��

�m � � � � �p�N

�m � i�e� not really such because they are constrained byU � DE � ��q� � K�p� � U � ��q�� Therefore it is clear that � � ��requires a proof and it does not reduce trivially to the law of large numberswhich is formulated for independent variables� We have in fact just discussedwhich extra assumptions are necessary in order to be able to show themicrocanonical ensemble orthodicity�

From a historical viewpoint the above treatment of the canonical ensem�ble is essentially the same as the original in Boltzmann� �Bo#�� the case ofthe microcanonical ensemble is somewhat di�erent and more involved thereason is that in Boltzmann the assumptions � � �� are only implicitlymade in fact Boltzmann studies the problem from a slightly di�erent view�point� He considers a priori a quantity that he identi�es with the amountdQ of heat that the system receives when the microcanonical parameterschange by dU� dV � In this way he shows that the microcanonical ensembleis orthodic even in a �nite volume� This is possible because the de�nitionsof dQ that he uses in the two ensembles are di�erent and in the languageused here they are consistent only in the thermodynamic limit �and only if� � �� are assumed�� But this is not the moment to attempt a philologi�cally correct treatment of Boltzmann s ideas �a treatment that is still quiteunsatisfactory in the literature� see x��

To conclude this section we can ask how strongly the orthodicity of thecanonical and microcanonical ensembles depends upon the hypothesis that� � �� and � � �� are good approximations to the partition sums �as �nitesums over cells in phase space�� and how strongly the orthodicity dependson the hypothesis that the system consists of only one species of identicalparticles�

Without exhibiting any analytic calculations we simply say that� in thecase that the integrals � � �� or � � �� are replaced by the sums that theyare supposed to approximate� orthodicity must be formulated di�erentlyin the canonical ensemble one has to interpret � as proportional to theinverse of the absolute temperature while in the case of the microcanonicalensemble one must de�ne the entropy directly via Boltzmann s formulaS � kB logN �U� V ��

One obtains in this way two models of thermodynamics� which are modelsin a sense which is natural although di�erent from the one so far used�Namely in the �rst case by setting T � �

kB�the expression �dU �pdV ��T is


an exact di�erential� but T is no longer proportional to the average kineticenergy� in the second case setting T�� dU � pdV ��dS the quantity Tis independent of the transformation that generates the variations dU� dVand the corresponding dS� Furthermore it is possible to prove that the twomodels of thermodynamics are equivalent� �Ru��The important and well�known universal identi�cation� �Cl�� Bo�� be�

tween the average kinetic energy with the absolute temperature is no longervalid in view of the role that this identi�cation played in the birth of statis�tical mechanics and in its developments one should regard this as a shockingmajor change� See Chap�III for a more detailed analysis of this point�Therefore the ensembles in which the partition functions are evaluated

without the �continuum approximation�� valid only when � � � � �or betterwhen �� hold� can still be used for the formal construction ofmodels of thermodynamics�However� as a consequence of the general considerations following � � � ��

in such cases it is not clear what the physical meaning of the thermodynam�ics that is constructed from the mechanical model could be a physicallycorrect investigation would in fact require� in such situations� using quantummechanics as a basis for the treatment�For what concerns the assumption of existence of only one species of par�

ticles in the systems considered so far we simply mention that orthodicitydoes not depend on this assumption� But there are some obvious changesthat one has to introduce in the formulation and in the combinatorial fac�tors to be used� As an example we just write the partition function for ageneral system with N� particles of species � and mass m�� N� species particles with mass m�� etc� Under the assumption that the cell size can beneglected we have

Z�� V � ��

N��N��

Zdp

�dq

�

h�N�

dp�

dq�

h�N�� e��

P�K�p

��q

��

� � ��

and the probability of a microscopic state will be

Y�

�dp

�dq

�

N��h�N�

�e��

P�K�p

��q

��Z�� V ��

The natural generalization to this case of the notion of orthodicity is checkedin exactly the same way as in the previous case of only one species of par�ticles�

x�� Equivalence between Canonical and Microcanonical Ensem�

bles�

In the above study of canonical and microcanonical ensembles Boltzman�n s constant appeared several times it was always denoted by the samesymbols� but it was to be regarded as a priori di�erent in each case�


In fact this is a universal constant kB � ��#� �� erg �K��The logical itinerary leading to the identi�cation of kB and to showing the

equivalence of thermodynamic models described by the orthodic canonicaland microcanonical ensembles is discussed in this section�Suppose �rst that the molecules do not interact� � �� i�e� consider the

microscopic model of a free gas� In this case it is easy to compute ex�plicitly the microcanonical and canonical partition functions� N � Z� in theapproximation in which cell size is neglected �see � � �� and � � ��One �nds� performing the integrals � � �� and � � �� in polar coordinates

in momentum space

N �U� V � �V N �

p mU

�N �p m�U �DE��N

�

N �h�N/��N�

�N

Z�� V � �V N

p �m��

�N

h�NN �

� ��

where /�d� � '�d� ��p�d

is the surface of the d dimensional unit sphereand '�x� is Euler s gamma function �i�e� '�x� � �x� ��The limits of � �� as N � �� V � �� with V�N � v� U�N � u

�xed� are easily studied via Stirling s formula '�x � �� xxe�xp

�x�� O��x�� or N � � NNe�N

p �N�� O��N�� and one �nds� see x �� and

� � ��

S � kB logN �U� V � �

� NkB�logV

N�

�

log

U

N� const � O�

logN

N��

F � �� logZ�� V � �

� �N��logV

N� �

log� � const � O�

logN

N��

On the basis of the discussion in x � � S has the interpretation of entropy inthe microcanonical ensemble and F of free energy� F � U �TS� see � � ��Hence we can compute the pressure in both cases

p

T� �

S

V�U � kB

N

V

�� O�

�

N��

microcanonical

p � ��F

V��

N

V� kBT v�� canonical

� ��

If NA is Avogadro s number �NA � �� molecules per mole� andN � nNA �with n � number of moles�� one sees that � �� establish thatthe perfect gas equation of state is pV � nRT in both cases� provided thevalue of kB is chosen the same in the two cases and provided it has thenumerical value

kB � R�NA � gas constant�NA � �#�� NA� erg �K��

� ��#� ��erg �K��


The speci�c heat at constant volume turns out to be� in the thermodynamiclimit �after an easy calculation� �nR

� for instance in the canonical ensemblethe average total energy� equal to the average total kinetic energy because� � �� is �

�NkBT and is volume independent� at �xed N � see � � �� andx �� below�As we see the two thermodynamics de�ned for the perfect gas by the two

microscopic models� canonical and microcanonical� coincide in the thermo�dynamic limit and they coincide with the experimentally known thermody�namics of a free gas� provided the constant kB is chosen in both cases as in� ��We now ask whether the coincidence of the thermodynamics de�ned by the

two statistical ensembles remains the same also for more general systems�This is the problem of equivalence of the microcanonical and canonical

ensembles� It is a fundamental problem because it would be a serious setbackfor the whole theory if there were di�erent orthodic ensembles predictingdi�erent thermodynamics for the same system� i�e� di�erent relations amongu� v� T� p� s� all compatible with the general laws of classical thermodynamicsalthough di�erent from each other�We shall see that �in general� for each given system there is equivalence �in

the thermodynamic limit� between canonical and microcanonical ensemblesif the constant kB appearing in the theory of the two ensembles is taken tobe the same�Once equivalence of the thermodynamics� de�ned either by the canonical or

by the microcanonical ensembles corresponding to a given system� has beenestablished we shall ask the further question of whether the constant kB thatappears as proportionality factor between temperature and average kineticenergy per degree of freedom is the same for all other systems� i�e� whetherthe numerical value � �� is system independent�The scheme of the proof of equivalence between canonical and microcanon�

ical ensembles� already used by Boltzmann and Gibbs� is the following� Set

N��U� V � �

ZE�p�q��U

dpdq

h�NN ��

Note that N �U� V � � N��U� V � � N��U � DE� V � and that the relationbetween N� and Z is simply given by

Z�� V � � �

Z �

Umin

dEe��EN��E� V � � ��

if Umin is the minimum of the energy and if Z�N are given by � � �� this is satis�ed by integrating � �� by parts over E� we treat here onlythe case in which the continuum approximation is accepted� �h � ��

� But one can check that the ensemble equivalence remains formally valid even if the cellsizes are not neglected provided the orthodicity notion is adapted to the new case asdiscussed in x��

� II� Statistical Ensembles

Hence� see x �

F �� V � � �� logZ�� V � �

� �� log� � �� log

Z �

Umin

e��EN��E� V � dE ��

The speci�c �i�e� per particle� thermodynamic quantities in the canonicaldistribution � with parameters �� V are� in the thermodynamic limit �V �� V�N � v �xed�

fc�� v�� limN��

�

NF �� V � canonical free energy

uc�� v�� limN��

U��

N�

�fc�

�� v� canonical internal energy

Tc��

kB��

�kB

K��

Ncanonical absolute temperature

� ��#�� v�V

Ncanonical speci�c volume

pc� limN��

P �� fcv

�� v� canonical pressure

sc�uc � fcTc

canonical entropy

where in expressing uc� pc as derivatives of the free energy fc via � � �� the operations of di�erentiation and of limit have been interchangedwithout discussion� because we proceed heuristically with the aim of ex�hibiting the essence of the mechanism of equivalence�The same thermodynamic quantities can be evaluated also in the mi�

crocanonical ensemble with parameters U� V � and of course they have ana priori di�erent de�nition

fm�um� vm� � �Tmsm � um microcan� free energy

um�U��

N�

U

Nm�c� internal energy

Tm�

�kB

K��

N�� smum

�um� vm��

m�c� abs� temperature

vm�V

Nm�c� speci�c volume

pm� P �� Tms

vm�um� vm� m�c� pressure

� �� sm� lim

N��kBN

log�N��U� V ��N��U �DE� V �

��

� limN��

kBN

logN��U� V � m�c� entropy

where the expressions for Tm� pm follow from � � �� the expression forthe free energy is the classical thermodynamic de�nition� while that of themicrocanonical entropy requires a digression�


In the theory of the microcanonical ensemble the value of DE is not spec�i�ed �and it is only subject to the condition that DE � U and that DE isa macroscopic quantity� i�e� DE�N ��

N�� De � �� Nevertheless the the�ory of the microcanonical ensemble would proceed in the same way evenif DE � U � Umin� i�e� as large as possible� and one would still obtain anorthodic ensemble� hence a model of thermodynamics in which the entropywould have the �new� value S � kB logN��U� V ��The function sm�u� v� � limN�� kB

N logN��U� V � is monotonic non de�creasing in u because such is� manifestly� N �U� V � and in reality one canshow that� in the cases we consider �i�e� stable and tempered potentials� see� � �� #�� it is strictly increasing �see Chap�IV� as we should wishbecause� if sm � sm� the derivative � sm�um�� should be equal to theabsolute temperature� which should be positive�Hence� at the dominant order in N �� and ignoring problems of exchange

of limits

N��U� V � � eNkB

sm�u�v�

N��U �DE� V �

N��U� V �� e

NkB

� sm�u�De�V �� sm�u�v�� e��N

� ��

and � � � as a consequence of the strict monotonicity of sm in u� so thatthe two limits in the last of � �� coincide and sm � sm� This showsalso the equivalence of the various versions of the microcanonical ensembledetermined by various choices of DE � NDe with De � ��Coming back to the equivalence between microcanonical and canonical en�

sembles we �x the constant kB in � ��#�� to be the same quantity andwe see that the problem can be formulated as follows if we establish a corre�spondence between the canonical state with parameters � � ��kBTc� v � vcand the microcanonical state with parameters u � um� v � vm such thatTc � �kB�� Tm and vc � vm then all the other quantities with thesame �name� �i�e� di�ering only by the label m or c� must coincide� Inthis way� because of orthodicity� all other thermodynamic quantities mustcoincide� Hence� if this coincidence really takes place� the two models ofthermodynamics de�ned by the two ensembles will coincide�The reason why the coincidence takes place is quite simple� if one neglects

matters of mathematical rigor and proceeds heuristically� For large N one�nds� by � �� and the �rst of � ��

Z�� vm� � �

Z �

Umin

e��EN��E� V � dE �

� N�

Z �

Umin

e��NueNsm�u�vm��kB du �

� constN�� exp

�N max

u��u �

�

kBsm�u� vm��

��

so that if the maximum is attained at a unique point u�� it must be that u�is such that � � �

kB�sm�u �u�� vm�� because the derivative with respect to u


must vanish in the maximum point u�� Furthermore

uc � U��N �

R�Umin

e��E�E�N�N��E� V � dER�Umin

e��EN��E� V � dE�

�

R�Umin

u e��usm�u�vm��kB�N duR�Umin

e��usm�u�vm��kB�N du� u�

� ��

because only the values of u � u� will give a leading contribution to theintegrals as N �� Equation � �� also con�rms the physical meaningof u� it is the average energy per particle� i�e� the internal energy perparticle�Recalling the relation remarked after � �� between u� and � and the

fact that uc � u�� we have

� ��

kB

smu

�uc� vm� ��

kBTm�uc� vm��

and choosing vc � vm and uc so that Tc � Tm�um� vm� it follows thatuc � um � u�� from the third of � ��#�� It remains to check that fm�um� vm� � fc�� vc�� this follows from � ��

which tells us that� for N ��

fc�� vm� � �� maxu

��u � sm�u� vm��kB� �

� ��uc � sm�uc� vm��kB� �

� �uc � Tcsm�uc� vm��

� �um � Tmsm�um� vm�� fm�um� vm�

because Tc � Tm� uc � um�The identity between the free energy� internal energy and absolute tem�

peratures implies �since the ensembles are orthodic� and therefore the usualthermodynamic relations hold� that of the entropies� so that the two en�sembles describe the same thermodynamics�

x�� Non Equivalence of the Canonical and Microcanonical En�

sembles� Phase Transitions� Boltzmann�s Constant

The derivation in x �� is classical but nonrigorous it can be made rigorousvia a more detailed analysis of the qualitative properties of the functionssm�u� v� and fc�� v� the central point of a rigorous proof of equivalence isin showing that sm�u� v� is �well approximated� �for N large� by S�U� V ��Nand� furthermore� it is a concave function of u and a convex function ofv� while fc�� v� is concave in both variables �� v� This implies that themaximum in � �� is actually reached at a point u� or� possibly� in aninterval �u�� u� where the function �u� sm�u� v� is constant in u�


A more detailed analysis of the question is postponed to Chap�IV it ishowever useful to mention that such an analysis requires making use ofthe stability and temperedness properties of the inter�particles interactionpotential �

As one can predict from the discussion that we have led� the proof �rigorousor not� of equivalence between the canonical and microcanonical ensemblesno longer works� in general� if the maximum in �� is reached on aninterval �u�� u�� u� � u rather than at a single point�

By the general properties of concave functions� one can see that this pos�sibility can be realized only for exceptional values of � �and precisely for aset of values forming �at most� a denumerable set�� This means that� forexceptional values of �� i�e� of the temperature� corresponding elements ofthe canonical and microcanonical ensembles may be not equivalent�

Such values of � are exceptional� if they exist at all� therefore it musthappen that as close as we wish to one of them� call it �� there exist values�� and �� which are not exceptional �� &� � ��For � � �� or � � �� there is equivalence of the corresponding elements

of the canonical and microcanonical ensembles� and in one case the internalenergy will be u� � u� and in the other it will be u�� u� having denotedby �u�� u� the interval on which the function �g�u� v� � ��u� sm�u� v��takes its maximum in u for � � �� as illustrated in Fig� ��

u u

�&�u � sm ��u � sm

uu� u�

Fig� �� Graph of the ��u� s�u� v� for di erent values of �

Hence we see that if for � � � the canonical and microcanonical statesare not� or may not be� equivalent then it must be that the internal energyuc�� v� shows a discontinuity jumping from u� to u when � is varied across�� Consequently also the speci�c entropy sc�� v� must show a discontinuitybecause fc�� v� � uc � Tcsc is necessarily continuous being convex� asmentioned above�

� Here the graphs will have a continuous �rst uderivative if the inverse temperature��sm�u

�v� T��m is continuous at constant v� this property is usually true but it is

nontrivial to prove it� We do not discuss this matter here� but in x�� we shall discussthe similar question of the continuity of the pressure as a function of the density atconstant temperature� In the Fig� �� we imagine that Tm is continuous �i�e� theplateau and the curved parts merge smoothly� �inside the black disks� to �rst order��


What has just been said� rather than being an obstacle to the microscopicformulation of thermodynamics� shows the possibility that statistical me�chanics can be the natural frame in which to study the phase transitionphenomenon� In fact we see that some of the thermodynamic quantitiescan have discontinuities in terms of others� exactly of the type empiricallyobserved in phase transition phenomena� where entropy and energy of twocoexisting phases are di�erent while the free energy is the same�

Hence cases in which there is no equivalence between corresponding ele�ments of the two ensembles� or more generally when there are correspondingbut nonequivalent elements in two orthodic ensembles� can be taken as sig�naling a phase transition this is in fact the de�nition of phase transitionthat is commonly accepted today�

From the point of view of Physics what happens in a case of nonequivalencebetween two elements of two orthodic ensembles with corresponding ther�modynamic parameters can be clari�ed by the following considerations� Ingeneral the states of an ensemble describe thermodynamic equilibria but mayfail to describe all of them� i�e� all the equilibrium phases �corresponding forinstance to a given free energy and temperature� or to given temperatureand pressure��

In other words� a given ensemble may be not rich enough to contain amongits elements � the statistical distributions that characterize all the purephases or their mixtures usually given a statistical ensemble E one will �ndamong the � � E a distribution describing a particular mixture of coexistingphases �if there are more phases possible with the same free energy andtemperature� but it may not contain the distributions describing the otherpossible phases or mixtrures�

This is precisely what can be seen to happen in the cases of the canonicaland microcanonical ensembles� at least in the few systems in which thetheory can be developed until such details are thoroughly brought to light�See Chap�V�

We can therefore conclude� in the case just examined of the canonical andmicrocanonical ensembles� that they provide equivalent descriptions of thesystem thermodynamics in the correspondence of the parameter values towhich no phase transition is associated� In the other cases the possiblenonequivalence cannot be considered a defect of the theory� but it can beascribed to the fact that� when equivalence fails� the elements of the twostatistical ensembles that should be equivalent are not because they describetwo di�erent phases that may coexist �or di�erent mixtures of coexistingphases��

One of the most interesting problems of statistical mechanics emerges inthis way it is the problem of �nding and studying cases of nonequiva�lence between corresponding elements of the canonical and microcanonicalensembles �or more generally of two orthodic ensembles��

We conclude this section by coming back to the question of the systemindependence of the Boltzmann constant kB � The above discussion onlyshows that the constant kB appearing in the theory of the canonical ensem�


ble must be the same as that appearing in the theory of the microcanonicalensemble� if one wants the two ensembles to describe the same thermody�namics �apart from the possible existence of phase transitions��It is� however� easy to give a general argument showing that kB must besystem independent and� hence� it has the value given by � �� computedfor the special case of a free gas� The idea is simply that we want our modelsof thermodynamics to also describe the same thermodynamics for a systemthat is part of a larger system�In fact putting into weak contact� mechanical and thermal� two systems that

are in thermal equilibrium �i�e� that have the same temperature� one buildsa composite system which� in the canonical ensemble� will be described bya distribution � with parameters �� v for the �rst set and by the parameters�� v�� for the second�We suppose for simplicity that each of the two systems contains only one

species of particle� The composite system will then be described by theproduct distribution � � �� because the two systems are independent andtheir mechanical interaction is supposed negligible �this is the meaning ofthe phrase �weak mechanical contact��On the other hand the distribution �� must be equivalent to a suitable

distribution � for the composite system� a distribution of equilibrium andcanonical� In fact we accept that the thermodynamic states of a systemcan be represented by the elements of an orthodic ensemble�� Hence if� and �� are two cells representing microscopic states of the two systems�� is proportional to exp��E�� E�� because the energy ofthe microscopic state �� is E�� E�� by the weak mechanicalinteraction hypothesis� Hence

exp��E�� E�� exp��E�� E��

for every pair of cells � and �� hence � � �� But � � ��k T � �� k� T � � � �� k T where T is the value� common

by the assumption of thermal equilibrium� of the temperature in the threesystems and k� k�� k are the three respective values of the constant kB �Hence k � k� � k i�e� k is a universal constant whose actual value kB can

be deduced� as was done above in � �� from the theory of a single specialsystem� namely that of the free gas which is the easiest to understand�

x�� The Grand Canonical Ensemble and Other Orthodic Ensem�

bles

It is easy to see that there exist a large number of orthodic ensembles�

�For instance the following generalization of the microcanonical ensemble�with DE � U � Umin� i�e� equal rather than small compared to U � Umin

� Unless� perhaps� there are phase transitions� an exceptional case that here we shallsuppose not to happen as we may imagine changing by a very small amount the thermodynamic parameters of the systems� still keeping thermal equilibrium�

�# II� Statistical Ensembles

�as assumed in x � �

�� N��U� V � ifE�� U

�� otherwise� ��

already considered in x �� after � �� is an orthodic ensemble� for thereasons discussed in x ��This ensemble is also called �microcanonical� �although perhaps improp�

erly because this name was introduced for the case DE � NDe� De � ��De � U�N�� But this is a somewhat trivial example of a new orthodicensemble�

�A di�erent and wide class of orthodic ensembles can be built by imaginingto �x other particles in positions q

�� q

�� and modify ��q�� see � ��

into ��q�

��q� � ��q� �

NXi�

Xj

�q�� q

j� � ��

where the sum over qj

runs over points qjexternal to the volume V inside

which the system particles are free to roam� The energy � has the meaningof potential energy of the system in the presence of particles �xed at pointslocated outside the container�As the shape or size of the container changes� when we vary V � we imagine

to remove the �xed particles whose positions fall into V �

Starting with the potential energy � �� we form the statistical micro�canonical or canonical ensembles with energy function E�p� q� � T �p� ��q��If the �xed external particles are distributed reasonably� e�g� so that each

unit cube only contains a bounded number of �xed particles� or a numberslowly increasing with the distance of the cube from the center of V �i�e� ifthe �xed particles are roughly distributed with uniform density� then itcan be shown �see Chap�IV� that the ensembles so obtained are orthodic�at least in the thermodynamic limit �V � �� V�N � v� U�N � u �xedor V � �� V�N � v� � �xed �respectively�� provided the interactionpotential satis�es the stability and temperedness of x � � If we do notwish to neglect the cells size then we should apply to such ensembles thecomments at the end of x � on the notion of orthodicity�The above new ensembles are called microcanonical or respectively canon�ical ensemble �with �xed particle boundary conditions�� It can be shownthat they are equivalent� in the absence of phase transitions� to the usualcanonical ensemble� in a sense analogous to that discussed in the previ�ous sections when comparing the canonical and microcanonical ensembles�i�e� they generate the same thermodynamics� in the thermodynamic limit��This can be done exactly along the same lines of argument that led to theequivalence between canonical and microcanonical ensembles in x ��


Other orthodic ensembles can be obtained by letting N or V vary� i�e� byconsidering simultaneously microscopic states describing systems with dif�ferent particles numbers N or occupying di�erent volumes V �

�An example which is very important in many applications is the grandcanonical ensemble its elements depend on two parameters � � � and ��They are probability distributions� on the cells � representing the statesof an N particle system in a given volume V and with N � �� ifE�� E�p� q� � K�p� � ��q� and if N�� number of particles in themicroscopic state � then

�� e��N��E��

0��

where the denominator is called the grand canonical partition function

0�� X�

e��N��E��

and the thermodynamic limit consists simply in letting V �� keeping �� xed�

�More generally one can replace ��q� with the potential energy ��q� de�scribed in � �� in this last case we talk about a grand canonical ensemble�with �xed particle boundary conditions��

�A further class of orthodic ensembles is provided by the pressure ensembleit also admits variants with �xed particles boundary conditions� In thisensemble one �xes N but the container V is thought of as variable andsusceptible of taking various volume values V� � V� V� � V� V� � �V� ��etc� the shape always remaining cubic�If � is a cell describing a microscopic state with N particles enclosed in a

container V �� and having energy E�� one de�nes for each value of thetwo parameters p � ��

�� e��pV ��E��

J�� p��

where the denominator is called the partition function of the pressure en�semble� and

JN �� p� �

�Xj�

X��V ��Vj

e��pV ��E��

The thermodynamic limit simply consists in letting N tend to in�nity�

Remark One can also imagine taking the containers to be susceptible ofassuming a continuum of values� e�g� any volume V keeping it� however�homothetic to a reference shape V �� e�g� to a unit cube�� in this case the

#� II� Statistical Ensembles

sum over the volumes should be replaced by an integral �V��R��

dV � Asimple application of the pressure ensemble will be found in x��#�

The theory of the grand canonical and pressure ensembles� as well as thetheory of the various ensembles with �xed particle boundary conditions�can be developed by showing their equivalence with the canonical� or mi�crocanonical ensemble� and this can be done via the method of the �maxi�mum value� that we have described in x �� it will work if the interparticlepotential satis�es the stability and temperedness conditions� see � � ��and � � ��#�� Fi��Ru��As a further example of the maximum value method� of very common use in

statistical mechanics� we deduce some of the properties of the grand canon�ical ensemble from the corresponding properties of the canonical ensembleand show their equivalence� Again we proceed heuristically� by ignoringproblems of mathematical rigor�If � is a generic element of the grand canonical ensemble corresponding to

the parameters �� one has� see � �� and the �rst of � ��#�

0��

�XN�

e��NX

��N��N

e��E��

��XN�

e��NZN �� V � ��XN�

e��Ne��Nfc��V�N� � � ��

��XN�

expV ��v�� v��fc�� v��

where in the last sum v � V�N and ZN �� V � is the canonical partitionfunction for N particles in the volume V and with temperature T � ��kB��Hence for V � �� and if v� is the value where the function ��v�� v��fc�� v�� of the variable v� attains its maximum we �nd

limV��

��V � log 0�� v�� v�� fc�� v�� #��

assuming that the maximum point v� is unique� Here v� satis�es �if onerecalls that by � ��#� pc � ��fc

�v �� v��

v��v�� v��fc�� v�

��vv�

� � � � � fc�� v�� v�pc�� v��

� �� On the other hand v� has the interpretation of grand canonical speci�c

volume vg because

v��g � N��V �

P�N�

NV e

��NZN�� V �P�N� e

��NZN �� V ��

�

P�N�NV ��e��v

��v��fc��v��VP�N� e

��v��v��fc��v��V ��V��

��V�� v��

II� Statistical Ensembles #�

due to the maximum of ��v�� v��fc�� v� being isolated and at thepoint v��Therefore from � �� and classical thermodynamics� one �nds the phys�

ical meaning of �

��N � F � pV � U � TS � pV � N�fc�� vg� � vgpc�� vg�

��

i�e� ��N is the Gibbs potential corresponding to the parameters �� vg��Furthermore from� ��#� one �nds that

limV��

�

Vlog 0�� pc�� vg� � ��

i�e� the grand canonical partition function is directly related to the canonicalpressure associated with the parameters �� vg��This suggests that the grand canonical and the canonical ensembles are

equivalent if the elements with parameters �� and �� vg� with vg � v��see � �� are put in correspondence �i�e� are thought to describe the samemacroscopic state�� This can be checked by setting� see � ��

ug � limV��

X�

��U��N��

Tg � limV��

� ��kB�X�

��K��N��

vg � limV��

X�

��V�N��

pg � limV��

X�

��p��

sg � �ug � lim��V � log 0�� Tg

and by showing the identity between the above quantities computed in thegrand canonical ensemble with parameters �� and the quantities with thesame name computed in the canonical ensemble with parameters �� vg��Using the fact that Tc�� v� � �

kB�and

Tg � limV��

P�N� e

��NZN �� V �Tc�� V

N

�P�N� e

��NZN �� V ��

kB��

we see that� for the same reason used in deriving vg � v� in � ��

ug � lim

PN e��NZN �� V �uc

�� V

N

�PN e��NZN �� V �

� uc�� v��

pg � lim

PN e��NZN �� V � pc

�� V

N

�PN e��NZN �� V �

� pc�� vg� �

� ��

# II� Statistical Ensembles

� �� clearly show that all grand canonical thermody�namic quantities coincide with the corresponding canonical quantities�As in the previous case of canonical versus microcanonical ensembles the

above analysis is not rigorous because it involves various interchanges oflimits and� furthermore� it presupposes that ��v�� v��f�� v�� has aunique isolated maximum �as a function of v�� but under the assumptionsof stability and temperedness of x � � � � �� #�� the problems ofmathematical rigor can again be solved� see Chap�IV�From the theory of the canonical ensemble� which we shall elaborate on

in more detail in Chap�IV� it follows also that the function �v��f�� v� isconvex in � and in v�� so that� with �few� exceptional values �i�e� at mosta denumerable family of values of �� the function ��v�� v��fc�� v��has a unique maximum point as a function of v�� For � outside thisexceptional set there is complete equivalence between thermodynamics ofthe equilibrium states in terms of the elements of the canonical and grandcanonical ensembles�For the other values of � �if any� the function ��v�� v��f�� v�� takes

the maximum value in an interval �v�� v�� as implied by the general prop�erties of concave functions� see the Fig� �� in such cases the descriptionsof states in terms of canonical or grand canonical distributions may benonequivalent� But the interpretation of the nonequivalence is again thatof occurrence of a phase transition non equivalence has to be interpretedby attributing it to the fact that the distributions in question describe twodi�erent equilibria that can coexist in thermodynamic equilibrium �i�e� theyboth have the same temperature and pressure� but di�erent speci�c volume�entropy� etc� in the same sense as discussed in x ��One of the main results of statistical mechanics which we wish to quote

with more detail has been that of showing that� at least in many interestingcases� there is complete equivalence between the ensembles which will becalled here enlarged ensembles such ensembles are obtained from a givenensemble of stationary distributions �like the canonical� microcanonical orgrand canonical� by adding to it all the distributions with boundary condi�tions of �arbitrarily� �xed external particles�In such larger ensembles it may still happen that two given states� corre�

sponding to the same values of temperature and pressure� may have di�erent

� Of course a denumerable set of values has zero length but it might be quite large inother senses� for instance it could be dense� Therefore this easy way of saying thatphase transitions are �rare is very unsatisfactory� But a more detailed analysis is verydi�cult and perhaps impossible at the level of generality in which we are discussingthe matter� More detailed statements� e�g� that the discontinuities in take place at�nitely many values of � and in the plane � � they occur on smooth lines� can bederived only when considering very special cases� see Chap�VI� One can ask whether the

pressure ��fm�v

�� p is a continuous function of v� if so v��v��fm�� v� will have

continuous vderivative and we could draw a �gure similar to Fig� �� in x�� Also inthis case� see the corresponding comment about the temperature in the microcanonicalensemble in footnote � in x�� it is nontrivial to show that p is a continuous function ofthe speci�c volume at constant temperature� in x�� we shall discuss this in more detail�in the case of hard core systems�


averages for other thermodynamic quantities �energy� entropy� speci�c vol�ume� etc� but it will happen that for every element of one ensemble thereis another in the other ensemble describing exactly the same macroscopicstate and thermodynamics� i�e� that associates the same values to all ther�modynamic quantities and even the same relative probability distributionto the most probable microscopic states�

In other words we can say that even the phase transition phenomenoncan be studied in an enlarged ensemble without worrying that in this wayone may �miss� some phases� because the enlarged statistical ensembles areoften rich enough to contain all possible phases and their mixtures�

This is the way to understand the nonequivalence between ensembles whichnormally describe correctly the thermodynamics of the system �i�e� that areorthodic�� One can� in fact� think that� given a state of a system� one canlook at an ideal macroscopic region which however occupies a volume lessthan the total� then the system in this volume can be regarded as a system inequilibrium with its surroundings� Such a system will have a �xed volumebut a variable particle number� One would thus describe it naturally inthe grand canonical ensemble� however the particles in the system haveinteractions with the identical particles that are outside the ideal volumeselected�

If one imagines taking a picture of the con�guration one will see a sample ofthe con�guration in the inner volume and one in the outer volume� Takingthe highly imaginative step of collecting only the pictures in which theexternal con�guration is the same we should still see statistically the samestate inside the ideal box this means that the state we see in the idealbox is determined by the state of the particles outside it provided they arechosen in a con�guration �typical� for the state that is being considered�If this is so we can expect that the grand canonical ensemble with �xedparticle boundary conditions can describe all possible states�

When there is more than one equilibrium state we can describe them byselecting at random a con�guration of the system and forming the grandcanonical distribution in a large volume with boundary conditions given bythe selected con�guration of particles�

Note that from this viewpoint the phenomenon of phase transitions ap�pears as an instability of the thermodynamic properties of a system withrespect to variations of boundary conditions for instance keeping the sametemperature and pressure but changing boundary conditions one can ob�tain di�erent values for intensive thermodynamic quantities like the speci�cenergy� the speci�c entropy� the speci�c volume� etc� i�e� by changing theforces that act near the boundary of our system we can change the macro�scopic state even if the system is very large �hence the boundary is far andrelatively small compared to the volume��

In a sense this is a further manifestation of the richness of statistical me�chanics such a complex phenomenon as a phase transition seems to �ndits natural theoretical setting� and the bases for its analysis� in the theory of�orthodic� ensembles�


In a given statistical ensemble macroscopic thermodynamic quantities ap�pear either as parameters of the ensemble� e�g� u� v in the microcanonicalcase� �� v in the canonical and �� in the grand canonical� or as quantitiesdirectly related to the ensemble partition function such as the entropy� thefree energy or the pressure in the above three cases� or they are related toderivatives of the partition function like the temperature� the pressure andthe energy in the three cases� respectively�It can be shown� see Chap�IV� that the �rst two types of quantities do

not depend on the boundary conditions �imagining the latter to be takenas �xed particle boundary conditions�� Hence a way of searching for phasetransitions in particular models �i�e� in systems obtained by assuming spe�ci�c choices for the interaction potentials� is to look for parameter values ofthe chosen statistical ensemble �e�g� u� v in the microcanonical case� �� v inthe canonical and �� in the grand canonical� �� p in the pressure ensemblecase� in correspondence of which the thermodynamic function associatedwith the partition function is not di�erentiable� �Ru��This is a method that has become classical it has� however� the defect of

not directly providing a microscopic description of the equilibrium statesdescribing the di�erent possible phases� It determines the location of thephase transition� in the thermodynamic parameter space of the ensembleadopted for the analysis but it does not analyze the characteristic physi�cal peculiarities of the possible microscopic distributions that describe thevarious phases�On the other hand� the study of the boundary condition dependence of the

equilibrium states of an �enlarged ensemble� is potentially richer in infor�mation and it can lead to a microscopic description of the phase transitionand phase coexistence phenomena� because each state of thermodynamicequilibrium is described in detail by a probability distribution of its micro�scopic con�gurations� The best understanding is obtained by examining indetail some simple case �there are not� however� many cases in which theabove statements can be followed and checked in detail� this will be thetheme of Chap�VI� where the Ising model for ferromagnetism will be dis�cussed in connection with the spontaneous magnetization phase transition�and Chap�VII where other simple models are discussed�Thus we have met two possible de�nitions of phase transitions� A systemshows a phase transition if a derivative of the thermodynamic function asso�ciated with the partition function of an orthodic ensemble has a discontinu�ity as a function of the parameters describing the elements of the ensemble�Alternatively a phase transition occurs if by changing the boundary condi�tions in the elements of an enlarged orthodic ensemble one can change� incorrespondence of suitable values of the parameters describing the elementsof the ensemble� bulk properties of the system�We conclude by mentioning that if one develops the thermodynamics model

associated with the pressure ensemble along the above lines one easily checks


the equivalence between pressure ensemble and canonical ensemble and one�nds that the Gibbs potential � � pv � u � Ts is related to the pressureensemble partition function by

�� limN��

�

Nlog JN �� p� � � ��

In fact one deduces� from the de�nition �recalling that V � is the volume ofthe reference container box� see the remark following � ��

J�� p� �

Zd V

V �e��pV ZV ��N� � ��

that � �N log JN �� p��

N�� minv��p v � �fc�� v� so that the mini�

mum is at vp such that �p � ��c�v �� vp�� Hence we see that p is identi�ed

with the pressure and� by � �� with the Gibbs potential�

x�� Some Technical Aspects

Some details concerning the derivation of the mathematical identities usedin x ��x � and some related matters will be provided in this sectionnamely we shall derive equations � � �� The reader wish�ing to delve into the subject can begin by consulting �Fi�� Mi�#�� Ru��La� �� LL� ��

�� It is certainly worth commenting on the step from the last of � � �� to� � ��In the last of � � �� one can make use of the independence of the integrals

performed with respect to the variables p from those performed with respectto the variables q� and the symmetry of the p�components of the integrand�

In this way one can replace mNv� by mNv� and eliminate the conditionv � �� then mNv� can be replaced by Np�

��m� thus taking advantage of

the symmetry of the p�

dependence in the three components of p��

Hence one can replace the integral on p�

that in � � �� is

Zv�

e��p�

��m mv� dp

�with

Ze��p

�

��m p�

�

�mdp

��

and a simple calculation shows that

Ze��p

�

��m p�

�

�mdp

��

�

�

Ze��p

�

��m dp

��

so that

p �XQ

N

�

s

S

�

Z�� V �

Zq��q

NV N

e��E�p�q�dq

�� dq

Ndp

�� dp

N

h�NN �

� ��


and the point q�

is� in each addend of � �� localized in Q �which issupposed so small that it has no importance where q

�exactly is inside

Q��We now imagine varying V from V to V � dV � increasing the volume by

displacing� along the outer normal� by � every area element of the surface�We see that logZ�� V � varies� since dV � S�� by

d logZ�� V � � � ��

�XQ

Ns�

Z�� V �

Zq��q

NV


�� dq

Ndp

�� dp

N

h�NN �

� dVXQ

N

Z�� V �

s

S

Zq��q

NV


�� dq

Ndp

�� dp

N

h�NN �

which� comparing with � �� proves � � ��

� � Another deduction calling for further details is the step from � � �� to� � �� By proceeding as in the derivation of � �� one �nds p dV starting from

the expression for p as the average P �� with respect to the microcanonicaldistribution with parameters �U� V �� Denoting by

R the integral over �p� q�

extended to the domain of the �p� q� such that E �DE � E�p� q� � E and�at the same time� q

�� dV � �QQ

p dV �N

N �U� V �

Z

�

p��

m

dp dq

h�NN ��

�N �U� V �

Z K�p�

dp dq

h�NN ��

having again used in the last step the symmetry of K�p� in p�� p

N�to

eliminate the factor N� and having writtenP

Q s�� Rq�dV � dq

�to obtain

a more elegant form �formally eliminating the summations over Q naturallyappearing� in conformity to its de�nition� in the expression of the pressure��To connect � �� with the derivatives of N we have to make more explicit

the dependence of N on U � by evaluating exactly the integral � � �� on thep variables in polar coordinates �which is an elementary integral��If /��N� is the surface of the unit sphere in �N dimensions and if we set

w�U� q� �q

m�U ��q�� we deduce

N �U� V � �

Zdq

h�NN �

�w�U� q��N � w�U �DE� q��N

�/��N�

�N� ��

hence

NU

�

ZV N

dq

h�NN �

�N

m�w�U� q��N�� w�U �DE� q��N��

�/��N�

�N� ��


and coming back to the original coordinates

�

NNU

��N

�

NZ

dp dq

h�NN �

�N �

�N

�

K�p��

�N

��

�N

�hK�p��i � ��#��

where hK�p��i is de�ned by � � �� and the integral � ��#� is extended tothe domain in which �U �DE � E�p� q� � U��If� instead� one had proceeded as in the derivation of � � �� in the canonical

ensemble case �see � �� above� one would have found

�

NNV

dV �N

NZ dp dq

h�NN ��

� N

�

�R K�p�

dp dqh�NN ��N

��

��

R K�p�

dp dqh�NN �

�R dp dqh�NN �

� � ��

where in the last step we multiply and divide by the same quantity and weuse the notation

R in � �� Then � �� and � �� imply

�

NNV

dV � Np dV

�� hK�p�i � ��

where hK�p��i is de�ned in � � ��Relation � � �� now follows from � � �� #� and � ��

�� We �nally deduce � � � � in the simple case �considered in x � � of aperfect gas� � � ��If we imagine dividing the six�dimensional phase space describing the in�

dividual particles of the system� into cells C having the form

C � set of the �p� q� in R� such that�k��q � �q� � q� � k��q � �q� k��p� �p� � p� � k��p � �p�

� � ��

and k� k� are two integer components vectors� it follows that the energy ofa single particle located in C is ��C�

��C� �

�X��

�

mk��p

� � � ��

Furthermore a microscopic state � of the system can be assigned by givingthe occupation numbers nC for each cell they tell us how many particlesoccupy a given cell� Then� without combinatorial or analytical errors �seex � �

Z�� V � �XnC��PCnCN

e��P

CnC��C� � � ��

## II� Statistical Ensembles

Taking L � V �� to be the side of the container and calculating� instead�the expression a1icted by obvious combinatorial errors

Z�� V � �XnC��PCnCN

�

�Q

C nC ��e��

PCnC��C� �

��

N �

�XC

e��C�

�N

��

N �

�L

�q

�Xk��

e��k��p��m

��N

�

��

N �

�L�p

h

�X��

e��k��p��m

��N

� � ��

� �N �

V Np

�m��N

h�N

��p

�m��pP�

k�� e��k��p��m

��N

which then leads to � � �� i�e� to the second of � �� since � �� ifh � � and if one approximates the sum in the last member of � �� with

the corresponding integralq

��m

Re��p

��m dp �committing in this way also

the above described analytic error��To compare � �� with � �� or � � �� it is necessary to decide

whether the values of nC that give the main contribution to � �� arethose for which nC � �� and in this case � �� are good ap�proximations of each other as well as of � � �� because the factor nC � hasvalue � in the majority of cases��We must therefore compute the average value nC of the quantity nC with

respect to the canonical distribution and the consistency condition� i�e� thecondition of negligibility of the combinatorial error will be nC � ��In the canonical ensemble� by de�nition � �� the probability of �nding

a particle� with known position� with momentum in dp is the Maxwell�Boltzmann law

e��p��m dp

�p

�m��

hence if � � N�V is the system density we shall �nd

nC � ��q��e��p��m ��p��p

�m��

�h�p �m��

� � ��

so that nC � �� for all cells C� if T � Tq with Tq given by � � � ��It is clear that the error that we call �analytic error� will be negligible ifp�� m�p � �� In the present context we did not �x separately �p and

�q nevertheless �q should be certainly chosen so that �q � �� averageinter�particles distance� otherwise it would not make sense to think of thesystem as built with particles as separate entities de�ned in the system�With this choice of �q� from �p �q � h� it follows that �p � h�� andone sees that the condition

p�� m�p� � is the same as � � � ��

#�

Chapter III�

Equipartition and Critique

��

�

III� Equipartition and Critique ��

x�� Equipartition and Other Paradoxes and Applications of Sta�

tistical Mechanics

One of the most well�known consequences of classical statistical mechanicsis the principle of equipartition of energy somewhat less well known is thatthis principle� after some shining initial successes� reveals itself as the sign ofstatistical mechanics inadequacy to solve important problems that fall intoits domain� Likewise other well�known important consequences of statisticalmechanics are a�ected by serious paradoxes and theoretical problems�Here we shall illustrate some signi�cant examples�

�I� The Free Gases Speci�c Heats

By using the canonical ensemble and assuming that the cells size h is verysmall �see x�� one easily computes the internal energy for a general modelin which each particle has � degrees of freedom and does not interact withthe others�The � degrees of freedom describe the three baricentric degrees of freedom

plus the �� internal degrees of freedom of the molecule internal motion�One supposes that energy is a quadratic form in the � conjugated momentap�� p�� p� and� possibly� in some of the internal position coordinates

E�p� q� �

�Xj�

p�j m

�

��Xj�

p�j Mj�)q�

�

�Xj��

�

�p�j � ��

j q�j � ��

where p�� p�� p�� q�� q�� q� are the momentum and position coordinates for theparticles baricenter� m is their mass while p�� q�� are the momentumand position coordinates describing the internal degrees of freedom and)q � �q�� q��Equation �� is the form that one expects for the energy of a molecule

which has a few internal degrees of freedom� precisely �� to which are as�sociated oscillatory motions around equilibrium positions �corresponding tothe values j � �� with respective proper frequencies ��j� theycan therefore be called oscillatory degrees of freedom� The �rst �� degreesof freedom describe what we shall call translational degrees of freedom� theposition variable corresponding to a translational degree of freedom is eithera position coordinate for the center of mass� varying in V � for j � �� oran internal angular coordinate� while as a rule the variables qj conjugatedto a momentum of an oscillatory degree of freedom will always be a variabledescribing an internal degree of freedom and it is best thought of as varyingin ��For instance if the gas consists of point atoms with mass m then � � �

and E�p� q� � K�p� � �p�� p�� p�� m� If the gas consists of diatomic

� III� Equipartition and Critique

molecules built with two atoms at a �xed distance �� then the kinetic energyis

K�p� q� ��

m�p�� p�� p��

�

�

�p��

�p��

�� sin� �

��

where m is the total mass and � the reduced mass� m � m� � m�� m�m��m� and p�� p� are the momenta conjugated to the variables and �respectively the latitude and the azimuth of the two linked atoms� In thiscase the variables q conjugated to the �rst three momenta are real variablesvarying in V while the other two variables are angular variables� There are�ve translational degrees of freedom and no oscillatory degree of freedom�For the perfect gases for which the total energy is a sum of the kinetic en�

ergies of the individual particles �i�e� the �srt term in �� the partitionfunction is

Z�� V � �

Zdp dq

h�NN �e��

PN

i�K�p

i�qi� ��

so that the average energy is computed by using the factorization of theintegrals and by calculating explicitly �rst the �Gaussian� integrals over thep s and then those over the q of the oscillatory degrees of freedom and �nallythose over the remaining q coordinates �that are trivial if performed afterthose over the p s��More generally if �� is the kinetic energy of a single molecule and if

)q are the translational position coordinates and "q are the oscillatory onesthen

U �

R �PNi�K�p

i� qi��e��

PN

i�K�p

i�qi�Q

i dpidq

iRe��

PK�p

i�qi�Q

i dpidq

i

�

�NR

dq�RK�p� q�e��K�p�q� dp d"q�R

d)q�Re��K�p�q� dpd"q�

� ��

�N��

Rd)q��

�Rd)q

�

� N��

��

This is an interesting relation because it is independent on the special formof �� i�e� independent of the coe%cients Mj�)q� �i�m�� it says that theinternal energy of a perfect gas is given by the number of degrees of freedomtimes �� kBT� �equipartition of the energy among the various degreesof freedom and between kinetic and potential energy� counting twice theoscillatory degrees of freedom because the latter contribute to the potentialenergy as well� One also says the �there is equipartition between kineticand internal elastic energy��The constant volume speci�c heat of a monoatomic gas and that of a gas

of rigid diatomic molecules are given� respectively� by

CV �U

T�

�

nR or

�

nR ��


where n � N�NA� with NA � Avogadro s number� n the number of molesof gas and R � kBNA is the gas constant�

Equation �� agrees well with the experimental results on rare�ed mono�atomic gases� the agreement is less good for the diatomic gases� even ifrare�ed�

In fact �� cannot be accepted on general grounds� not even formonoatomic gases �even if rare�ed�� because it is known that gases con�sist of atoms with very many degrees of freedom� mostly oscillatory� andtheir speci�c heat is� nevertheless �

�nR� For instance neon could be thoughtof as built with � protons and neutrons and �� electrons� i�e� it would have�� degrees of freedom� of which #� are oscillatory��

But even the simple case of a truly diatomic molecule in which all internaldegrees of freedom are neglected except the three describing the relativeposition of the two atoms is conceptually unclear if one assumed rigidity ofthe distance between the two atoms then the speci�c heat would be �

�nR�if� instead� we admitted that the distance between the two atoms oscillatesaround an equilibrium position �which is more �realistic�� then the speci�cheat would be �

�nR because the degrees of freedom would be �� one of whichoscillatory�

It appears� therefore� that �things go as if� some of the internal degreesof freedom were less important than others� they are �frozen� and do notcontribute to the energy equipartition� Which� therefore� would not be validin general� in spite of it being an extremely simple consequence of the theoryof the canonical ensemble�

�II� The Speci�c Heat of Solids�

Another success�failure of classical statistical mechanics is the theory of thespeci�c heat in crystalline solids� A crystalline solid can be modeled as asystem of particles oscillating elastically around ideal equilibrium positionsarranged on a regular lattice� e�g� a square lattice with mesh a �to �x ideas��

It is known from the elementary theory of oscillations that such a systemis described in suitable normal coordinates by the Hamiltonian

H �Xk

�

�p�k � ��k��q�k� �p

k� qk� � R� ��

where the sum runs over the triples k � �k�� k�� k�� of integers with ki ��

pN�� if N is the number of atoms of the crystal �which we assume

cubic and with side �pN � L�� and

��k�� c��Xi�

�� cos

akiL

�

��

with c being the sound propagation velocity in the crystal�

�� III� Equipartition and Critique

If we could compute the system properties by using the canonical ensemblethen the internal energy could be computed as

U �

��

R Pk �p�

k� ��k��q�

k� e��H�p�q� dpdqR

e��H�p�q� dp dq�

�N

��#�

��

because the sum over k concerns �N values and the calculation proceedsas in the case of the discussion of energy equipartition� with the di�erencethat now all the �N degrees of freedom are oscillatory�Therefore the speci�c heat of a crystal should be

C � �NkB � �nR ��

if n is the number of moles� and this is quite well satis�ed at high temper�atures �above the solidi�cation temperature� but below liquefaction� and isknown as the law of Dulong�Petit�If however one takes into account that a typical model of a conducting

solid consists of N ions on a lattice and N electrons forming a free electrongas� then one �nds that instead one should perhaps expect a speci�c heatof �nR � �nR

� �Experiments show that the speci�c heat of crystals at high temperature

indeed conforms to the Dulong�Petit law� At lower temperatures insteadthe speci�c heat approaches �� according to a general principle called thethird law of Thermodynamics�Hence classical mechanics produces erroneous predictions also for a crys�

talline solid it looks as if some degrees of freedom are frozen because theydo not contribute to the speci�c heat �in other words their contribution tothe internal energy is the same as that which they would give if their tem�perature could be considered zero and staying constantly so in all the systemtransformations which is not possible because then the system would notbe in thermal equilibrium�� Furthermore at lower temperatures the crystaloscillations seem to become less and less describable by classical statisticalmechanics because the speci�c heat deviates from the Dulong�Petit law� andtends to ��

�III� The Black Body�

A thermodynamic theory of radiation can also be developed on the basisof the theory of ensembles� and one reaches disturbing and upsetting con�tradictions with the experimental observations as a consequence of classicalstatistical mechanics developed in the previous chapters�In fact it was in the theory of the black body where� historically� the

contradictions were felt most and led to the origin of quantum mechanics�Consider a cubic region V �lled with electromagnetic radiation in ther�

mal equilibrium with the surrounding walls with which it is supposed toexchange heat� We describe the electromagnetic �eld by the vector poten�tial A and the relations �which are implied by Maxwell s equations in the


vacuum�

E � � �

c

A

t� H � rot A � div A � � ��

where c is the velocity of light� c � �� cmsec�� It is well�knownthat the motion of such a �eld is described by the Lagrangian

L ��

#�

ZV

�E� �H�� dx ��

regarded as a function of A� !A�If L is the side of the volume V occupied by the radiation� which for

simplicity it is convenient to consider with periodic boundary conditions�by identifying the opposite sides of V �� it will be possible to write A interms of its Fourier expansion

A�x� ��

�pL�

Xk

�X��

A��k� e��k� eikx ��

where k � �L � and � is an integer component vector� and e��k� are two

polarization vectors� with unit length� and orthogonal to k and to each other�One �nds

L ��

Xk

�X��

��

��c�!A��k��

��k�A��k��

��

Therefore the evolution in time of the �eld in the cavity can be describedby the Hamiltonian function

H ��

Xk

�X��

��c�p��k��

k�

��q��k��

��

��

Xk

�X��

� p��k�� k�c� q��k��

��

where the pairs �p��k�� q��k�� or � p��k�� q��k�� p

�c��p��k��

q��k��p

�c�� are canonically conjugated coordinates� equivalent becausethe transformation �p� q� � � p� q� is canonical�Hence an electromagnetic �eld in a cavity V can be regarded as a system

of in�nitely many independent harmonic oscillators�It is� therefore� very tempting to describe this system by statistical mechan�

ics and to say that at temperature T the microscopic states of the systemwill be distributed according to a canonical distribution and� hence� theprobability of �nding the oscillator with labels �� k�� i�e� with polarization� and wave vector k� in the cell C � dp��k�dq��k� is

e�� p

��k��c�k�q��k�� dp��k�dq��k�p��k��c��

� ��


It is clear that by assuming �� one assumes that the cell size is negligi�ble this usually introduces the two types of errors that have been discussedin x �� and x �� In the present case the combinatorial error is absent be�cause this time the oscillators are pairwise distinct� However if � is largethe error due to having neglected the cell size by considering p��k� andq��k� as continuous variables is still present and it might be substantiallya�ecting the results�If one accepts �� the average energy per oscillator will be kBT � by

the equipartition argument in �I� above� because each oscillator representsan oscillatory degree of freedom �see ��Therefore it follows that� if � � jkjc� � is the frequency of the wave with

wave number �k� the quantity of energy L�u�d� corresponding to the oscilla�tors with frequency between � and ��d� is related to the number of integervectors n such that � � jnjc�L � � � d� via

L�u�d� ��

� � �number of jnj such that jnjc�L � �� d��

�

�

�L�c

�� L

cd� � L� #�

c�� d�

��

where in the �rst step the factor after �� is there because� for each�k there are two oscillators with di�erent polarizations and equal averageenergy� kBT � Hence the Rayleigh�Jeans� formula emerges

u� �#��

c�kBT ��

which is manifestly in disagreement with experience� becauseR��

u� d� � �and a radiating cavity� in thermal equilibrium� would have in�nite energy�Experimentally the distribution �� is observed only if � is small� and

for large � the observations are in contrast with the energy equipartitiontheorem because one �nds that u� approaches � very quickly as � tends toin�nity�We see that classical statistical mechanics in the above three applications

leads to paradoxes and wrong predictions� In the next section we shall seethat the paradoxes disappear if the constant h measuring the cells size ish �� and it will be possible to say that all contradictions that appearin classical statistical mechanics arise when� to simplify the formulae byreplacing summations with integrals� errors of an analytic and combinatorialnature are introduced� see also x �� x � and x �� by taking h � ��

x�� Classical Statistical Mechanics when Cell Sizes Are Not

Negligible

In x�� and in the previous chapters we always neglected the size h ofthe phase space cells representing the microscopic states of the system� As


pointed out repeatedly� important errors are introduced in so doing �see x ��x �� which we shall see are ultimately intimately related to the paradoxesdiscussed in the previous sections�The main error� however� is due to the fact that if the cells can no longer

be thought of as points then one should simply not use classical statisticalmechanics� The previous section shows that the theory leads to resultsthat in turn permit us to test whether the cells in phase space can reallybe regarded as points the disagreement between theory and experimentalresults implies that the cell size is actually accessible to experiment andthis must necessarily lead us to reformulate the very principles of classicalmechanics and therefore of statistical mechanics�To realize how drastic could be the changes of the Thermodynamics of

a system in a �quantum regime� in which h cannot be neglected one canjust proceed by assuming as valid the description of the system in termsof cells in phase space and evaluate more accurately the partition sums ofthe various ensembles� avoiding committing the combinatorial and analyticerrors that we have described above and that are really negligible only inthe limit as h� ��Consider as a �rst example a free gas of identical particles with no internal

degrees of freedom� and let C be a generic cell of the six dimensional phasespace in which the states of the single particles can be described let thevolume of C be ��p �q�� h��Since the identical particles are indistinguishable then the microscopic con�

�gurations � are determined by the numbers nC of particles that� in thecon�guration �� occupy the cell C� Then

E�� P

C nCe�C� total energy

N�� P

C nC number of particles��

where e�C� is the energy of a particle in the cell C�Let us study the system in the grand canonical ensemble� where the calcu�

lations are somewhat simpler� The partition function is then

0�� XfnCg

e��P

CnCe��

PCnCe�C� ��

where� for each C� nC � �� see x �� We perform the summations explicitly� thus avoiding the combinatorial

and analytical errors whose e�ects we are investigating� We �nd

0�� YC

�

�� e��e�C�� e�

PClog��e��e�C��

and the probability that nC � n can be immediately computed� see � ��

p�n�C� �e��n��n e�C�

�� e�� e�C��

�# III� Equipartition and Critique

The equation of state is deduced by expressing � as a function of the density� and of � via

� � ��

V

XC

Xn

n p�n�C� ��

V

Xn�C

n e��n��ne�C�

�� e��e�C��

and then by replacing � with �� in the grand canonical expression ofthe pressure� By recalling that in the grand canonical ensemble the pressureis directly related to the partition function� see � �� we get

� p��

Vlog 0��

V

XC

log�� e��e�C��

and the total energy per unit volume u� is

u��

V

XC�n

n e�C�e��e�C��n

�� e��e�C��

��

V

XC

e�C�e��e�C��

�� e��e�C��

��

To appreciate the di�erence between �� and the classical perfectgas properties it is convenient to imagine that e�C� � p�� m if C is a cell

with center at the point �p� q� and� hence� to neglect the variability of p�� min C�The latter approximation implies

�p�� Z

d�p

h�log�� e��p

��m��

��

Zd�p

h�e��p

��m�

�� e��p��m�

��

v�� #��

u��

Zd�p

h�p�

m

e��p��m�

�� e��p��m�

Integrating the �rst of �� #� by parts one gets the relation

�p��

��u� ��

The neglect of the variability of p�� m in C introduces an error� however onecan check� without di%culty� that it does not alter the qualitative propertiesof �� which we shall discuss shortly �the approximation onlysimpli�es the analysis� to some extent��The most relevant phenomenon is the Bose condensation the �� show

that the parameter � must be such that �� minC e�C� � �� Hence� as


appears from �� #�� the maximum density �� of the system seems tocorrespond to � � �

��

Zd�p

h�e��p

��m

�� e��p��m�

��

which looks incorrect because the density can be prescribed a priori� byassigning the number of particles� hence it cannot be bounded above�But the density can be larger than �� because �� does not have

sense if � � �� e�C� � �� Interpreted literally �� for � � �� shows thatthe particle number in a cell C with e�C� � � isP�

n� ne��nP�

n� e��n � � d

d��log

�Xn�

e��n �

�d

d��log�� e��

�e��

�� e��

� �

��

The correct interpretation of �� and of the last remark is that thecells with e�C� � � can contribute the quantity �� to the density �� atmost however the remaining part of a larger density� � � �� is due� if� � �� to the particles that are in the cells C with e�C� � � � Note thatthere are many such cells because they must only have � momentum butthe spatial centers of the cells can be anywhere in the container V �This in fact means that the most appropriate way to describe the states of

this system should be the canonical ensemble� But from the above discussionwe can imagine describing a state with density � � �� in the grandcanonical ensemble by setting � � � and then by imagining that ��Vparticles are in the cells C with e�C� � ��It is important to remark that since �� for � �� the phenomenon

of Bose condensation is always important at low temperature if the totaldensity is kept �xed� And it is clear that the particles that are in the cells Cwith e�C� � � have zero momentum and therefore they do not contribute tothe internal energy nor to the pressure nor to the speci�c heat at constantvolume�In particular if we wish to examine the speci�c heat at constant volume

when T � � we can note that� as soon as T is so small that �� theinternal energy becomes

U � u�V � V

Zd�p

h�p�

m

e��p��m

�� e��p��m

� V �T ��

�only cells with e�C� �� contribute� with

� �

Zd�x

h�x�

e�x�

�� e�x�� m��k

��B � ��


Hence

Cv �u�T

� constant T �� if � � ��

which shows how in the perfect gas that we are studying the equipartitionresult Cv � �nR� is no longer true instead one �nds Cv � � for T � ��At low temperatures equipartition fails if one takes h �� seriously�Another example in which h cannot be neglected is the case in which the gas

particles are imagined to interact in a very simple way� conceivable althoughnot usual in classical mechanics suppose that the particles �repel� eachother in the sense that they cannot occupy the same cell in phase spaceso that one cannot �nd two or more particles in a given cell� The unusualnature of this force is expressed by its dependence on velocity �because itgenerates a �hard core� in phase space��In the latter case the partition function is �� with the condition thatnC � �� Hence

0�� YC

�� e��e�C��

and the probability that nC � n is� instead of ��

p�n�C� �e��e�C��n

� � e��e�C��n � ��

and �� #� change accordingly�This gas does not resemble at all the classical perfect gas and at low tem�

perature it exhibits the phenomenon of Fermi condensation� one sees in factthat

p�n�C��

�� if e�C� � �� if e�C� � ��

so that at low temperature only the cells with p�� m � �� are occupiedtheir momenta �ll a sphere in momentum space �Fermi sphere�� Note thatif � � � the system density tends to � as T � �� If one wants to keep aconstant density while T � � one must �x � � �� In fact if � � � thedensity is such that

��

Zd�p

h�e��p

��m�

� � e��p��m�

��

��

�h�

p� m�� #��

Hence if � � � and the density stays constant �i�e� � � �� one �nds theinternal energy and the speci�c heat at constant volume via the relations

U � V

Zp�

m

d�p

h�e��p

��m�

� � e��p��m�

CV ��U

T

�V

� V

Zp�

m

d�p

h�kB�

�� p�� m�e��p

��m�

�� e��p��m��

��


and an elementary analysis of the integrals leads to the asymptotic formula

CV �V T as T � � ��

with a suitable �� Hence also this system behaves in a di�erent way ifcompared to the classical perfect gas at low temperature� In particular�� shows that equipartition of energy no longer holds �because CV ��V �nR� ��The conditions under which a behavior emerges re$ecting the fact that h

can no longer be considered negligible� and the classical perfect gas showsproperties that are completely di�erent from those just exhibited �withoutneglecting the size of h and� of course� only for small T � have been discussedin Chap�II� � � � � and x �� We just recall that we obtained an estimateof the value of the temperature below which the e�ects of the nonvanishingcell size begin to be felt as

Tq � h��mkB��

One can check� on the above formulae� that the latter value for Tq coincides�as we should expect� with the value of the temperature such that ��q� � �in the �rst case and such that �� in the second�It is common to say that the condition T � Tq is the condition that the per�

fect gas does not present degeneration phenomena due to the nonnegligiblesize of h�It is not di%cult to realize that the degeneration due to the fact that h

is appreciably �� can be the mechanism that permits us to avoid all theparadoxes due to energy equipartition�For instance in the theory of a crystal� the electron contribution to the spe�

ci�c heat is negligible because the value of the temperature below which theelectron gas presents degeneration phenomena �with consequent smallnessof the speci�c heat� see �� or �� can be estimated on the basisof �� and it gives a very high value of Tq�By using �� and m � �� g� � � �� cm�� density of the

free electrons in iron� one �nds Tq

Tq � ��kB�q � �� K � ��

More generally one can think that� if a given system consists of variousparticles� each with several internal degrees of freedom� then at a giventemperature only some degrees of freedom are nondegenerate equipartitionthen takes place �between� them� while the other particles remain in a de�generate state and therefore produce novel phenomena� among which thelack of contributions to the speci�c heat�A very interesting example is that of black body radiation theory in fact

the black body is a system with in�nitely many independent degrees offreedom� most of which are in a state of extreme degeneracy �see below��


so that the equipartition of the energy takes place only between a �nitenumber of degrees of freedom�In x�� we saw that a radiating cavity can be regarded as a set of in�nitely

many harmonic oscillators with Hamiltonian ��

H ��

�X��

Xk

� p��k�� c�k� q��k��

where p��k� and q��k� are canonical variables�The canonical distribution attributes to the con�gurations in which the

oscillator with polarization � and wave number k is in the cell C��km�n with

center � p��k�� q��k�� m�p� n �q�� m�n integers�� the probability

p�C��km�n� �

e�� m

��p�c�k��q�n��Pm�n e

� �� m

��p�c�k� n��q��

where we do not neglect the dimensions of C��km�n� and we take seriously the

canonical ensemble� forgetting that its use is doubtful when the cell sizesare not negligible and that in such cases statistical mechanics should becompletely reformulated�By repeating the analysis followed to obtain �� see x � and x �� one

easily �nds the condition under which the size of h is negligiblep��p� ��

p� cjkj�q � � � ��

Without explicitly �xing the values of �p and �q we see that �� willimply� in particular �multiplying corresponding sides of the two conditions�that � is too large for a classical statistical description of the oscillatorswith frequency � if

� c jkj �p �q � �hcjkj � ��h� � � ��

where � � cjkj� ��We must therefore expect that� given h� the high�frequency oscillators �withjkj � ��hc� or h� � �� kBT � will be degenerate� i�e� they cannot bedescribed without taking into account the cells sizes� Note that since �can be as large as we want there will always be frequencies � for whichh� � �� kBT �If we take h � �� erg � sec and T � �� K �temperature at the

surface of the Sun� one sees that the degenerate frequencies are all thosegreater than

�� h� � �� cycles sec��

which can be compared� for the purpose of an example� with the fre�quency of green light �where the Sun spectrum has its maximum� �green �� cycles sec��


The latter numerical values explain why the degeneration phenomenon hasbeen so �easy� to observe� or �so conspicuous�� in the black body radia�tion and why it has plaid such a big role in the development of quantummechanics�The average energy of a nondegenerate oscillator is� by energy equipartition�kBT � �� while if we do not neglect the possibility of degeneration thisenergy is

u�� k� �Xm�n

�

�n��p� � c�jkj�m��q��p�C

��kn�m�� #��

as expressed by �� The quantity of energy in the radiation with frequency between � and� � d� is then �see ��

L�u� d� ��

c�d�L�

�X��

u�k� ��

where jkj � ��c� If � � ��h equipartition holds as one can compute

explicitly using �� #�� and �� is simply

u� �#��

c��

To discuss the high�frequency case � � ��h it is necessary to �x �p and�q but in classical mechanics one cannot give a clear criterion for choosing�p or �q� Hence for concreteness we shall choose �p and �q so that�

�p �q � h�p � �cjkj�q � � ��q

��p �

p ��h

�q � ��ph� ��

��

with �� Although this is a �natural� choice because it makes approx�imately equal the two addends in �� #� for m � n � � �exactly equalif � �� it is nevertheless arbitrary� The results are qualitatively inde�pendent of the choice of � but their quantitative aspects do depend on itsvalue�From �� one deduces with a brief analysis of the series onm and n� that if �h� � � and & � � min� ��

u�k� �� "h�e��h� "h � � & � h if ��

u�k� �� "h�e��h� "h � �h if � ��

so that �� yields �for �� and "h � �h �� the distribution� identicalto the Wien�s distribution�

u� � #��

c�"h�e��h� ��


which shows that the energy present at high frequency is far below theequipartition value and� in fact� the total energy of the electro�magnetic�eld in thermal equilibrium is �nite� unlike what would happen if everyoscillator had the same average energy� Of course �� cannot reallybe taken seriously because� as repeatedly remarked already� the very factthat we do not neglect the size of h shows that it would be necessary toreinvestigate the basic laws of motion� based on �� that we are using�A further indication that �� cannot be considered a correct distri�

bution is seen also by noting that by changing by a small amount the cellshape �e�g� take � � or �� in �� one would �nd a quantitativelydi�erent result�For instance Planck used phase space cells C

��kn �for single particles� with

the shape of an elliptic annulus de�ned by

�n� ��h� � �

�p��k�� c�jkjq��k�� nh� n integer � � ��

and area h� i�e� he imagined that the cells were de�ned by the value of theenergy �and more precisely of the action to which� in this case� the energyis proportional� rather than by the momentum and position� Note that thisshape is �very� di�erent from the parallelepipedal shapes used so far�In this way �� and �� are replaced by

p�C��kn � �

e��h�n

�� e��h��

u�k� r� ��Xn�

nh�e��nhv

�� e��h��

h�e��h�

�� e��h�

��

which leads to the Planck distribution

u� �#��

c�h�

e�h� � ��

for the black body radiation�Obviously on the basis of classical statistical mechanics it is impossible to

decide which is the correct radiation distribution we can only say that ifin fact phase space cells cannot be chosen smaller than a minimum size�then it will be impossible to accept equipartition and� on the contrary� thehigh�frequency oscillators will have a very low average energy�The experimental result that radiation in thermal equilibrium conforms to

the Planck distribution is an indication of the non�inde�nite divisibility ofphase space� And the black body is a system particularly apt to reveal thediscrete structure of phase space� because it consists of an in�nite numberof oscillators with frequency � greater than an arbitrarily pre��xed value�� and therefore it contains an in�nite number of degenerate oscillators if his positive� no matter how small� In fact degeneracy happens to be alreadyvisible in �everyday life��


It is also possible� and of major interest� to investigate which would be thepredictions of a strict intepretation of radiation and speci�c heat theories interms of classical mechanics� i�e� assuming h � � in spite of our argumentsin Chap�I and above� on the �unphysical nature� of such an assumption�This is however a very di%cult task and many open problems remain�

Therefore I can only quote here a few papers that after the work �FPU��have tried attacking the problem and brought a wealth of new ideas andresults on the behavior of large assemblies of purely classical point particlesin a situation in which the temperature is lower than the value �� where problems with a classical interpretation begin to appear� See �GS� ��BGG#�� and for the more recent developments see �BGG�� Be�� Be��

x�� Introduction to Quantum Statistical Mechanics

In a sense quantum statistical mechanics is very similar to classical sta�tistical mechanics this should come as no surprise as both theories aim atexplaining the same macroscopic phenomena�As we have seen in x�� some of the main phenomena that receive their ex�

planation in the framework of quantum mechanics �like the low�temperaturespeci�c heat of solids or the black body radiation or the perfect gases spe�ci�c heats� can in a very qualitative and empirical sense be guessed also inclassical statistical mechanics and historically this actually happened� andsparked the genesis of quantum mechanics�Phase space no longer has a meaning and one only thinks of observable

quantities which are described mathematically by some �few� not necessar�ily all� which would lead to conceptual problems� �VN�� BH�� Be#��linear operators on a Hilbert space� usually in�nite dimensional� On thispoint a long discussion could be started by arguing that this is in fact notreally necessary and the dimension might be chosen �nite and its size wouldthen become a parameter that would play in quantum statistical mechanicsa role similar to the phase space cells size h in classical mechanics�However� since no �crisis� is in sight which would lead to a new mechanics�

at least no crisis that is as obvious and as universally recognized as a problemlike the black body radiation laws were in the early days of the twentiethcentury� we shall not dwell on the exercise of trying to understand howmuch the theory depends on the dimension of the Hilbert space or� for thatmatter� on the continuity of the space of the positions that particles canoccupy �which one may� also� wish to challenge��statistical ensembles are de�ned in terms of the Schr�odinger operator de�

scribing the observable energy and usually denoted H � But their elements ��rather than as probability distributions of phase space� are de�ned as rulesto compute the equilibrium averages of observables �which is essentiallywhat they are used for also in classical statistical mechanics��A ensemble E will� then� be a collection of rules � each of which allows

us to compute the average value that an observable has in the macroscopicequilibrium state � � E the element � should be stationary with respect to


quantum mechanical time evolution� as in the corresponding classical cases�The mathematical notion necessary to de�ne a �rule to evaluate averages�

of observables represented by self�adjoint operators� and therefore the analogof the classical probability distributions on phase space� is that of densitymatrix� If A is an observable and H is the energy operator that correspondsto N particles in a container V � one de�nes the canonical ensemble as thecollection of all the density matrices which have the form

� � const e��H ��

and the average value of the observable A in the macroscopic state repre�sented by �� parameterized by � and V � as in the analogous case ofthe classical statistical mechanics� is de�ned by

A �TrA e��H

Tr e��H��

where Tr is the trace operation�As hinted above� for all practical purposes �most� operators can be regarded

as big auto�adjoint matrices of large but �nite dimensions� so that the tracemakes sense after some practice one in fact understands how to avoidannoying errors and pitfalls linked to this view of the operators� much in thesame way in which one learns how to avoid di�erentiating non di�erentiablefunctions in classical mechanics�Thermodynamics models are deduced from the �quantum� canonical par�tition function

Z�� V � � Tr e��H ��

and now ��kB is interpreted as the temperature� while the free energy isde�ned by f�� v� � lim

V��

V logZ�� V � in the limit V �� V�N �v �thermodynamic limit��Note the the absolute temperature is no longer de�ned as proportionalto the average kinetic energy rather it is identi�ed as proportional to theparameter �� that appears in �� see also x �� and x�� for a relatedcomment on this di�erence� In some respects this is the really major noveltyin quantum statistical mechanics�One can also de�ne quantum microcanonical or grand canonical ensembles

and check their equivalence� sometimes even rigorously under suitable extraassumptions like stability and temperedness �see � � �� #�� Ru��For instance� considering N identical particles of mass m in a cubic con�

tainer V � the Hilbert space is the space L��VN � of symmetric �or an�

tisymmetric� square integrable functions of the N position coordinates�q

�� q

N�� and the energy operator is

H � � &h�

m

NXi�

�qi

� ��q� ��


where ��q� �P

i�j�qi � qj� is the potential energy of the interaction�

&h � h� � if h is Planck s constant and �qi

is the Laplace operator with

respect to the i�th particle coordinate and with suitable boundary conditions�e�g� periodic or Dirichlet boundary conditions��

The symmetry or antisymmetry of the wave functions is imposed to takeinto account the speci�c quantum nature of the particles which can be eitherbosons or fermions� the latter corresponding to a system of particles which�besides the interaction energy � in �� also have the extra interaction�classically nonstandard but quantum mechanically very natural� that notwo particles with the same momentum can occupy the same position�

The stability notion for the interaction � is important in quantum statisti�cal mechanics as much as it is important for classical statistical mechanics�An interaction is called quantum mechanically stable if there is a constantB such that the Schr�odinger operator H � �� for N identical particlessatis�es� for all N � ��

H � �BN ��

where the inequality holds in the sense of the operators �i�e� for any nor�malized quantum state j� i � h� jH j� i � �BN��

It is interesting and important to note that the inequality �� can nowbe valid even if the in�mum inf ��q� of � equals �� because the potentialbecomes �� at � distance� In fact one can no longer separate the potentialand kinetic energy as independent quantities the indetermination principlein fact forbids concentrating too many particles in too small a box withoutgiving to them a high kinetic energy� Hence there is the possibility that thedecrease in potential energy due to too many close particles �contributinga large negative potential energy if �� or �� is compen�sated by the increase of kinetic energy necessary to achieve the con�nement�Whether this really happens or not depends on the system �mainly on thebosonic and fermionic nature of the particles� and has to be quantitativelychecked� It will be brie$y discussed in Chap�IV�

The case in which the system contains several species of identical particles istreated as easily as in classical statistical mechanics� In the latter case it wassu%cient to introduce suitable combinatorial coe%cients to take the identityof the particles into account� see � � �� in the quantum case oneshall simply require the symmetry or antisymmetry of the wave functionswith respect to the permutations of the positions of identical particles�

For instance a system of N� electrically charged particles with charge �eand of N� particles with charge�e interacting with the Coulomb force wouldhave� in the classical canonical ensemble� the partition function

Z�� V � ��

N��N��

Ze��PN�

i�p�i��m

PN�N�

iN��p�i��m�

�

e��q� dp dq

h��N�N��

��

��# III� Equipartition and Critique

with

"��q� �

N�Xi�

N�Xj�

�e�jqi� q

jN�j �

�

N�Xi�j�i�j

e�

jqi� q

jj�

�

N�Xi�j�i �j

e�

jqiN�

� qjN�

j �

N�N�Xi�j

�qi� q

j�

��

where is a potential describing a nonelectric force between the particlesand m�m� are the masses of the two species�In the quantum case one has instead Z�� V � � Tr exp��H� where H is

the Schr�odinger operator

H � � &h�

m

N�Xi�

�qi� &h�

m�

N�Xi�

�qiN�

� "��q� ��#��

considered as an operator acting on the space of functions f�q�� q

N��

qN��

� � � � � qN�N�

� symmetric or antisymmetric with respect to the permu�

tations of the �rst N� variables or of the second N�� but with no symmetryproperty with respect to �mixed� permutations�The statistics� as one often calls the symmetry properties of the wave

functions� with respect to the permutations of their argument plays anessential role in the theory� From the classical viewpoint adopted in x��above we already had a glimpse of the phenomena that may make quantumstatistical mechanics quite �strange� even from a qualitative point of view�at least at low temperatures� The reason is that the statistics can be in�terpreted as a special �simple� further interaction �i�e� no interaction in thecase of bosons and a repulsive interactions in phase space for the fermions�compare �� and �� But the statistics may play a role even at ordinary temperatures for

instance electrically neutral systems in which particles interact only viaCoulomb forces are unstable in classical statistical mechanics� at all temper�atures� for the trivial reason that the Coulomb potential between particlesof opposite charge is unbounded below near the origin� But in quantumstatistical mechanics they are stable if the charged particles satisfy Fermistatistics or if the bosons have charges of only one sign� See Chap�IV fora discussion of the importance of stability in statistical mechanics even insystems in which no charged particles are present�

x�� Philosophical Outlook on the Foundations of Statistical Me�

chanics

Contemporary �i�e� AD �� equilibrium statistical mechanics can be saidto be in an ideal conceptual stage of development�

� Which form the space on which the Hamiltonian acts as an operator


�� There seem to be no fundamental theoretical problems after the irre�versibility of macroscopic evolution has been shown to be compatible withthe reversibility of microscopic dynamics this understanding already cameabout at Boltzmann s time in terms of the existence of time scales of verydi�erent orders of magnitude over which irreversibility and reversibility canmanifest themselves� It was put in a rigorous mathematical form by Lan�ford� see x��# and this should have paci�ed the stubborn nonbelievers inthe incompatibility between microscopic reversibility and macroscopic irre�versibility �it did so only very partially� in fact��

� � The paradoxes to which classical statistical mechanics leads have beenunderstood in terms of quantum e�ects and the conditions of applicabilityof classical statistical mechanics have been correspondingly precisely formu�lated� see �� and x ��

There are still many questions to be understood on the dynamics of theapproach to equilibrium in many�particles systems and to develop reliable�and universally recognized as correct� methods to evaluate the time scalesrelevant in the phenomena of approach to equilibrium� see Chap�I� Chap�IX�

The ergodic problem is still not well understood particularly in systemsclose to mechanical equilibrium positions �as in oscillations in crystals�where it might even be conceivable that the ergodic hypothesis really failsin a substantial way� �FPU�� This is so in spite of the major successachieved by Sinai in proving the ergodicity of a really interesting physicalsystem �two balls in a periodic box� �Si�� and its extension to many ballsin a box �KSiS�� On the whole the scarce understanding of nonequilibriumphenomena is re$ected also in major problems in the kinetic theories of gasesand liquids and of the related transport phenomena� �Co��Co��Do�#��

The importance of the latter question has been strongly stressed by L�Galgani and by his collaborators who have devoted to the subject severalimportant studies which led to a much better understanding of the relevanceof the �probable� lack of ergodicity on time scales as long as the life of theUniverse� The investigations stem from� and develop� �forgotten� remarksof �founding fathers� like Jeans� �GS� �� BGG�� Be��

But open problems abound also in equilibrium statistical mechanics�

The central problem of equilibrium statistical mechanics is perhaps thetheory of phase transitions and of the corresponding critical points� Thereis no evidence of fundamental di%culties and recently some clari�cationhas been achieved in the phase transition phenomenon as a phenomenon ofinstability with respect to boundary conditions� or of sensitive dependence ofthe equilibrium state on the boundary conditions� see Chap�II and Chap�V�

Via simple soluble models� see Chap�VII� it has been shown how even thesimplest models of mechanical systems �like systems of magnetic spins ona lattice� can show nontrivial phase transitions and� in fact� very interest�ing ones� Nevertheless very important phenomena� such as the liquid�gastransition or the crystal�liquid transition� are not really understood�

In fact there is no model that could be treated avoiding approximations


which are really out of control and which describes one such transition�By �out of control� I mean approximations that have to be conceptuallyregarded as parts of the model itself because their in$uence on the resultscannot be estimated �without hand waving�� See� however� �Jo�� and thevery recent �MLP�#� for very encouraging steps in the right direction�

A meager consolation comes from the reassuring con�rmation of the theo�retical possibility �i�e� consistency� of such transitions accompanied by thedevelopment of many approximate theories that are continuously generated�and used in concrete applications the ultimate goal for a wide class ofscientists��

The �rst among such theories is the mean �eld theory which until the��s was the only available theory for the study of phase transitions� Thisis a simple theory� see Chap�V� but somewhat too rough �so as to predictphase transitions even in systems which can be shown to have none� likeone�dimensional systems with short�range interactions��

The theory of phase transitions has undergone important developmentsmainly for what concerns the theory of critical phenomena in the contextof which new approximate theories have been developed which provide the�rst real novel theoretical proposals after mean �eld theory� they are knownas the renormalization group approach of Fisher� Kadano� and Wilson��WF� �� Wi#�� We cannot deal with such developments in this monographthe reader will �nd a modern acounts of them in �BG�� Fi�#��

Another important phenomenon of equilibrium statistical mechanics is thatof metastability and it is still not well understood its theory involves dealingwith ideas and methods �and di%culties� characteristic both of the evolutionand of the equilibrium problems� Here we shall not deal with this matter�see �LP�� CCO�� MOS��

Another class of not well understood phenomena are equilibrium andnonequilibrium phenomena in charged particles systems until recently itwas even qualitatively unclear how a neutral system of charged particles�i�e� matter� could stay in thermodynamic equilibrium� notwithstanding thestrong intensity and long range of the Coulomb interaction� see �Fe#�� fora detailed analysis of the basic mechanism� Until very recently only phe�nomenological theories for phase transitions were available� essentially basedon the same type of ideas at the roots of mean �eld theory �for instanceDebye�s screening theory��

Recently the problem of stability of matter �i�e� of proving a lower boundproportional to N on the energy of N charged particles with zero totalcharge� has been satisfactorily solved in the framework of quantum sta�tistical mechanics� �DL�� LL� ��LT�� but the problem of a quantita�tive understanding of the thermodynamic equilibria in neutral aggregatesof charges and of the related screening phenomena remains open� �Fe#��Li#�� Of course there are �plenty� of very elaborate and detailed phe�nomenological theories� but here we mean that they are not fundamentaland that� to be developed� require further assumptions �besides the funda�mental assumption that equilibrium states are described� say� by the canon�


ical ensemble� that are only justi�ed on a heuristic basis necessary to bypassotherwise non�by�passable �technical� di%culties�

For instance in the theory of molecular gases one usually postulates thata given system just consists of identical particles� with given time�invariantproperties �molecules�� that interact with each other via e�ective forces dueto screened electromagnetic interactions� This is clearly an approximation�that we empirically think of as perfectly adequate� which obviously ignoresan important part of the problem namely that the molecules are formedby atoms which are formed by nuclei and electrons �forgetting protons�neutrons� quarks� etc�� and the possibility that they dissociate� ionize orreact chemically� Therefore one may wish to see a microscopic explanationof why in a range of densities and temperatures �absolutely crucial for ourlives� matter presents itself mostly bound into complexes which have thesize of isolated atoms and molecules this is still not understood althoughimpressive progress has been achieved in the �eld� �Fe#��

quantum statistical mechanics not only solves the conceptual problem ofthe stability of matter� �Li#�� but it also introduces the possibility of a the�oretical understanding of a large variety of new phenomena typically relatedto the quantum nature of microscopic physics super$uidity and supercon�ductivity are typical examples� So far such phenomena are understood onlyon the basis of phenomenological theories close in spirit to the mean �eldtheory of phase transitions� �BCS�� Br�� and Chap� �� of �Fe� �� Buta deeper theory has still to be developed� In fact one can say that in quan�tum statistical mechanics all the problems of classical statistical mechanicsare present� usually in an unsolved form even when the corresponding classi�cal problems are solved� and new problems that do not even exist in classicalmechanics become analyzable theoretically�

It does not appear that any of the problems that are not understood arenot understandable in the framework of statistical mechanics �classical orquantum as the case applies� no fundamental problem seems to have atheoretical description that is in con$ict with experimental results� Thisparadisiac atmosphere may not last for long �its stability would be verysurprising indeed� but as long as it lasts it gives us great peace of mindwhile still o�ering us a wide variety of fascinating unsolved problems�

Finally we mention that statistical mechanics is related to many branchesof mathematics� particularly probability and information theory that havereceived a great in$ux of new ideas from the theory of phase transitions andof ensembles� �Ru�� and the theory of dynamical systems that has receivedin$ux from the theory of approach to equilibrium� �RT�� Ru�#a�� Do�#��Combinatorics has been greatly widened by studies of the exactly solublemodels in statistical mechanics� �Ba# ��

Many problems in ordinary or partial di�erential equations have their originin statistical mechanics which has also inspired several developments in thetheory of turbulence� �Fr�� and in the theory of quantum �elds� �BG��

One can say that the present state of statistical mechanics is perhaps com�parable to the state of mechanics at the moment of its triumphal applica�


tions to celestial mechanics and to ordinary mechanics at the end of the ��s and the beginning of the #��s� No obvious contradiction with ex�periments has yet come up and nevertheless many simple and interestingphenomena remain to be explained by the theory� A sign of vitality is alsoto be seen in the fact that statistical mechanics continues to generate newand deep mathematical problems one can perhaps say� as a nontautologi�cal statement� that physical theories are sources of interesting mathematicalproblems only as long as they are really alive and faced with di%culties thatare not purely technical�

��

Chapter IV �

Thermodynamic Limit and Stability

��

�

IV� Thermodynamic Limit and Stability ��

x�� The Meaning of the Stability Conditions

The stability and temperedness conditions �see � � �� #��

��q� �Xi�j

�qi� q

j� � �BN stability

j�q � q��j � Cjq � q�j�� for jq � q�j � r� temperedness

��

for suitable constants C � �� r� � �� have to be imposed on theinteraction potential in order to insure the existence of the limits in � ��#�� or � �� de�ning fc� sm or pgc respectively� i�e� the thermodynamicfunctions associated with the partition functions of various orthodic ensem�bles� The conditions are particularly interesting because� besides the abovemathematical role� they have a simple� and profound� physical meaning�The existence of the limits is a necessary requirement to have orthodicity

and equivalence of the thermodynamic models de�ned by the di�erent en�sembles� as discussed in Ch�II� We have seen that if the above limits existthen the ensembles de�ne the same model of thermodynamics for a givensystem� Therefore to understand the signi�cance of the stability conditionsit is convenient to examine their meaning in the thermodynamics model de��ned by one of the ensembles and we shall choose the canonical ensemble�where the analysis is simplest� The following analysis also illustrates someof the typical methods that are used in statistical mechanics�

�a� Coalescence Catastrophe due to Short�Distance Attraction� The �rstcondition in �� can be violated in several ways� One possible way iswhen the potential is negative at the origin we always assume that isa smooth function for q �� and� in the case at hand� we also assume that is smooth at the origin�Let � � � be �xed so small that the potential between two particles at

distances � � is � �b � �� Consider the canonical ensemble element withparameters ��N for a system enclosed in a �cubic� box of volume V � Wewant to study the probability that all the N particles are located in a littlesphere of radius � around the center of the box �or� for that matter� aroundany pre��xed point of the box��The potential energy of such a con�guration is � � �b�N� � � b

�N� �be�

cause there are�N�

�pairs of particles interacting with an energy � �b�

therefore the canonical probability of the collection C of such con�gurationswill be

Pcollapse �

RC

dpdq

h�NN �e��K�p��q��R dpdq

h�NN �e��K�p��q�

� � ��h� �N ��N

N � e�b ��N�N��R dq

h�NN �e��q� ��

�� IV� Thermodynamic Limit and Stability

where we see that the contribution to the integral in the numerator canbe considered as due to two factors� One is the value of the integrandfunction� whichwe call an �energy� factor� and one is the volume of thecon�guration space where the function takes the value considered� which wecall an �entropy factor� or a �phase space factor�� The �rst is � dependentwhile the second is not�The phase space is extremely small because the con�gurations that we

consider are very special nevertheless such con�gurations are far moreprobable than the con�gurations which �look macroscopically correct��i�e� con�gurations in which the particles are more or less spaced by theaverage particle distance that we expect in a macroscopically homogeneouscon�guration namely �NV ��

��

The latter con�gurations will have a potential energy ��q� of the order ofuN for some u� so that their probability will be bounded above by

Pregular �R dpdq

h�NN �e��K�p�uN�R dpdq

h�NN �e��K�p��q�

�V N

h�NN �e��uNR dq

h�NN �e��q� ��

and we see that

�� the denominators in �� are �of course� equal and

� � the phase space factor in the numerator in �� is much larger thanthe corresponding one in �� i�e� V N against ��N �� at least in the �ther�modynamic limit� V �� N �� NV � v��

However� no matter how small � is� the ratioPregularPcollapse

will approach � as

V � �� NV � v�� extremely fast because e�bN

�� eventually dominatesover V N eN logN �This means that it is far more probable to �nd the system in a microscopic

volume of size � rather than in a con�guration in which the energy has somemacroscopic value proportional to N note that in a free gas� for instance�where �� b � �� the situation is the opposite� and in general� if the stabilityproperty in �� holds� the above argument also does not apply�This catastrophe can also be called an ultraviolet catastrophe as it is due to

the behavior of the potential at very short distances it causes the collapseof the system into con�gurations concentrated in regions as small as weplease �in the thermodynamic limit��

�b� Coalescence Catastrophe due to Long�Range Attraction� This is a moreinteresting catastrophic behavior� because of its physical relevance� It occurswhen the potential is too attractive near�� To simplify matters we supposethat the potential has a hard core� i�e� it is �� for r � r�� so that theabove discussed coalescence cannot occur and the system cannot assumecon�gurations in which the density is higher than a certain quantity �cp �� called the close packing density�


The catastrophe occurs if �q� �gjqj�� g� � � �� for jqj large� Forinstance this is the case of matter interacting gravitationally� if k is thegravitational constant� m is the particles mass �assuming an identical par�ticles system�� then g � km� and � � �In this case the probability of �regular con�gurations�� where particles

are at distances of order �� from their close neighbors� is compared withthat of �catastrophic con�gurations�� with the particles at distances r� fromtheir close neighbors to form a con�guration in �close packing� �so that r�is equal to the hard core radius�� Note that in the latter case the systemdoes not �ll the available volume and leaves empty a region whose volumeis a fraction

cp� cp

V of V �

A regular con�guration will have a probability �in the canonical ensemblewith parameters ��N and if L is the diameter of V � proportional to

Pregular CV N

h�NN �eg ��N

RL

�jqj��dq

��

because the energy of interaction of a single particle in a medium with

density � is� to leading order in L�� R L�gjqj��dq � �gL�� in �� C

is a normalization constant �i�e� it is the reciprocal of the canonical partitionfunction��Likewise if we consider a con�guration in close packing and we dilate it by a

factor �� we obtain a con�guration in which each particle can be moved

in a small sphere of radius O��cp �� and we can call such con�gurations

catastrophic or collapsed as they occupy only a part of the volume allowed�no matter how large the latter is �i�e� no matter how small � is comparedto �cp��In the canonical ensemble with parameters ��N the probability of the

catastrophic con�gurations can be bounded below by

Pcatastrophic � C��cp ��N

h�NN �eg ��N cp��

��RL

�jqj��dq

��

where the constant C is the same normalization constant as in �� andagain we see that the catastrophic con�gurations� in spite of their verylow phase space volume� have a much larger probability than the regularcon�gurations� if � � �cp and � is small enough because the exponentialin the energy factor� in �� grows almost as �

��cpV��

��V��

provided � � �cp and � is small enough�A system which is too attractive at in�nity will not occupy the volume wegive to it but will stay con�ned in a close packed con�guration even in emptyspace�This is important in the theory of stars stars cannot be expected to obey

�regular thermodynamics� and in particular will not �evaporate� becausetheir particles interact via the gravitational force at large distance� Starsdo not occupy the whole volume given to them �i�e� the universe�� they donot collapse to a point only because the interaction has a strongly repulsive

��# IV� Thermodynamic Limit and Stability

core �even when they are burnt out and the radiation pressure is no longerable to keep them at a reasonable size� a reasonable size being� from ananthropocentric viewpoint� the size of the Sun��

�c� Evaporation Catastrophe this is a another infrared catastrophe� i�e� acatastrophe due to the long�range structure of the interactions like �b�above� it occurs when the potential is too repulsive at � i�e� �q� �gjqj�� as q �� so that the temperedness condition is again violated�

Also in this case the system does not occupy the whole volume it willgenerate a layer of particles sticking in close packed con�guration to thewalls of the container� Therefore if the density is lower than the closepacking density� � � �cp� the system will leave a region around the center ofthe container empty� and the volume of the empty region will still be of theorder of the total volume of the box �i�e� its diameter will be a fraction ofthe box side L with the value of the fraction not depending on L as L��

The proof of this statement is completely analogous to the one of theprevious case� except that now the con�guration with lowest energy will bethe one sticking to the wall and close packed there� rather than the one closepacked at the center�

Also this catastrophe is very important as it is realized in systems ofcharged particles bearing the same charge the charges adhere to the bound�ary in close packing con�guration and dispose themselves so that the elec�trostatic potential energy is minimal� We cannot� therefore� expect that thecharges that we deposit on a metal will occupy the whole volume they willrather form a surface layer minimizing the potential energy �i�e� so that theCoulomb potential in the interior is constant�� They do not behave thermo�dynamically for instance� besides not occupying the whole volume given tothem� they will not contribute normally to the speci�c heat�

x�� Stability Criteria

There are simple criteria that make sure that the conditions �� aresatis�ed� The �rst condition is satis�ed� in general� if � � one callsthis case the repulsive potential case� although this is a somewhat improperde�nition because � � does not imply that is monotonically decreasing�which would in fact generate a repulsive force in the usual sense of theword�� In this case one can take B � ��

Another interesting case is that of a smooth potential which has a non�negative Fourier transform )� In fact in this case

��q�� q

n� � ��n �

�

nXi�j�

�qj� q

i� �

� ��n �

Z)��k�

�

nXi�j�

ei�k��q

i�q

j� � ��n

��


becausePn

i�j� ei�k��q

i�q

j�

� jPnj� e

i�k�qj j� � �� This will be called the

positive de�nite potential� or positive type� case and one can take B � ��Of course a potential that can be expressed as a sum of a positive potential

and of a positive type potential is also stable� A remarkable case of apotential that can be expressed in this way is a potential such that� forC�C �� r� � �� it is

�q� �C� r�jqj�� for jqj � r�

j�q�j �C �� r�jqj�� for jqj � r� ��

Such a potential is sometimes called a Lennard�Jones potential although itis more general than the potential that was originally introduced with thisname �see x�� The proof of the possibility of representing as a sum of a positive potential

and of a positive type potential can be found in �FR�� but the stability ofa potential satisfying �� can be checked directly very simply� �Mo��Given� in fact� a con�guration q

�� q

nlet r be the minimum distance

between pairs of distinct points r � mini�j jqi� qjj� Suppose that the pair

of closest particles is q�� q

�and that r � �

�r�� then

��q�� q

n� � ��q

�� q

n� � �r� �

Xjqj�q

�j r�

�qj� q

��

where the equality would have held had we summed over all the j s� i�e� alsoover the j s such that jq

�� q

jj � r��

Around each of the qj

we can draw a cube Qj with side rp��

and qj

being

the vertex farthest away from q�� Since any two points among q

�� q

nhave a distance � r the cubes thus constructed do not overlap� and theirunion is contained in the complement of the sphere jq

�� qj � r� � r

� � r��

furthermore

jq�� q

jj��

p� ��

r�

ZQj

dq jq�� qj��

so that the sum in �� is bounded below� for some C� � �� by

�C � �p

� ��

r�

Zjqj r�

�

r��

jqj�� dq � �C�

�r�r

��

and ��q�� q

n� � C

�r�r

�� C�

�r�r

�� q

�� q

n� so that the sum

of the �rst two terms is bounded below by some �C� � �� provided theassumption that the con�guration q

�� q

nis such that r � �

�r� holds�

� � IV� Thermodynamic Limit and Stability

The case r � ��r� is easier because the density is bounded above by

#r�� and the interaction decreases summably at � a repetition of theabove considerations yields simply that ��q

�� q

n� � ��q

�� q

n� �Pn

j� �q�� q

j� � ��q

�� q

n� � C� for a suitable C� � �� Hence if b �

max�C�� C�� then ��q�� q

n� � �b � ��q

�� q

n� � �bn and stability

is proved�

We conclude by noting that �� are su%cient stability and temperednessconditions� But in general they are far from being necessary�

In fact one would like to prove that certain systems not ful�lling �� arenevertheless stable and have a well�de�ned thermodynamics� equivalentlydescribed by the canonical ensemble or by other ensembles�

For instance a gas composed of electrically charged particles and with �total charge is an example of a system for which we would like to provestability� and even orthodicity of the classical ensembles�

Note that it is easy to see that a gas of charged classical particles is notstable just consider the con�guration in which pairs of opposite charges areput very close to each other� while the pair centers of mass are essentiallyequispaced� this con�guration has an energy which can be made � �bN forall b s� by pulling the pairs close enough together�

If� however� the particles also have a hard core interaction besides theCoulomb interaction� so that pairs of particles cannot be closer than somer� � �� the system becomes stable as remarked by Onsager� In fact thepotential of interaction between the charges will be the same as that whichthey would have if replaced by uniform balls of charge ej uniformly dis�tributed in a sphere of radius �

�r� around the j�th particle� Then by usingthe fact that there is a hard core

� �

��nXi�j

eiejjqi� q

jj �

��nXi�j

Zfqig�fq

jgdq dq�

ei�qiej�q

j

jq � q�j ��

where �qi

does not vanish only in the sphere with radius r�� around q

iand

it is constant there and with integral �� here the integration domain is theproduct of two balls fq

ig� fq

jg of radius �

�r� centered at qi

and qj� Hence

� � �nXi�

Zfqigdq

e�i��qi

jq � qij �

�

��nXi�j

Zfqig�fq

jgdqei�q

iej�q

j

jq � q�j � ��

If the number of species is �nite it is clear that the sum in the �rst line canbe bounded below by �nb� if n is the total number of particles� The doublesum is simply proportional to

Zd�k

j)��k�j��k�

� � �� #��

IV� Thermodynamic Limit and Stability � �

where )��k� is the Fourier transform ofP

i ei�qi�q�� and we have used that

the Fourier transform of the Coulomb potential is proportional to �k��Hence �� #� shows that a gas of a few di�erent species of charged par�ticles� interacting also via a hard core potential �and possibly any furtheradditional stable potential�� is stable and one should also note that stabilitydoes not even depend on the system being neutral�This example shows that stability can occur under situations more general

than �� and the conditions in �� may fail and nevertheless thesystem may be stable�A similar remark can be made for the temperedness condition ��

is su%cient for temperedness but the violations of temperedness describedabove and leading to the above infrared catastrophes may be absent inspecial systems� For instance in a gas of charged particles that is over�allneutral this is what really happens� see �LL� ��

x�� Thermodynamic Limit

A way to check that we are not missing some other basic condition onthe potential is to show that� if the stability and temperedness conditionsare satis�ed� the thermodynamic limits exist and the basic ensembles areequivalent�We have considered in Ch�II the existence of the thermodynamic limit of

the entropy

s�v� u� � limV��

NV�v�� U

V�u

�

NkBlog

ZH�U

dp dq

h�NN ��

where h�N is the size of a phase space cell� see Ch�I and Ch�II�And� at least at a heuristic level� we have seen that the existence of the limit

�� the microcanonical entropy� is the key to the proof of the existenceof the limits for fc� pgc� see � ��#�� and for the equivalence of thethermodynamics models based on the classical ensembles microcanonical�canonical and grand canonical�Therefore we shall discuss the problem of the existence of the thermody�

namic limit in �� i�e� the problem of the existence of the entropy in themicrocanonical ensemble this is in some sense harder than the problem ofshowing the existence of the corresponding limits in the canonical or grandcanonical ensembles� but it has the advantage of implying the results forthe other ensembles as heuristically discussed in Ch�II the argument giventhere is easily turned into a proof� under very general conditions�What follows is important not only because of the results that it establishes

but also because it illustrates some of the basic techniques used in appli�cations of statistical mechanics� In spite of its technical nature the readercould be interested in following it as it provides a good understanding ofthe physical meaning of the conditions �� and of their relation with theextensivity properties of thermodynamic functions�

� IV� Thermodynamic Limit and Stability

Before proceeding it is necessary to warn however that the following argu�ment looks at �rst a bit subtle� but it is in fact quite straightforward andthe intricacy is only due to the fact that we must make sure that whenwe divide the number of particles by certain factors �often but not always � � or #� we get an integer number� If one does not pay attention to thiscondition then the proof becomes trivial �although strictly speaking incor�rect�� Another source of problems is that we need to say that the partitionfunction corresponding to a region that is the union of two subregions isthe product of the partition functions corresponding to the subregions� aproperty that would be obvious of the regions were separated by a corridorof width larger than the interaction range� But the regions that we haveto consider touch each other and we must produce corridors by shrinkingthe regions and therefore we must compare partition functions relative toa region and to the smaller region obtained by cutting out of it a layer ofwidth �

�r� around its boundary�

The reader should� on a �rst reading� simply disregard the technical detailsrelated to the above counting and corridor problems and see that ��and the consequent ��#� hold at least approximately� and the existenceof the limit then follows on the special sequence of cubes Bn with side nL� �for a �xed L� � � and identifying temporarily the cubes Bn and B�

n��subsequently �� i�e� esentially still �� implies both existence overarbitrary sequences of cubes and shape independence�

One could invoke the fact that the numbers of particles are so large that achange of the particle number by a few units or the taking out of a layer ofwidth �

�r� around the boundary of a region that is becoming in�nite� makesno di�erence �on physical grounds�� but this is precisely the point� as wemust show that this is correct�

To simplify the discussion we shall suppose that the interaction has �niterange� i�e� it vanishes for jqj � r��

�A� Ground State Energy Convexity as a Function of the Energy

We �rst consider a special sequence of boxes and a special sequence ofvalues of N�U � The boxes will be cubes B�

n with side size L�n � nL� � r�where L� is an arbitrarily chosen unit of length� the boxes B�

n are containedin the cube Bn with side size Ln � nL� and stay away r�

� � at least� fromits boundaries� The volume jBnj is �nL�

��

B�n

Bn�


Fig� ��

The �gure illustrates the boxes Bn� and B�n in the corresponding two.

dimensional case �for simplicity�� the region Bn is represented by the �rst�quarter� �i�e� the lower left square�� the shaded areas represent the corri�dors of width �

�r� between the boundaries of the four copies of B�n and the

corresponding copies of Bn�Given a density � � � we suppose �rst that it is dyadic� i�e� that it has the

form � � m ��sL�� with m� s positive integers� Hence if n � s the numberNn � �jBnj is an integer and we can de�ne the �ground state energy� e atdensity � by

e � inf

��q�� q

Nn�

jBnj � �B� ��

where� the in�mum is taken over all n � s and over all con�gurationsq�� q

Nnin B�

n and the last inequality is a consequence of the stability�

�� Making a di�erence between B�n and Bn is a convenient device that

allows us to simplify some minor points in the forthcoming analysis and itshould not be regarded as an issue� The quantity e can be thought of asthe minimum energy density of the particle con�gurations with numericaldensity ��Note that if �� are two dyadic densities then also � � �

� �� isa dyadic density� Furthermore if we consider a box B�

n� we see that itcontains # � � boxes B�

n separated by at least r��If we put ��jBnj particles in four of the boxes B�

n and ��jBnj in the other�� it is clear� from the de�nitions� that the energy density of any suchcon�guration is � �

�e � jBnj� ��e � jBnj so that

e � �

�e � � e ��

In fact it is easy to see that e �� because �� holds together with

the easily checked e � ��maxjqj �� q� if � is small enough��

Therefore if we de�ne e� � � the function �� e is continuous on the setof dyadic densities on which it is uniquely de�ned �convex functions on thedyadics are continuous� and it can be extended by continuity to all real ��dyadic or not� We shall call A the convex region above the graph of theextension to all � s of �� e �

�B� Upper Bound on Entropy�

Let A be the region � � � and e � e � let �� e� � A be an �interior�point with � dyadic and e � e � Given �� e� V � � V let� to simplify the

� The quantity e� is de�ned to be � if one cannot �t jBnj particles inside B�n� this

may happen if the interaction contains a hard core part�

� This is a bound of the energy of a con�guration of points regularly spaced by ��


notations� N�� e� V ��def� N��U� V � with N� the microcanonical partition

function� � �� N � �V � U � eV � In the following V � will always be Vdeprived of a small corridor near its boundary�De�ne� for the � s that are integer multiples of ��nL�� a correspondingmicrocanonical entropy

�n�� e� ��

jBnj logN�� e� B�n� �

�

jBnj log

ZK�p��q��e jBnj

qj�B�n

dp dq

h�NN ��

Then �n can be bounded �because the potential � satis�es � � �BNn ��B�jBnj� by

�n�� e� � �

jBnj log/��Nn�

�p mjBnj�e� e �

��Nn

h�NnNNnn e�Nn

p �Nn

jB�njNn �

� � log��me��

�� e� e �

��

h��

��

where /�m� � p�m

'�m� ��

is the surface of the unit sphere in m dimen�sional space and no confusion should arise between the energy density eand the e �equal to the base of the natural logarithms� arising from usingStirling s formula for Nn� and for the gamma function� The above inequal�ity is an essential consequence of the stability which� in this proof� is usedhere for the �rst time�

�C� Quasi Convexity of the Entropy in Finite Volume�

The function �n�� e� has the simple property that it is �almost� convex�Suppose in fact that � � �

� �� and e � �� e��e�� with �� multiples

of ��n�� n� ��n� respectively� and e� � e � � e� � e � �hence e � e ��Then we can can look at the box B�

n� and at # copies of B�n that �t in it

separated by a corridor of width r��In four of the boxes B�

n we put N�� jBnj particles in a con�gurationwith energy � e�jBnj and in the other four we put N�� jBnj particlesin a con�guration with energy � e�jBnj� then N � ��N�� N�� Then weshall have in B�

n� exactly N � �jBn�j particles with energy � ejBn�jand clearly

N�� e� B�n�� N�� e�� B

�n��N�� e�� B

�n��

where the N � in the de�nition ofN�� see �� is very important because inderiving �� one uses the fact that there are N �

N��N��ways of selecting

which among the N particles are put in each of the eight boxes�Relation �� can be written in terms of the �n and it becomes

�n�� e� � �

��n�� e�� n�� e�

��


It will be convenient to have �n�� e� de�ned not only for dyadic ��i�e� multiples of ��nL�� but for any real �� Since �n�� e� is actuallyde�ned for � � m

��nL�� this can be simply achieved by de�ning �n�� e� for

the other values of � by linear interpolation�The linear interpolation is very convenient and natural because it satis�es

the bound �� since the latter is convex in �� e� furthermore it has� by�� the further property that if �n�� e� is nondecreasing in n � n� foreach � a multiple of ��n�L�� then it is also nondecreasing in n � n� forall ��

�D� Monotonicity of Entropy as Function of the Container Size

The property of �n�� e� of being nondecreasing in n� for dyadic �� is thekey property to the analysis� in fact since� as remarked above� the box B�

n�

contains eight boxes B�n separated by a corridor of width r� we see �from

�� with � � �� and e � e� � e�� that

�n�� e� ��

jBn�j logN�� e� B�n��

� �

jBn�j logN�� e� B�n��

��n�� e�

��#��

where again we make essential use of the N � in the de�nition �� ofN�� in the same way as in the derivation of �� By �� n�e� �� isuniformly bounded in n�Hence the limit as n � � of �n�� e� does exist for all � dyadic and fore � e � But the sequence �n�� e� will also converge for the nondyadic � sbecause �n�� e� is �almost convex� in the sense of �� and in fact theconvergence will be uniform on every closed set in the interior of A� Thelatter property follows from elementary considerations on convex functionsmonotonically convergent to a limit� Clearly �� imply that thelimit function s�� e� will be convex and bounded in the interior of the regionA� Hence it will be continuous in the same region�

�E� Independence from the Special Sequences of Density Values and of Grow�ing Containers�

We now want to show that we can free ourselves from the special sequenceof boxes and of densities that we have considered�Consider a family of cubic boxes B with side L � � containing N par�

ticles in con�gurations with energy � E and suppose that NV ��L�� and

EV ��L�� e with �� e� in the interior of A� Below we use equivalently thenotation V � jBj � L��Let �� and �� be supposed dyadic� let e� � e� � with �� small�

We can divide the box B into boxes Bn of side Ln � nL�� Their numberwill be the cube of the integer part � LLn � and they will cover a volume inside


B whose complement has size � #L�Ln� The corresponding slightly smallerboxes B�

n will therefore cover a volume

� L� � #L�Ln � �L

Ln��#L�

n

r�

� jBj�� #LnL� #

r� Ln

��

If n�L are so large that

�� #

LnL� #

r� Ln

��

� � ��

we see that by �lling each box B�n with Nn � ��jBnj particles we would put

in B more than N particles �N � �jBj��Hence if we �ll each box B�

n with Nn particles until � NNn�Nn particles are

located and if we put in one of the remaining boxes a suitable number � Nn

of particles we shall have located exactly N particles inside B� They areput down in con�gurations with energy � e�jBnj in each box among the�rst � NNn

�Nn and in a regular but arbitrary con�guration in the last box� tocover as many points as necessary to reach the total correct number N �forinstance on a square lattice with density higher than ��Then it is clear that if L is large enough we shall have �lled the box B

with N particles with total energy � E � e V and with overall density ��at the same time we shall have shown the inequality

�

jBj logN�� e� B� � �

jBj�

logN�� e�� B�n��

NNn

� � Cn�

��

where Cn bounds the logN�� e�� coming from the partition function of theparticles in the last box Cn is the maximum of logN��

�� e�� for �� mjBnj

for � � m � Nn and e� suitably large� Once more the N � in the de�nitionof the microcanonical partition function �� is essential�This means� by taking into account the arbitrariness of n� that

"�� e�def� lim inf

L��NL�

�� EL�

�e

�

jBj logN�� e� B� � �n�� e��n�� e��

��

because �jBjCn��L�� and �

jBj�NNn

��L��

�jBnj �

Then the arbitrariness of e�� as well as the continuity of the function�� e� imply that "�� e� � �� e��But we can clearly repeat the argument by exchanging the role of B andBn namely we take a very large L so that �

jBj logN�� e� B� is very close

to the lim sup as L�� NL� � �� EL� � e of �jBj logN�� e� B� and we take

a very large n so that Ln � L and show by the same type of argument that

�� e� � lim supL��

NL�

�� EL�

�e

�

jBj logN�� e� B� � ��


This will imply

�� e� � limL��

NL�

�� EL�

�e

�

jBj logN�� e� B� � ��

completing the proof of the existence of the microcanonical entropy oversequences of cubic boxes�The extension to sequences of parallelepipedal boxes with all sides tending

to � at comparable speeds can be done along the same lines and we shallnot discuss it�

�F� Box�Shape Independence�

We just mention that the argument can be perfected to show that thelimit �� exists and equals �� e� over much more general sequences ofboxes�Imagine� in fact� paving space with cubes of side � � L� �one of which

is always centered at the origin� to �x the arbitrariness due to translationinvariance�� call B�� the union of the cubes of the pavement that containpoints of B� and B�� the union of the cubes of the pavement entirelycontained in B� Then we say that B tends to � in the sense of Fisher ifthere exist A�� such that

jB��j � jB��jjBj � A �

� �

diamB

��

��

This means that the box B grows keeping the surface small compared tothe volume �homothetic growth of a box with smooth boundary triviallysatis�es �� with � � ��Then it can be shown that� for �� e� in the interior of A� �� holds on

any sequence B �� in the sense of Fisher�If the potential has a hard core the same argument as above applies except

that the interval of variability of � is no longer �� but �� cp�� i�e� itis a �nite interval ending at the close packing density�The �nite range condition can also be eliminated and replaced by the

temperedness condition ��A complete treatment of all these remaining cases can be found in �Fi��

�Ru��The function �� e� is trivially related to the function s�v� u� of Ch�II

s�v� u� � v ��v�� v��u� ��

so that we see that s�v� u� is convex in v� at �xed u� and in u at �xed v��

� In fact by taking the second v�derivative of vf�v�� one sees that the convexity of f��implies that of vf�v��

� # IV� Thermodynamic Limit and Stability

This is also physically interesting as it shows that the derivative��s�u

�v

�

T�� is monotonic nonincreasing in u ��positivity of the speci�c heat� atconstant volume� and likewise behaves

��s�v

�u

� pT �

One can also see that the latter quantities are � �� as demanded by thephysical interpretation of T� p as� respectively� the absolute temperature andpressure� In fact one can prove the relations

�n�� e � �� n�� e� � � log��

e� e � � � �

�sv

�u��v��

��

which together with the convexity properties immediately imply the posi�tivity of T� p�

�G� Continuity of the Pressure

Consider the pressure as a function of the density at constant temperature�This is a function that is most conveniently studied in the grand canonicalensemble� Since pressure is a convex function of the chemical potential� �its derivative is monotonically nondecreasing �because convexity means thatthe �rst derivative is nondecreasing�� hence its graph has� at most� countablymany upward jumps and countably many horizontal plateaus� A verticaljump corresponds to a value of the chemical potential where the right andleft derivatives of the pressure are di�erent� while a horizontal plateau cor�responds to a straight segment in the graph of the pressure �drawing a fewschematic graphs is very helpful here��The de�nition of grand canononical partition function implies� by di�er�

entiating it with respect to �� that its �.derivative is the density� Thento say that there are no horizontal plateaus in the graph of the density asa function of the chemical potential is equivalent to saying that the graphof the pressure as a function of the chemical potential contains no straightsegment�A horizontal plateau in the graph of the density as a function of the chemi�

cal potential means that there are several chemical potentials correspondingto the same density�� Hence there are several pressures corresponding to

� Because �p � �Vlog "�� and "�� the grand canonical partition function� see

Ch�II� is " �P�

n��e��n

n�Z�n�� with Z�n the canonical partition function for n parti

cles� hence in general the logarithm of a sum of quantities cec�� with c � � is a convex

function of as one checks that the second derivative of this sum with respect to isnonnegative�

� Note that the derivative of the pressure with respect to the chemical potential is thedensity and� therefore� it should be strictly positive so that the pressure is strictlyincreasing with the chemical potential� But one has to show that the density is positiveif the chemical potential is negative enough� this is not so easy and it will be shownin complete generality in x�� see the sentence preceding �� where the relation � e�� O�e�� is given and interpreted as saying that the density is proportionalto the activity z � e�� at small activity i�e� at negative enough chemical potential�


the same density� i�e� a vertical discontinuity in the graph of the pressureas a function of the density�It is therefore interesting to see that such discontinuities of the pressure

as a function of the density at constant T are in fact not possible underthe only condition that the potential is superstable� see Ch�V� �� for ade�nition�A simple proof� due to Ginibre� �Gi�� can be given for systems with hard

core or� alternatively� with purely repulsive pair potential� The general caseof a supertsable potential is harder� �DM��Ru��

De�ne the activity as zdef� e��

p� m��

�h�� where the second factor

comes from integrating explicitly the kinetic part of the energy so thatthe grand canonical partition function 0 �in volume V and temperatureT � �kB�� can eventually be written

0 ��Xn�

zn

n�Zn�

see below� Then we estimate

��

�

�p

��

�

�p

��

��

��

�z�p

z

�

z

z

�

at constant � in the grand canonical ensemble it is �p � �V log 0� The

physical interpretation of � is clearly that of isothermal compressibility�If we can show that �� can be bounded away from �� for z � e�� in any

�nite interval then � is bounded away from zero in any �nite interval andthe graph of the pressure as a function of the density is continuous �in factLipshitz continuous� and as a function of the chemical potential it cannotcontain any straight segment�

The grand canonical density is � � hniV with hni def�

P�n� n

zn

n�Zn� � where

Zn �

ZV n

dx� � � � dxn e��P

i�j��xi�xj� ��#��

with 0 �P�

n��n�z

nZn� Hence� if �p � V �� log 0� then

� � z�p

z�hniV

and zhniz

� hn�i � hni��

furthermore

��

�

�p

��

�

�

�p

z��

z��

Hence

��

�z�p

z�z�

z��

�

�

hnihn�i � hni� ��

and we need a lower bound on hn�i�hni�hni �


Consider systems with hard core interactions� i�e� with a pair potential�x � y� equal to �� for jx � yj � a for some a � �� or systems with

repulsive potential � �� Let �q�def� �q� if �q� � � and �q�

def� �

otherwise�The key remark is that Zn��Zn� � Zn��Zn � D for some volumeindependent and continuous function D � D�� Hence in the hard corecase one can take D � e�B

R��e��x�� dx � �� if �B is a lower bound

for the energy of interaction �� of one particle with any number of others��

Setting ��X� the potential energy of the con�guration X this fol�

lows from the Schwarz inequality� abbreviating �x�� xn�def� X and

��X� x�def�Pn

j� �xj � x� � �B� we �nd

Z�n� �

� ZdXe��X�

Zdxe

��P

n

j��x

j�x��

� � Z dXe��X�� Z dX dx dye��X��X�x��X�y��

� Zn

ZdXdxdye��X�x�y��e��x�y� � � � ��

� ZnZn� � Zn

ZdXdxe��X�x�

Zdye��X�y�� e��x�y��

� ZnZn� � Zn

ZdXdxe��X�x�

Zdye�B�� e��x�y��

� ZnZn� � ZnZn�D� �� Zn�

Zn�� Zn�

Zn�D

where D � e�BR

�� e��y��dy� Therefore� using again the Schwartz

inequality �plus the normalization propertyP

nznZnn��

�hni�� zD�� Xn�

�

n�zn

Zn0

�zn�Zn�

znZn� nzD

��

�Xn�

�

n�zn

Zn0

�zn�Zn�

znZn� nzD��

��

so that developing the square the r�h�s� becomes

�Xn�

zn

n�

�

0

�z�Z�n�

Zn� z�DnZn� � z� n�D� Zn

�

� Therefore B � � if there is a hard core or if � � �� while if the potential is ofLennard�Jones type� or more generally a superstable potential� with an attractive partthere is no such B and the theory is more di�cult� see �Ru��


and using the last of �� this is bounded above� if pn � �zn�n�� Zn�0��by

�Xn�

zn

n�

�

0

�z�Zn� � z�DZn� � z�DnZn� � n�z�D�Zn

��

�

�Xn�

��n � ��n � �pn� � zD �n � ��pn��

� zDn�n � ��pn� � z�D� n�pn�

�

� hn�i � hni� zDhni� zD�hn�i � hni� � z�D�hn�i �

� �� zD��hn�i � �� zD�hnihence one �nds

hn�i � hni�hni � �

� � zD��

which proves a lower bound on � and� therefore� the continuity of the pres�sure as a function of the density at constant temperature�


�

��

Chapter V �

Phase Transitions

��

�

V� Phase Transitions ��

x�� Virial Theorem� Virial Series and van der Waals Equation

Van der Waals theory is one of the earliest and simplest applications ofclassical statistical mechanics ��#�� see �VW##�� Nevertheless it bringsup one more of its conceptual problems� although not as deep as the criticalproblems of Chap�III� It clearly indicates that one has to give up the naivehope that the theory of phase transitions and phase coexistence could beeasily quantitatively accessible�The classical approach starts from the virial theorem �Clausius�� Consider a

real gas with N identical particles with mass m in a spherical �for simplicity�container with volume V � suppose that the microscopic interaction potentialbetween two particles at distance r is a Lennard�Jones potential

�r� � ��r��r�� r��r�

��

where � is the interaction strength and r� is the diameter of the molecules�Let the force acting on the i�th particle be f

i� multiplying both sides of

the equations of motion m�qi

� fi

by � ��qi we �nd

��

NXi�

mqi� �q

i� ��

NXi�

qi� f

i

def�

�

C�q� ��

��

and the quantity C�q� is the virial of the forces in the con�guration q� notethat C�q� is not translation invariant because of the presence of the forcesdue to the walls writing the force f

ias a sum of the internal forces and of

the external forces� due to the walls� the virial C can be expressed naturallyas sum of the virial Cint of the internal forces �translation invariant� andof the virial Cext of the external forces� By dividing both sides by � andintegrating over the time interval �� one �nds� in the limit � � ��

hT i ��

hCi ��

which is read by saying that the average kinetic energy equals half theaverage virial of the forces�The virial naturally splits as the sum of the virial due to the internal forcesCint and that due to the external ones Cext� The virial of the external forcesis simply

hCexti � �pV ��

where p is the pressure and V the volume� Equations �� and ��constitute the virial theorem of Clausius�A quick proof of �� is that the external forces act only on the boundary

of the �spherical� box B containing the system they send back into the

�� V� Phase Transitions

container particles that try to get out� The average force that a surfaceelement d� of the walls exercises over each colliding particle is �pne d� if pis the pressure and ne the outer normal� Hence hCexti � p

R�B d�� ne��

p�V by Green s volume formula� A more re�ned argument leading to thesame result is possible��

Since the average kinetic energy is ��

��N we see that the equation of stateis

�

�� pv �

hCinti�N

��

if �� kBT and T is the absolute temperature and v the speci�c volume�The two relations �� and �� together with their corollary ��

constitute Clausius� virial theorem� The equation �� is essentially theequation of state� In the case of no internal forces it yields �pv � �� theideal gas equation�Van der Waals �rst used the virial theorem to perform an actual compu�

tation of the corrections� Note that the internal virial Cint can be written�if f

j�i� �q

i

�qi� q

j�

Cint �

NXi�

Xi�j

fj�i

� qi� �

Xi�j

qi�qi� q

j� � �q

i� q

j� ��

which shows that the contribution to the virial by the internal repulsiveforces is negative while that of the attractive forces is positive� To evaluatethe average of �� we simply use the theory of the ensembles and chooseto use the canonical ensemble� as it is more convenient�

� The force due the spherical container boundary can be represented as�

�x� � �

Z�B

d�� ne��F �x� �� a�

where F is a nonnegative scalar� ne�� is the outer normal to the boundary �B of B at

�� The function F is not zero and very intense only in a very tiny region near the origin�

so that �x� is not zero only very close to the boundary� We are really interested in thelimiting case in which the force F is a Dirac function� which represents the ideal caseof a perfect wall with no width�The virial of the external forces necessary to con�ne the system inside the box B is

�hXi

xi � �xi�i �

Z�B

d��ne�� h

Xi

F �xi � ��i �b�

where h�i denotes the time avergae� and having replaced ne�� xi by ne�� because of

the locality property of F �exact if is a delta function�� The average hP

iF �xi � ��i

is � independent because of the assumed spherical symmetry and it represents the force

exerted per unit surface area near �� i�e� it is the pressure p so that the average virial is

pR�B

d��ne�� pV � For an extension to nonspherical containers see �MP��


One could proceed by using time averages and the ergodic hypothesis�i�e� the microcanonical ensemble� but the result would be the same� Onecould also proceed by simply taking the few time averages that we reallyneed and argue that their value should coincide with the one we calculate as�suming the gas essentially free and adding corrections that take into accountthat close particles interact� This would be far weaker than the ergodic hy�pothesis and it was the path followed by van der Waals� however� unlike theequidistribution assumption� it does not lead easily to a systematic powerseries expansion in v�� of the corrections�

In the canonical ensemble the average internal virial is� taking into accountthe symmetry in q

iand denoting �q��q

�� q

N� �

PNj� �q

�� q

j�

hCinti �

�N

�Zdq

�dq

��

N �e��q

��q��q

N��q

��q

N��

� e��q��q�� q�

�q�� q

�� q

�� q

��

��

which can be rewritten �using an integration by parts� as

hCinti �

�N

�Zdq

�dq

��

N �e��q

��q��q

N��q

��q

N��

� ��

�q

�

�e��q��q��

� � �q�� q

��

Zdq

�dq

�

�

�� #��

�Z

dq��

�N � ��

�e��q��q��

�e��q

��q��q

N��q

��q

N��

�Z

dq�dq

�dq

�

Zdq

�� dq

N

�N � �� e��q� �q��qN ��q

��q

N��

� �e��q��q�� q

�� q

�� q

�

�q�� q

��

where the q�e��q

��q

�� has been replaced� before integrating by parts� by

q��e��q��q��

�to avoid boundary contributions in the integrations �in

fact e��q� is � at q � � when� eventually� we take the limit as V ��To rewrite ��#� in a better form it is useful to introduce the notion

of correlation function the k�points correlation function ��q�� q

k� is

de�ned so that ��q�� q

k�dq

��q

k

k� is the probability of �nding k particlesin the in�netisimal volume elements dq

�� dq

k� Hence� in the canonical

ensemble

��q�� q

k� � Z��N

Zdx� � � � dxN�k

�N � k��e��q��qk�x��xN�k� ��

where the normalization ZN is the canonical partition function� Note that��q

�� q

k� is not normalized to �� in fact

RV��q

�� q

k�dq

�� dq

k�

��# V� Phase Transitions

N�N � �� N � k � �� It is a simple matter of algebra to check that

hCinti ��

�

Zdq

�dq

�

�e��q��q��

�e��q��q��q

�� q

��

��

�

Zdq

�dq

�dq

�

�e��q��q��

��

� q�

�q�� q

�� q

�� q

��e��q��q��q

�� q

�� q

��

��

It can be shown� see x�� below� that the correlation functions of order kare� for � small� analytic functions of � divisible by �k and proportional to

e��q��qk�� in fact

��q�� q

k� � �k e��q��qk� �� F��q� � ��F��q� � � � ��

so that �� can be used as a starting point for a systematic expansion ofthe equation of state in powers of �� We do not discuss here the possibilityof the expansion �� not because it is di%cult� but because it wouldlead us into a technical question that certainly was not worrying people atthe time the above analysis was performed� we defer it to x��The physical meaning of the correlation functions of order k shows that

they should be proportional to �k and their de�nition �� shows that

they ought also to be proportional to e��q��qk�� Hence it is quite clearthat unless some integrals diverge� �� already allows us to evaluate the�rst correction to the gas law� We simply neglect the third order term in

the density and use ��q�� q

�� e��q��q�� in the second order term�

But there is no apparent reason for the integrals to diverge they contain

the factors�e��q��q��

�and q

��q

�� q

�� which tend to zero at large

arguments so that the divergence sources should be quite subtle� Abouthundred years after the original work of van der Waals the actual conver�gence of the series in �� and of the virial series has been mathematicallyproved� We shall discuss it from a modern viewpoint in the following x��Then

�

hCinti � V

�

��I�� V O��

where I��

R �e��q��

�d�q and the equation of state �� becomes

pv � I��v � O�v��

The calculation of I can be performed approximately if �� i�e� at�high temperature�� by imagining that �r� � �� i�e� e��r�� for r � r� and e��r� � � � ��r� for r � r�� One has

I � �

Z r�

�

��r�dr � �

Z �

r�

�r��r�dr �

� ��v� ��

��v� �

� ��b� �a�

��


with

v� ��

��r�

�� b � �v�� a ��

��v� � ��

Then it follows that pv � av � b

�v � �� so that

�p �a

v�� v � ��

b

v�

�

��

�

�� bv

�

�� O�

�

�v��

or �p � av� ��v � b�� O�v�� which gives the equation of state up to

O�v�� and for �� i�e� at high temperature and low density�It is in fact possible to compute� or at least to give integral representations

of the coe%cients of arbitrary order of the virial series

�p � v��

�Xp�

cp��v�p ��

and one can even show that the series converges for � small and v large�i�e� high temperature and small density� if the stability and temperednessconditions discussed in Chap�II� Chap�IV� �� hold�Equation �� can be compared with a well�known empirical equation

of state� the Van der Waals equation

��p � a�v��v � b� � � or �p � An��V ��V � nB� � nRT ��

where� denoting Avogadro s number by NA�

A � aN�A� B � bNA� R � kBNA� n � N�NA � ��#��

It is clear that �� and �� coincide up to quantities of O�v��hence �once an explicit form for like �� has been assumed as a gooddescription of the system� ��#�� show us how it is possible toaccess the microscopic parameters � and r� of the potential via measure�ments detecting deviations from the Boyle�Mariotte law �pv � � of therare�ed gases

� � �a�#b � �A�#BNA� r� � ��b� �� B� �NA��

Equation �� is� however� empirically used beyond its validity region�very large v� i�e� very small density� by regarding A�B as phenomenolog�ical parameters to be experimentally determined by measuring them neargeneric values of p� V� T � The result is that the values of A�B do not �usuallyvary too much� and� apart from this small variability of A�B as functions ofv� T � the predictions of �� have been in reasonable agreement with ex�perience until� as the precision of the experiments increased over the years�serious inadequacies eventually emerged�


A striking prediction of �� taken literally� is that the gas undergoesa �gas�liquid� phase transition with a critical point at a temperature Tc�volume vc and pressure pc that can be computed via �� and are givenby �see x�� table ��

RTc � #A� �B� Vc � �B �n � ��

The critical temperature is de�ned as the largest value Tc of the temperaturefor which the graph of p as a function of v is not monotonic decreasing� thecritical volume Vc is the value of v at the horizontal in$ection point occurringfor T � Tc�At the same time this is very interesting as it shows that there are sim�

ple relations among the critical parameters and the microscopic interactionconstants �� kBTc and r� �Vc�NA��

� � #�kBTc�� r� � �Vc� �NA��

if the model �� is used for the interaction potential � see the table inx�� On the other hand� �� cannot be accepted acritically not only be�

cause in its derivation we made various approximations �essentially neglect�ing O�v�� in the equation of state�� but mainly because for T � Tc thefunction p is no longer monotonic in v� and the latter is a thermodynamicfunction that in Chap�IV and Chap�II has been shown to be monotonic non�increasing as a consequence of the very general convexity of the free energy�evaluated for instance in the canonical ensemble fc�� v�� as a function ofv� i�e� �fc�v

� � � so that �p�v � �fc�v� � ��

If� nevertheless� the isotherms of �� are taken seriously even for T �Tc� by interpreting them as describing metastable states� then the �correct�equation of state can be obtained by noting that p as a function of v hasa horizontal plateau �vl� vg� in the situations in the Fig� �� Here theplateau associated with the represented isotherm is drawn� hence the densityundergoes a jump from vl to vg as the pressure decreases and vl� vg areinterpretable as the speci�c volumes of the liquid and of the gas�

vl vg

p

�

�

Fig� ��

v


The horizontal plateau must be drawn so that the areas �� are equal�The reason is that the reversible thermodynamic cycle obtained by havingthe system go through a sequence of transformations along the plateau andback along the curved parts of the isotherm would yield an output of workrepresented by the di�erence between the areas �if run in an appropriatedirection�� However it would be a Carnot cycle at constant temperaturewhich� by the second principle of Thermodynamics� should instead yield �work�This is the well�known Maxwell construction that� as we see� is motivated

in a rather obscure way because it is not clear whether is is really possible toperform the above Carnot cycle since it is at least doubtful� �LR�� that theintermediate states with p increasing with v could be realized experimentallyor even be theoretically possible �see� however� the theorem in x�� The van der Waals equation� re�ned and complemented by Maxwell s rule�

nevertheless provides a simple picture for the understanding of the liquid�gastransition in statistical mechanics� But it predicts the following behavior

�p� pc� � �V � Vc�� T � Tc

�vg � vl� � �Tc � T �� for T � T�c��

which are in sharp contrast with the experimental data gathered in thetwentieth century� For the simplest substances one �nds instead � � �� An accurate measurement of � and � is very delicate and this explains why�

for a long time� the equation of van der Waals has been considered a �goodrepresentation� not only for a high�temperature low�density gas regime butfor the liquid�gas transition regime as well� To gain an idea of the orders ofmagnitude of the constants A�B� hence of the microscopic interaction data�see the table at the end of x�� One should stress that the disagreement between theory and experiment

that we are discussing has a rather di�erent meaning and implications ifcompared with the discussions in Chap�III� The disagreement here is due tobad approximations �such as having neglected higher�order corrections inv�� in �� or such as having assumed that the virial series convergedeven for values of v� T close to the critical point��Here the disagreement does not involve fundamental questions on the foun�

dations of the theory it only involves the analysis of whether a certainapproximation is reasonable or correct� or not�One should remark� last but not least� that the blind faith in the equation of

state �� is untenable also because of another simple remark nothing inthe above analysis would change if the space dimension was d � or d � �but in the last case� d � �� one can easily prove that the system� if theinteraction decays rapidly at in�nity� does not undergo phase transitions�a fact usually known as Landau�s argument� see x�� in �LL�� and whichcan be made into a mathematical theorem proved� as such� by van Hove��VH�� see x��# below�


In fact it is now understood that the Van der Waals equation representsrigorously only a limiting situation� in which the particles have a hard coreinteraction �or a strongly repulsive one at close distance� and a furthersmooth long�range interaction very small but with very long range� Thisis discussed in x�� As a �nal comment it is worth stressing that the virial theorem gives in

principle the corrections to the equation of state in a rather direct andsimple form as time averages of the virial of the internal forces� Since thevirial of the internal forces is easy to compute if one knows the positionsof the particles as a function of time we see that the theorem provides amethod for computing the equation of state in numerical simulations� Infact this idea has been exploited in many numerical experiments� in whichthe �� plays a key role�

x�� The Modern Interpretation of van der Waals� Approxima�

tion

Suppose that the system has an interaction potential �r� � hc�r�r�� r�r�� where hc�r�r�� vanishes for r � r� and is �� for r � r� �hardcore potential�� while � is a smooth function with short range �i�e� eithereventually equal to � as r � �� or tending to � exponentially fast asr � �� say�� Here � is a dimensionless parameter which is really used toset a variable value of the range to ��r��In other words we assume that� apart from the hard core� the particles

interact via a potential which is very long range as � � � but� at the sametime� it becomes very weak�When � is very small and the density of the system is �xed to be � we see

that the energy of interaction between one particle and the remaining oneswill be essentially entirely due to the particles that are very far apart theclose ones� being �relatively� few� will therefore contribute a small amount tothe energy� because the strength of the potential is very weak� proportionalto ��The energy of a particle in the force �eld of the others is in fact� if � is

the ��independent� integral of ��r�r��

�

Z��

r

r�� d�r � �

Z�

r

r�� d�r

def� ��

so that the energy of a con�guration in which the hard cores of the particlesdo not overlap will be essentially given� at least for small �� by

U ��

N � � � ��

The quantity � � is sometimes called the mean �eld at density ��The last relation allows us to compute immediately the canonical partition

function� Let V � Nv� be the volume available to each particle� i�e� the


total volume minus the volume occupied by the impenetrable hard cores ofthe particles v� is of the order r�� and it will be taken� to be in agreementwith �� to be

b � �v� � ��

��r�

��

Then if the energy of a con�guration is well approximated by �� thecanonical partition function is approximately

Z�� v� �

Ze��

p�

�m� �� N�

d�Nq d�Np

h�NN ��

�� m

�h�

� �N� e��

�� N �V �Nb�N

N �

��

where a somewhat uncontrolled approximation is made about the q integra�tions� as clearly the integral over the con�gurations of N particles that areconstrained to stay at a distance r� from each other is a highly non trivialquantity only naively can one hope to approximate it by �V �Nb�N � evenif we allowed simple adjustements of the value of the empirical �excludedvolume� b to improve the approximation�

If one accepts �� then the equation of state can be computed straight�forwardly� In fact the free energy fc� see � ��#�� is given by

�� fc�� v� � limN��

�

NlogZ�� v� �

� log�p

�m��h��

v�� log�v � b�e

��

leading� by di�erentiation� to

� p�� v� � ��fcv

��

v��

�

v � b��

which coincides with �� thus providing an alternative interpretationof the van der Waals equation and motivating the quali�cation� which isusually given to it� of mean �eld theory�

The above discussion shows that the van der Waals equation can be exactonly if the interaction has extremely long range and at the same time justweak enough to have a nonzero integral � so that it is correspondinglyso small that any individual particle contribution to the potential energyof a �xed particle is negligible apart� of course� from the hard core energywhich� unsatisfactorily� is taken into account by replacing the integral overthe con�gurations of non overlapping cores by �V �Nb�N �

In fact the latter approximation can be eliminated by replacing �� by


the more accurate

Z�� v� �

Ze��

p�

�m �� N� d�N

h�NN �dq �

��r �m

�h�

��Ne�

�� NZ��v��

Z��v� �

Zjqi�q

jjr�

d�Nq

N �

��

where the con�guration integral Z��v� �which is ��independent� over thenonoverlapping hard core con�gurations fq

ig in the volume V is not com�

puted�Equations �� then imply that the free energy and the equation of state

of our gas are

��f�� v� � ��f�� v��

v�

�p�� v��

v�� P��v� �� #��

where

P��v�def� �

v

�

NlogZ��v� ��

is the �temperature�independent� product of � times the pressure p� ofthe hard core gas� It replaces its crude approximation P��v� � �p��v� ��v � v��

�� in �� It is not di%cult to see that the ��independence of P��v� implies that� if� � �� i�e� if the potential has a long range attractive �i�e� negative� tail�then �� #� will have� at low temperatures� a graph which is qualitativelysimilar to that of �� with a � � �hence like Fig� ��Thus the equation of state �� #� will show phase transitions� and also the

phenomena of negative compressibility and metastability�The negative compressibility can be eliminated by Maxwell s rule� But one

is still left with the unpleasant feeling that somehow one is doing somethingwrong� This is clearly signaled by the fact that in spite of the improvementsin the approximations we are still getting a pressure that is a nonmonotonicfunction of the speci�c volume �if � is large enough� i�e� if the temperatureis low��At least in one�dimensional gases the excluded volume problem is trivial

and one can simply check that �p�� v� � P��v� is indeed �v�v� with v� � r��

and� therefore� this is clearly a contradiction because we are getting a nonmonotonic pressure in a situation in which the theory of x�� does apply�and implies convexity of the free energy� i�e� monotonicity of the pressure�Continuing to denote by P��v� the temperature�independent product�p�� v� of � times the pressure of the pure hard core gas� the followingresult sheds a great amount of light on the intricacy of the above situation�showing that the presence of a negative compressibility region is an artifactof the mean �eld approximation


Theorem Suppose that we �x � � � and we call p�� v� �� the canonicalpressure �in the thermodynamic limit� for the gas interacting with the po�tential

�r� � hc� rr�

��

��r

r�

��

where hc � � for r � r�� hc � � for r � r�� is a smooth potentialrapidly decreasing at � and with integral � � �� Let also P��v� be the ��independent� product of � times the pressure of the pure hard core gas �inthe thermodynamic limit�� i�e� the pressure of the gas in which the particlesinteract only via the hard core potential �� Then

� p�� v�def� lim

�� p�� v� ��

h� �

v�� P��v�

iMaxwell rule

��

where the subscript �Maxwell rule� means that in the regions where theright�hand side is not monotonic in v �existent if � � � and � is large�the pressure p�� v� is obtained with the help of the Maxwell constructiondiscussed in x ��Thus we see that there is indeed a �rm foundation to Maxwell�s rule which

does not rest on dubious Carnot cycles the van der Waals equation becomesrigorously valid in the limit in which the attractive tail of the potentialbecomes very weak but with so long�range that the mean potential ��mean�eld�� see �� that it generates in a point has a �xed value�If the dimension is d � � the hard core gas pressure P��v� is rigorously�p� � �v � v�� and the equation of state becomes exactly the Van derWaals equation� In higher dimensions �p� � �v � v�� is only an approxi�mation �no matter how v� is chosen�� but the basic fact that the equation ofstate is a trivial modi�cation of a reference� �simpler� �so to speak�� system�the hard core gas� together with Maxwell s rule remains valid�One can also say that the van der Waals equation arises when one inter�

changes two limits the thermodynamic limit and the limit of in�nite range� � �� It is obvious that if instead of taking the limit V � � �rst andthen the limit � � � one did the opposite then the equation of state wouldhave been p � p� and the attractive tail would have given no contribution�The potentials like �� are called Kac�s potentials� �HKU�� and one

can say that the above theorem plays a role analogous to that of Lanford stheorem for the Boltzmann equation� see x��# and x�� in both cases astatement that has approximate validity becomes exact in a suitable limit�And in both cases the statement seems incompatible with obvious propertiesof the system �reversibility in the �rst case and strict convexity in �nitevolume systems of the free energy in the second�� although of course the�rst case concerns a far more fundamental problem than the second�But both cases are instances of a method of analysis that has been devel�

oped very much in the twentieth century� in which one tries to understandsome properties that cannot be exactly true in normal situations but thatbecome exactly true in suitable limiting situations thus leading to a more


or less detailed understanding of why they may look true even when thelimit is not taken� Of course a complete theory should also come togetherwith estimates �and possibly reasonable ones� of how far we are� in concretesituations� from the limiting cases �i�e� how big are the corrections on thequantities on which we might be interested��

The method is a modern interpretation of the basic conception of Boltz�mann on the relation between the apparent continuum of reality� as weperceive it and input it in most of our models or theories for its interpreta�tion and understanding� and the possibly intrinsic and deep discrete natureof reality and of our own thinking�

This is exempli�ed by the quotation in x�� and in many others amongBoltzmann s writings� for instance

�The concepts of di�erential and integral calculus separated from any atom�istic idea are truly metaphysical� if by this we mean� following an appropriatede�nition of Mach� that we have forgotten how we acquired them��

And I cannot resist the temptation of more quotations� as this is reallymusic for the mind

�Through the symbols manipulations of integral calculus� which have becomecommon practice� one can temporarily forget the need to start from a �nitenumber of elements� that is at the basis of the creation of the concept� butone cannot avoid it��

or

�Di�erential equations require� just as atomism does� an initial idea of alarge �nite number of numerical values and points �� Only afterwardsit is maintained that the picture never represents phenomena exactly butmerely approximates them more and more the greater the number of thesepoints and the smaller the distance between them� Yet here again it seemsto me that so far we cannot exclude the possibility that for a certain verylarge number of points the picture will best represent phenomena and thatfor greater numbers it will become again less accurate� so that atoms do existin large but �nite number� see p� � in �Bo��

and

�This naturally does not exclude that� after we got used once and for all tothe abstraction of the volume elements and of the other symbols �of calculus�and once one has studied the way to operate with them� it could look handyand luring� in deriving certain formulae that Volkmann calls formulae forthe coarse phenomena� to forget completely the atomistic signi�cance of suchabstractions� They provide a general model for all cases in which one canthink to deal with �� or ��

��

elements in a cubic millimeter or even withbillions of times more� hence they are particularly invaluable in the frameof Geometry� which must equally well adapt to deal with the most diverse


physical cases in which the number of the elements can be widely di�erent�Often in the use of all such models� created in this way� it is necessary toput aside the basic concept� from which they have overgrown� and perhaps toforget it entirely� at least temporarily� But I think that it would be a mistaketo think that one could become free of it entirely��

The latter sentence� p� �� in �Bo�� reminds us that the evaluation of thecorrections is of course a harder problem� which it would be a mistake to setaside� even in the above case of mean �eld theory� In fact the correctionsare quite important and somehow even more important than the mean �eldtheory itself� which will remain as a poor idealization of far more interestingcooperative phenomena�

It should also be noted that the above analysis does not allow us to solvea fundamental question can classical statistical mechanics predict and de�scribe phase transitions� We have seen that the van der Waals theory isno proof that when no in�nite range mean �eld limit is taken �i�e� in mostinteresting cases� then a system can show phase transitions�

This clearly emerges from the remark that if � is not � then in d � � one canprove� as a theorem �see above� and x��#�� that the system cannot undergoany phase transition whatsoever �and the pressure is strictly monotonic inthe speci�c volume� �no plateau� at all�� and nevertheless the above theoremalso holds in d � �� where in fact it leads precisely to the Van der Waalsequation �with the volume of the box being replaced by the length of thebox and similar obvious changes��

It is therefore important to see whether genuinely short�range models �no� around� generate equations of state with phase transitions� This will bethoroughly discussed in Chap�VI in simpli�ed models� because in the casesin which one would like to have results the problem is still open� and we shallsee that in the simpli�ed models phase transitions are possible even whenthe interactions have short range and the analysis will leave little doubt �infact no doubt at all� that phase transitions are possible in classical statisticalmechanics� without the necessity of introducing any new assumptions or newphysical laws�

We should however mention that important breakthroughs seem to be un�der way see �Jo�� and �LMP�#��

The standard approach to the van der Waals theory �also called mean �eldtheory� can be found in �CC�� p� #�� A more re�ned and interestingformulation is in �VK�� A precise and very clear theory is in �LP�� The�rst precise understanding �and full proof in particular cases� of mean �eldtheory is in �HKU�� in a series of papers reproduced� with introductoryremarks� in �LM�� A more phenomenological but very interesting andoriginal theory is in the book �Br�� where the most common phase tran�sitions are treated from the unifying point of view of the mean �eld theory�The original work of van der Waals has been reprinted� �VW##��

� Lucretius would not have said it better�


x�� Why a Thermodynamic Formalism�

In the next sections we devote ourselves to a more detailed analysis of theframework in which phase transitions could be placed and of the techniquesthat one may envisage to apply towards an understanding of their proper�ties� It will be a rather abstract analysis� usually called �thermodynamicformalism� that plays in statistical mechanics a role akin to that played bythe Hamiltonian formalism in mechanics�

One does not have to recall that the formalism of Hamilton� in itself� doesnot make mechanics problems any easier than other formalisms� Howeverit has become a tautology that it is a very appropriate formalism to de�scribe mechanical phenomena� The same can� or should� be said about thethermodynamic formalism�

The theory of orthodic ensembles provides us with a model of Thermody�namics but� strictly speaking� only in the limit of in�nite volume� In thissituation one also obtains equivalence between the various ensembles� seeChap�II�

The elements of the orthodic ensembles describe in great detail the struc�ture of the thermodynamic phases �i�e� macroscopic states�� well beyond thesimple microscopic de�nition of the classical thermodynamic quantities� andeven provide us with the �surprising� possibility of computing theoreticallysome relations between them �e�g� the equation of state�� Every elementof a statistical ensemble describes details of the microscopic con�gurationsthat are typical of the corresponding phase� because it gives the probabilityof each individual microscopic con�guration�

The problem of the �thermodynamic limit� theory is that of establishinga formalism in which it becomes possible to make precise and sharp variousstatements that we have made so far� on intuitive or heuristic grounds� andthus lay the grounds for a deeper analysis and for deeper physical questions�

We shall only consider the case of classical statistical mechanics� in whichone neglects the size of Planck s constant h�

What follows� as stated at the end of the previous section� is a formalismas with all formalisms it has interest only because it provides a naturalframe �as experience taught us� in which the discussion of the most impor�tant questions and applications can be situated� This is not the place toargue that this is the best formalism others are possible and in the endequivalent� But we need a formalism just in order to formulate precise ques�tions� suitable of being given quantitative answers� The amount of work tobe done will be independent of the formalism used �of course��

It is well known that for each class of problems the formalism in which theyare formulated often has a clarifying and unifying role the emergence of a�good� formalism is often successive to the solution of important problemsin the �eld� This seems to be the case of the thermodynamic formalism andthe following few sections should be understood from this viewpoint�

As an example of the problems that it would be premature to formulatewithout a clear formalism in which they �t one can quote


�� describing the spatial correlations between particles in a gas�

� � describing �and in fact de�ning� the surfaces of separation betweendi�erent� but coexisting� phases�

�� understanding the formation and dissociation of gas molecules or atomsinto their more primitive constituents in stationary state situations� andother cooperative phenomena�

A initial question is in which sense an element of a statistical ensembledescribes a probability distribution on phase space once the limit of in�nitevolume has been taken� We shall consider here only the grand canonicalensemble representation of the equilibrium states� because it is somewhateasier to discuss than the canonical or microcanonical� We examine the caseof a system of identical particles with mass m enclosed in a cubic box V �The particles interaction will be assumed to take place via a potential

satisfying at least the stability and temperedness conditions �� thatare necessary according to the analysis of Chap�IV in the theory of theensembles i�e� ��q

�� q

n� �

Pi�j �q

i� q

j� � �Bn ��stability�� and

j�r�j � Cjrj�� for jrj � r� � � ��temperedness�� B�C� � � ��To avoid several technical problems it will also be convenient to suppose

that the potential has a hard core with diameter r�� i�e� it is de�ned as thesum of a smooth potential plus a singular potential which is �� for jrj � r�so that �r� � �� for jrj � r�� This has the physical signi�cance that twoparticles cannot be closer than r�� but a large part of what will be discusseddoes apply� with suitable modi�cations �and several open problems left�� tothe case of a superstable potential� This is a potential such that there aretwo constants A�B � � such that

��q�� q

n� � �Bn � An��V if q

�� q

n� V ��

where V is an arbitrary cubic volume containing an arbitrary number n � of particles located at q

�� q

n� The Lennard�Jones potential� see ��

is a typical example of a superstable potential� However the potential � ��the free gas model� is not superstable �although it is trivially stable�� see�Ru�� for a general theory of such potentials�Let V be a cubic volume and consider the element ��V � of the grand

canonical ensemble with parameters �� and with particles con�ned inV � � ��kBT � kB � Boltzmann s constant and T � temperature� � �chemical potential� see x �� The probability of �nding n particles in themicroscopic state dp

�� dp

ndq

�� dq

nin the distribution ��V � is

��V ��dp�� dq

n� �

e��E�p�q��n�

0��

dp�� dp

ndq

�� dq

n

n�h�N��

where E�p� q� � T �p� � ��q� �Pn

i� p�i� m � ��q

�� q

n� and 0 is the

grand canonical partition� see x �� We want to take the limit of �� as


V �� and interpret it as a probability distribution on the in�nite systemcon�gurations that one reaches in this way�One begins by giving a precise de�nition of the in�nite system con�gura�

tions� then comes the problem of giving a meaning to the limit as V � �of �� and �nally one will want to characterize the distributions that arefound by following this limiting procedure starting from �� or start�ing from the more general grand canonical distributions with �xed externalparticle boundary conditions� the latter were introduced� as generalizationof �� in x �� see � ��

x�� Phase Space in In�nite Volume and Probability Distribu�

tions on it� Gibbs Distributions

It is natural to de�ne the phase space M in in�nite volume as the spaceof the sequences �p� q� � �p

i� qi��i� of momenta and positions such that

in every �nite �cubic� volume there are only �nitely many particles� calledlocally �nite con�gurations if we consider systems of particles with hardcore of diameter r� � � this will be �automatic�� as the only con�gurationsq that we have to consider are those with jq

i� q

jj � r�� for i �� j�

However� to take into account microscopic indistinguishability the con�gu�ration space will not be M � but the space M obtained from M by identifyingsequences �p� q� di�ering by a permutation of the particles�A probability distribution on M is in general de�ned so that the following

question makes sense what is the probability that by looking in a given �cu�bic� volume V � one �nds in it exactly g particles with momenta in dp

�� dp

g

and positions in dq�� dq

g�

Therefore the probability distribution � will be characterized by the func�tions fV ��p

�� p

g� q

�� q

g� such that the quantity

fV ��p�� p

g� q

�� q

g�dp

�� dp

gdq

�� dq

g

g��

is the probability just described� The functions fV � will be called the localdistributions of � the factor g� could be included in fV � � but it is customarynot to do so since the particles are indistinguishable and this factor simpli�escombinatorial considerations�By using the functions fV � it will be possible to evaluate the average value

of a localized observable� localized inside the volume V � this is� by def�inition� a function on phase space that depends on �p� q� � M only via

the state of the particles located in V �� Adopting the convention that

fpgg� fqgg def� p�� p

g� q

�� q

g� if F is such a local observable we can

write its average as

hF i ��Xg�

ZR�g�V g

�

F �fpgg� fqgg� � fV ��fpgg � fqgg�dfpgg dfqgg

g��


Consider a probability distribution like �� describing a particle systemenclosed in a �large global� volume V that we suppose cubic� With a �xedV � � V �we think here of V as huge and V � as much smaller�� one cancompute the probability that inside V � the con�guration �p� q� will consistin g particles in dp

�� dp

gdq

�� dq

g� Once the appropriate integrals are

performed one will �nd necessarily an expression like

f�V �V � �p

�� p

g� q

�� q

g�dp

�� dp

gdq

�� dq

g

g��

It is then natural to de�ne the limit as V �� of the probability distribution��V �� as the distribution � on M characterized by the localdistributions

fV ��p�� p

g� q

�� q

g� � lim

V��f�V �V � �p

�� p

g� q

�� q

g� ��

provided the limit exists for each V ��It can be shown that if the interparticle potential is superstable� see

�� and hence a fortiori if it has a hard core� then the limit �� exists�at least along subsequences of any sequence of volumes V with V �� Inthe hard core case this is an almost obvious �compactness argument� �i�e� a�free� argument based on abstract nonsense��

The same remains true if ��V � is replaced by a more general element of agrand canonical ensemble with �xed external particle boundary conditions�provided the external particle density �does not grow too fast with theirdistance to the origin�� see x ��The latter condition means that� �xing a length unit � �arbitrarily�� the

number n�� of external particles in a box � with side size � does not growtoo fast with the distance d�� O� of � from the origin� e�g� it satis�es

n��

d�� O��d��O��

This condition is automatically satis�ed if the interaction has hard core� ifit is not satis�ed then it is not di%cult to �nd a con�guration of externalparticles such that the above limit does not exist �or is �unreasonable��

� When one imagines the volume of the �global container increasing to � keeping thesize of the region V � that is under scrutiny �xed�

� On the contrary it is highly nontrivial� when true� to prove the existence of the limitV without restricting V to vary along a �suitable subsequence�

� Consider in fact a system of particles interacting via a LennardJones potential� ��which is � �b for distances between a and �a� Let V be a cubic container and distributeoutside V � at distance exactly a� M � Nc external particles� c � �� Inside the volumeV suppose that there are N particles with N � V � � �� With such boundarycondition the canonical distribution has a thermodynamic limit in which there are noparticles in any �nite region with probability � �i�e� fV � � � for all V �� In fact onechecks that� putting allN internal particles in a corridor of width a around the boundary�one gets a set of con�gurations with energy � O��NNcV �� and phase space volume

O��aV ��NN �� Then a comparison argument similar to those of x�� to study thevarious catastrophes applies� Likewise one can discuss the corresponding example in thegrand canonical ensemble �or the microcanonical��


Therefore we can de�ne the set of Gibbs distributions on the phase space Mas the set of all possible distributions that are obtainable as limits of conver�gent subsequences� in the sense of �� of grand canonical distributions��V � with periodic boundary conditions or with �xed external particleboundary conditions whose density does not grow too fast at in�nity in theabove sense�The distributions that are obtained in this way will de�ne the equilibriumphases of the system �see x �� and are not necessarily invariant under trans�lations� i�e� such that for every displacement � � R�

fV ��p�� pg � q� � �� qg

� �� fV ��p�� p

g� q

�� q

g� � ��

Except in the �important� special case in which periodic boundary coondi�tions are used� translation invariance symmetry is broken by the fact thatthe system is� before the thermodynamic limit V �� enclosed in a �nitebox V � and it is not necessarily true that the invariance is �restored� by themere fact that we send V ��The physical phenomenon related to the above �possible� spontaneousbreakdown of translation symmetry is the possibility of the existence ofthermodynamic states in which pure phases coexist occupying� for instance�each half of the total space allotted to the system� being separated by amicroscopically well�de�ned surface one should think here of a liquid inequilibrium with its vapor�Therefore we shall distinguish the set of Gibbs distributions G�� from

its subset G�� G�� consisting in the distributions which are invari�ant under translations� i�e� which have local distributions satisfying ��If � is a translation�invariant probability distribution on M and if S �

�S�� S�� S�� are the translations by one length unit� in the three directions�i�e� S��p

i� qi��i� � �p

i� qi

� e�� with e�� being the unit vectorin the ��th direction�� then the triple �M�S� �� is� according to a well es�tablished terminology� a dynamical system� this is a useful fact to bear inmind� as we shall see on several occasions�One could� of course� de�ne the Gibbs distributions by starting from distri�

butions of the canonical ensemble �or microcanonical or any other orthodicensemble� with �xed external particle boundary conditions�By so doing one would generate the problem of the equivalence of the en�

sembles �see Chap�II� in the sense that one should show that the totality ofthe Gibbs distributions on M built starting from the grand canonical en�semble distributions with �xed external particles boundary conditions doescoincide with the totality of the Gibbs distributions �on M� built start�ing with microcanonical or canonical distributions with periodic or �xedexternal particle boundary conditions�The analysis of the latter question is di%cult it is essentially complete only

in the case of hard core systems� �Do�#a��Do� �� LR�� but it is somewhatincomplete in the �general� case of superstable potentials� see �Ru��La� ��Ge�� Nevertheless there is no evidence that there might be conceptualproblems on such matters�


x�� Variational Characterization of Translation Invariant Gibbs

Distributions

If we restrict our attention to the translation�invariant Gibbs distributions� � G�� also called homogeneous phases� then an alternative and inter�esting variational characterization of them is often possible�The �rst simple remark� that stems immediately from �� or from

its variants with di�erent �xed external particles boundary conditions�is that if � � G�� then the momenta distribution is Maxwellian�i�e� fV ��p

�� p

g� q

�� q

g� can be written as

fV ��p�� p

g� q

�� q

g� �

e��

gPi�

p�i��m

p �m��

�g fV ��q�� q

g� ��

where the factor in the square root is introduced because it provides an ob�viously convenient normalization� making f a quantity with the dimensionof an inverse length to the power �g since the quantity

�� p

�m��

is a �momentum�� Sometimes one de�nes instead �� p

�m��h��including in it also the factor h��g that appears in � � �� with this choicefV� would be dimensionless�The probability distributions on phase space M with local distributions

that depend upon the momenta as in �� are called Maxwellian distri�butions� The problem is therefore that of characterizing fV � so that thedistribution de�ned by �� is in G�� Going back to a �nite total volume a well�known argument shows that

�� satisfy a variational principle� More precisely let �p� q� abbreviate�p

�� p

n� q

�� q

n�� and write �� as

f�p�� p

n� q

�� q

n� dp

�� dp

ndq

�� dq

n� fn�p� q� dp dq ��

and set

En�p� q� �nXi�

p�i

m��q

�� q

n� � Tn�p� � �n�q� � ��

��

Then consider the functional J�f� � S�f� � �U�f� � ��N�f� de�ned onthe functions f

J�f�def�

�

V

�Xn�

Z�fn�p� q� log fn�p� q�

dp dq

n��

� �

V

�Xn�

Zfn�p� q��En�p� q� � �n�

dp dq

n��

��

V

�Xn�

Zfn�p� q��log fn�p� q� � �En�p� q� � ��n�

dpdq

n��


By the Lagrange multiplier method one checks that J�f� is stationary �ac�tually a maximum� on the set of the f � � such that

�Xn�

fn�p� q�dpdq

n��

if� still with the notation �p� q� � �p�� p

n� q

�� q

n�� the functions fn

satisfy

�� log fn�p� q�� En�p� q� � �n�� constant ��

i�e� if f is given by �� It is natural� at this point� to introduce the space of all translation invariant

distributions � on M that have a Maxwellian momentum distribution andto de�ne on this space the following functionals the �speci�c volume�� the�total energy� and the �potential energy� corresponding to the interparticlepotential � and the �entropy��We denote such functionals by v�� u�� u�� and s�� respectively�

and we write them �rst in the general case and then we shall consider theexpression that they assume when the fV � have the Maxwellian form ��To simplify we also abbreviate the notation for the local distributions in

the volume V �� see �� as

fV ��g�p� q� � fV ��p�� p

g� q

�� q

g�

fV ��g�q� � fV ��q�� q

g�

��#��

where �p� q� stands for �p�� p

g� q

�� q

g� and dpdq � dp

�� dp

g

dq�� dq

g� Then the speci�c volume of � will be de�ned by

v�� limV ��

�

V �

�Xg�

ZgfV ��g�p� q�

dpdq

g��

� limV ��

�

V �

�Xg�

ZgfV ��g�q�

dq

g��

��

With the notations in �� the total energy will be

u�� limV ��

�

V �

�Xg�

Z�Tg�p� � �g�q�� fV ��g�p� q�

dpdq

g��

� limV ��

�

V �

�Xg�

Z�

�

�g � �g�q�� fV ��g�q�

dq

g��

� �

�v�� u��

��


where in the �rst step the gaussian� hence trivial� integrals over the momentap have been performed explicitly and in the second step we used ��Likewise the thermodynamic entropy is

s�� limV ��

� �

V �

�Xg�

ZfV ��g�p� q� log fV ��g�p� q�

dpdq

g��

� limV ��

� �

V �

�Xg�

Ze��Tg�p�p

�m��h��g fV ��g�q��

� ��Tg�p�� g

log� �m��h�� log fV ��g�q��

dq dp

g��

� limV ��

� �

V �

�Xg�

ZfV ��g�q�

dq

g��g�

� ��g

� �g

log � �m��h�� log fV ��g�q��

�� v�� log � � em��h�� s��

All limits above do exist in the case of systems with hard core potentialsto prove this the techniques are similar to those used in Chap�IV to discussthe existence of the thermodynamic limit� The limits� however� exist undermuch more general conditions that we shall not discuss here�We now maximize� on the space of the Maxwellian translation�invariant

distributions � on M �with inverse temperature �� i�e� having the form�� the functional

s�� v�� u��

and let � p�� denote the supremum of �� We proceed by quoting only results that are valid in the case of hard coresystems� to avoid discussions on the more general superstable case �for whichsimilar� but less satisfactory results can be obtained�� the general discussionbeing somewhat technical� �Ru�� If has a hard core one has

� p�� max�

�s�� v�� u��

��

and the maximum is reached exactly on the translation�invariant Gibbs dis�tributions � � G�� and only on them� see �Do�#a��LR�� One cancheck that the meaning of the maximum value p�� is that of �pressure��leaving aside mathematical rigor this is� in fact� quite clear from the dis�cussions in x �� and above��The variational property �� has been heuristically based on the men�

tioned check �see �� that the functional �� leads� in a �nite

� In the quoted papers one considers lattice systems� see x�� below� but the techniquesand results can be extended to hardcore systems quite straightforwardly�


volume container with �open boundary conditions� �i�e� no �xed particlesoutside the container�� to the element �� of the grand canonical ensem�ble� However a remarkable feature of �� is that nevertheless it happensthat the solutions of the maximum problem �� contain� as well� thetranslation invariant Gibbs distributions that can be obtained by imposinggeneral �xed external particle boundary conditions and subsequently con�sidering the thermodynamic limit of the distributions so obtained� �Do�#a��Do�#b�� Do� �� LR��It can be checked �this is a simple but nontrivial theorem� that the set of

translation�invariant probability distributions that realize the maximum in�� i�e� the set denoted G�� of the Gibbs distributions with inversetemperature � and chemical potential �� form a convex set �i�e� �� G�� implies a�� a�� G�� for all a � �� Furthermorethe convex set is actually a simplex� i�e� such that every � � G�� can berepresented uniquely as a convex superposition of extremal distributions inG�� The statistical interpretation of a �convex� superposition of two probability

distributions is that of a mixture hence the meaning of the latter describedproperty is interesting� It says� in other words� that if the extremal distri�butions of G�� are interpreted as the pure homogeneous �i�e� translationinvariant� phases� then all the other elements ��homogeneous phases�� inG�� are mixtures of pure phases and they can be represented as such ina unique way�

For instance if G�� contains only two extremal elements � and �� the�rst representing the �liquid phase� and the second the �gaseous phase��then every other distribution in G�� can be represented as a� � �� a�� with � � a � �� and a has the interpretation of fraction of mass of theliquid phase�

It is remarkable that it is possible to prove that the extremal states � ofG�� enjoy the property of ergodicity in the sense that the above de�ned�see x�� corresponding dynamical systems �M�S� �� are �ergodic� and theyare the only points in �M�S� �� with this property� see �Ru��The ergodicity property is the natural generalization of the notion intro�

duced in the discrete evolution cases of the systems in Chap�I� We considera family of commuting invertible transformations� S � �S�� Sn� acting

� We shall only consider here and in the rest of the book measurable transformations�measurable functions� measurable sets� These are rather delicate notions� on the brink ofthe imponderable because to �nd nontrivial examples of nonmeasurable correspondingobjects one needs the sinister axiom of choice� However if one wants to discuss notionslike ergodicity in systems that are not regarded as discretized� abandoning Boltzmann s wise discrete conception of the world� one must say a few words on measurability�The spaces M�M � � � � that we consider here and later will all have a natural notion of�closeness bewteen points� a topology in Mathematics� typically a metric can be de�nedon them �this metric can be de�ned but often it is not really useful so that it is notalways explicitly de�ned as there is little doubt about what it could be�� Therefore itmakes sense to de�ne open sets� One declares all of them measurable� more generallythe smallest family of sets that contains all the open sets and that is closed under theoperations of countable union� complementation and intersection is by de�nition the


on a space M so that setting Sk � Sk�� Sk�� Sknn � with k � �k�� kn�an n�ple of integers� we can de�ne Skx for x �M � We say that a probabilitydistribution � on M is S�invariant if for every measurable set E � M it is��SkE� � ��E�� The triple �M�S� �� is called a discrete dynamical systemand Sk is called a �translation of k by S� �here S�

j is the identity map��AA�#�

De�nition �ergodicity� Let �M�S� �� be a discrete dynamical system� it is�ergodic� if there are no nontrivial constants of motion� i�e� no measurablefunctions x� F �x� on a phase space M which are invariant under �trans�lation by S� and which are not constant as x varies excluding� possibly� aset of zero ��probability�

The de�nition can be extended in the obvious way to the case in whichS is a continuous �ow� i�e� k � Rn and Skj are commuting transformations

which satisfy the group property Skj Shj � Shkj for all h� k � R and S�

j isthe identity map�The above dynamical system with M being the phase space points of an

in�nite system� with the �evolution� S being the spatial translations andwith � being a Gibbs state� is ergodic if one cannot �nd observables that aretranslation invariant and at the same time not constant �outside a possibleset of zero ��probability��If we sample the system con�gurations from an ergodic distribution � we

must �nd that the translation�invariant observables always have the samevalue� Thus for instance the global density �or speci�c volume� will alwayshave the same value on all con�gurations that are sampled with distribution��We see that because of the above�quoted theorem of unique decomposabil�

ity of the elements of G�� into extremal distributions the extremal pointsof G�� deserve the name of pure phases as there is no way to see thatthey consist of di�erent con�gurations by measuring global� translation�invariant� properties that they enjoy�If� instead� a probability distribution in G�� is not pure but it is a

mixture of� say� two pure states with di�erent densities and with coe%cients

family of measurable sets� or Borel sets� A mesurable transformation S is a map of Minto M � such that S��E is measurable for any measurable E� Therefore it makes senseto say that a function �i�e� a map of M to R� is measurable� A probability distribution�or a �normalized measure� � on M is a function de�ned on the measurable sets withvalues ��E� � � and which is additive �i�e� if E � �� En and the En s are pairwise

disjoint� then ��E� �P

��En�� and such that ��M� � �� Given a distribution � onMone calls ��measurable any set in the smallest collection of sets� closed under countableunion� complementation and intersection� that contains the measurable sets as well asany other set that can be enclosed into a measurable set with � �measure� the latterare called �� measure sets� Likewise we can de�ne �measurable functions and �measurable maps� The above notion of �measurability should not be confused with theprevious notion of measurability� Why the name �measurable� because a measurablefunction of one veriable is the most general function for which it is possible to set up�in principle� a table of values� i�e� a function that can be approximated by piecewiseconstant functions�


a and �� a then by sampling the system con�gurations we may get eithera con�guration of the dense phase or one of the rare�ed phase� so thatthe global density� which is a translation�invariant observable� can have twodistinct values �each with a probability of occurrence in samples given by aand �� a respectively�� so that it is not constant�An important general theorem for dynamical systems is Birkho��s ergodictheorem� �AA�#�

Theorem �Birkho�� Let �M�S� �� be a discrete dynamical system and let ,be a cube of side L� Then for all ��measurable functions f on M the limitlimL��

j�jP

k� f�Skx� � f�x� exists apart� possibly� from a set of x�s

of zero ��probability� Hence if �M�S� �� is ergodic f�x� is a constant for��almost all x and� therefore� it is equal to

RM��dy�f�y��

The above statement can be also formulated for the case in which S is a$ow �i�e� k � Rn is a continuous vector� and it is also a valid statement�Another consequence of the ergodicity is that particles located in two far

apart cubes are observed as if they were independently distributed� at least onthe average over the boxes locations� This is also a property that intuitivelyshould characterize the physically pure homogeneous phases��

x�� Other Characterizations of Gibbs Distributions� The DLR

Equations

Via the variational principle �� one �nds all the translation invariantGibbs distributions� but on physical grounds� as remarked� we expect thatthere may also exist� under suitable circumstances� nontranslation�invariantGibbs distributions� i�e�� with the notation of x�� in general we shall havethat G�� contains G�� but it does not coincide with G�� Therefore it is useful to look also for other characterizations of Gibbs states

which do not �discriminate� the nontranslation�invariant states� Such a

� A simple abstract argument proves the statement� Let � � G�� be ergodic anddenote by �#� the average over the con�gurations x of the number N�x� of particlesin the unit cube # with respect to the distribution �� let �#�#�� be the average ofthe product of the number of particles in the unit cube # times that in the unit cube#�� The translation invariance of � implies that �#� is independent of the locationof the unit cube # and that �#�#�� depends only on the relative position of the unitcubes # and #�� If ! is a large volume paved by unit cubes # the average number

of particles in ! will be h Nj ji � �

j j

P��

�#� and the average hN�N��

j j�i will be

�j j�

P��

�#�#�� Given a con�guration x� the limit as ! ofN�xj j

will exist� possibly outside of a set of con�gurations with �probability �� since thislimit �when it exists� is obviously translation invariant as a function of x� it mustbe a constant �possibly outside a set of � probability� by Birkho s theorem above�

nonconstancy would be against ergodicity�� For the same reason alsoN�x�N�x��

j j�will

have a limit equal to that ofN�x�

j j�which has to be constant and therefore equal to ��

Hence �j j�

P��

�#�#�� j j

P��#��

�� which means that �#�#��

�#��#�� on the average over #�#��


characterization is possible and is suggested by a heuristic argument basedon the �nite volume grand canonical distribution ��V � without external�xed particles� �� We ask given V � � V what is the probability of �nding inside V � exactlyg particles in the positions q

�� q

gknowing that out of V � the particles

are located in the positions q�� q�

��

Denoting by fV ��q�� q

gj q�

�� q�

�� fV ��g�qjq�� the density of this

conditional probability� where q abbreviates �q�� q

g� and q� abbreviates

�q�� q�

�� then it is immediate to deduce from �� that

fV ��g�qjq�� ge

��g��q

��q

g��

Pg

i�

Pj��

��qi�q�

j��

normalization��

where �� m�� g �see �� and the normalization is deter�

mined by imposing the condition that fV� de�nes a probability distribution�i�e� that �X

g�

ZfV ��g�qjq��

dq

g��g� � � ��

This relation depends on the total volume V only because q�� q�

�� i�e� the

particles of the con�guration external to V �� are constrained to be in V �i�e� in the global container of the system �and outside V ��It is therefore natural to de�ne� as an alternative to x�� x�� a �in�nite

volume� Gibbs distribution on M with parameters �� as a distribution� on M Maxwellian in the momenta and for which the probability for theevent in which the particles inside a �xed �nite volume V � are in q

�� q

g�

conditional to knowing that the particles outside the box V � are in q�� q�

��

�with any momenta� is given by �� without any restriction that theparticles at q�

�� q�

�� be inside a larger container V �because the latter has�

now� to be thought as in�nite��This reading of �� is known as the DLR equation and it was proposed

as a very general de�nition of Gibbs state �in the thermodynamic limit� byDobrushin� Lanford� Ruelle� �Do�#�� LR��This is important because one can establish� quite generally� the theo�

rem that Gibbs distributions� de�ned as the probability distributions onM which are Maxwellian in the velocities �with the same inverse tempera�ture parameter �� and which satisfy �� coincide with the distributionsin G�� de�ned via the thermodynamic limit in the previous sections�whether or not they are translation invariant�This is a theorem that holds as stated in the case of hard core systems� its

validity in more general situations still presents a few technical problems tobe understood although various weak versions of it exist in most cases ofinterest �e�g� in the case of superstable potentials�� LP��


x�� Gibbs Distributions and Stochastic Processes

By integration of the momentum coordinates p� the probability distri�butions � on the �in�nite volume� phase space M de�ne correspondingprobability distributions on the space M� of the position con�gurationsof in�nitely many particles q� Thus Gibbs distributions integrated overthe momentum variables �which can be �disregarded� as playing a trivialmathematical role from the point of view of the description of the states�although they are physically very important�� provide us with an interestingclass of distributions on M� which we shall still simply call Gibbs distribu�tions �rather than using a pedantic distinction between Gibbs distributionsand con�gurational Gibbs distributions��

In general the probability distributions � on M� are known in probabilitytheory as stochastic point processes because a point q �M� in fact describes

a family of points� i�e� particles located in q�� q

�� in R�� if q � �q

�� q

��

The remark permits us to give a new physical interpretation to severalresults of the general theory of point stochastic processes� and mainly itinduces a translation of problems relevant for physics into interesting math�ematical problems in the theory of point stochastic processes�

The issue that is� perhaps� central is to show that there exist simple choicesof the interparticle potentials � assumed with hard core for simplicity�and of the parameters �� for which the variational principle or the DLRequations admit more than one solution�

This is the same as the problem of the existence of phase transitions in ahomogeneous system of identical particles in fact we have argued that thephysical pure phases that can coexist can be identi�ed with the solutions ofthe variational principle or of the DLR equations�

We have seen above that the van der Waals theory provides us with ana%rmative answer to this issue� however it is rather unsatisfactory and�to date� there is still no example that can be treated without uncontrolledapproximations �i�e� without introducing ad hoc hypotheses at the �rightmoment�� The above nice thermodynamic formalism might be empty� afterall but this possibility is really remote� and it is certainly not realized inmodels that are somewhat simpler than the ones so far used for continuousgases see Chap�VI and the recent breakthrough in �LMP�#��

Other remarkable problems that arise in the theory of stochastic processesand� independently� in the theory of phase transitions are related to ques�tions of scale invariance�

From experience and from the phenomenological theories of phase transi�tions not only does the hypothesis emerge that the liquid�gas transition re�ally takes place whenever the interaction potential has� besides a repulsivecore� an attractive tail� but also the hypothesis that such a transition has acritical point ��c� �c� where the Gibbs distribution �and the correspondingstochastic process� � has special scaling properties� �Fi�#�� BG��

More precisely imagine that we pave the ambient space R� with a lattice ofcubes QL

n � with side L and parameterized by three integers n � �n�� n�� n��


so that the cube QLn consists of the points with coordinates nhL � xh �

�nh � ��L� h � �� De�ne the family of variables �i�e� �functions� on phase space� in proba�

bility theory language� or �observables� in physics language� �n on M

�n � ��particles number in QLn�� v��L��L��

where � is a parameter to be chosen�One gets a stochastic process� i�e� a probability distribution on a space ofstates consisting of the sequences f�ng indexed by n � Z�� in which the�states at the site n� are real numbers labeled by n and de�ned by ��

By these phenomenological theories of the critical point� we may expectthat� in the limit as L � � and if � is suitably chosen� the stochasticprocess describing the distribution of the variables �n tends to a limitingprocess such that� in the limit� the �n can be represented as

�n �

ZQLn

��x� dx ��

where ��x� are random variables this is a �stochastic process on R�� withhomogeneous correlation functions� i�e� for every k and x�� x�� xk

h��x�� xk��i � homogeneous function of �x�� xk� ��

if h�i denotes the operation of evaluation of the average value �also called�expectation� in Probability Theory� with respect to the distribution of therandom variables ��Since no nontrivial examples of point stochastic processes with the above

properties are known �or� better� were known until recently� one under�stands the interest� even from a purely mathematical viewpoint� of the the�ory of phase transitions which in its heuristic aspects provides a solution tovarious problems related to the existence and structure of stochastic pro�cesses� The heuristic results suggest in fact very challenging mathematicalconjectures� and some ideas for their understanding �often only partial�� sothat the subject continues to attract the attention of many� �Wi#�� WF� ��Ga�� Fi�#��It seems fair to say that the tumultuous development of statistical me�

chanics and of the theory of phase transitions has literally revolutionizedthe theory of probability as well�We conclude by mentioning �we come back on this point later� that so far

we have only discussed the properties of the Gibbs states as equilibriumstates� but without ever introducing the dynamics� We have regarded them

More generally one calls stochastic process a probability distribution on a space of statesconsisting of families of variables� called random variables� indexed by an arbitrary label�and both the label and the labeled variables can be in any space� a ghastly generality�


as dynamical systems with respect to space translations and we have seenthat this leads to the mathematical de�nition of pure phase� One maywonder whether one could obtain similarly interesting notions by regardingthe Gibbs states as stationary states for time evolution as well �i�e� fortranslations in time�� This is a much harder question and we defer discussingit to Chap�IX�

x�� Absence of Phase Transitions� d � �� Symmetries� d �

�A� One dimension� After the above general analysis and after setting up aformalism well suited for our programs we return to more concrete questions�We begin by showing that� as already stated several times� one dimensionalsystems with �nite range interactions cannot have phase transitions of anysort� unless it is considered in the somewhat unphysical situation of havingzero absolute temperature� Again we limit ourselves to the simple case ofhard core interactions and call r� the hard core size �so that �r� � �� ifr � r��

We shall use here as a de�nition of phase transition the presence of a hor�izontal segment in the graph of the pressure as a function of the speci�cvolume at constant temperature� But other de�nitions could be used� e�g�the inequivalence of some of the ensembles and the dependence of the ther�modynamic limit on the boundary conditions� discussed in x �� and above�in the present chapter�

Consider �rst the case in which the potential vanishes beyond r � r�where r� is the hard core radius this case is particularly easy and is calledthe �nearest neighbor� interaction case�

It is best to use the pressure ensemble� see x �� with thevolume V taking the continuum of values between � and ��

Then the partition function in the pressure ensemble� see � �� is

JN �� p� � ��N

Z �

�

dL

L�

Z��L�

dq� � � � dqNN �

� e��pL �N��Yj�

e��qj��qj�

��#��

where ��def�q

�m�h� �

We shall use the fact that the interaction cannot extend beyond the nearestneighbor and we label the particles �� N so that q� � q� � � � � � qN � Inthis way we restrict the integration domain by a factor N �� Thus� extendingthe integral to the region � � q� � q� � � � � � qN � L we get rid of theN �� present in the de�nition of the partition function�

The momentum integration yields the square root in front of the integral �itis raised to the power � rather than the usual � because the space dimensionis now �� The length L� is an arbitrary dimensional factor �see � ��

Then we note that L � q� ��PN��

j� �qj�� qj��

�L� qN and introducing


the variables qj� � qj as independent variables� we see that

JN �� p� � ��N�

��p�� L�

�Z �

�

e��pqe��q�dq�N��

��#� ��

so that the thermodynamic limit limN�� N log JN �� p� is

�� p� � log�r �m

�h��Z �

�

e��pqe��q� dq�� #��

Equivalence between pressure ensemble and canonical ensemble is workedout along the same lines in which in x �� equivalence between canonical andgrand canonical ensembles �hence orthodicity� were derived�One �nds� as mentioned in x �� that the quantity p can be identi�ed with

pressure and �� p� can be identi�ed with the Gibbs potential u� Ts� pv�see � �� and � � ��kBT � Moreover the equation of state is derived�by using the orthodicity� from the thermodynamic relation ��p �� v�

Relation ��#�� implies that the Gibbs potential �� p� is analytic in �� pfor �� p � �� and it is strictly monotonic in p so that the relation ��p �� v

implies that pressure is analytic and strictly monotonic �decreasing� in vhence the equation of state cannot have any phase transition plateau�The above analysis is a special case of Van Hove�s theorem� which holds

for interactions extending beyond the nearest neighbor� see Appendix ��A�below� and it played an important role in making clear that short�rangeone�dimensional systems cannot undergo phase transitions� �VH�� Furtherextensions can be found in �GMR��If one adopts the de�nition of phase transition based on sensitivity of

the thermodynamic limit to variations of boundary conditions one cangive a more general� conceptually simpler� argument to show that in one�dimensional systems there cannot be any phase transition if the potentialenergy of mutual interaction between a con�guration q of particles to theleft of a reference particle �located at the origin O� say� and one con�gura�tion q� to the right of the particle �with q�O � q� compatible with the hardcores� is uniformly bounded�The argument� due to Landau� is simply that� in this case� the distribution

of the con�gurations to the right of a point and to the left of it are essentiallyindependent hence by changing the con�guration of �xed particles outsidea box one does not alter appreciably the probability distribution inside it�This is so because the weight of a con�guration q� consisting of a part q

�to the left of the origin and of a part q

�to the right of it� is the exponential

of ��H � if H is the energy of the con�guration� But the energy of such acon�guration is a sum of two quantities �large� of the order of the volumeoccupied by the con�gurations� which are the sum of the energies that eachof the two parts q

�and q

�would have� in the absence of the other part�

plus the mutual energy� The latter is� however� bounded independently ofthe choice of q

�and q

��


In itself this does not immediately imply that there can be no dependenceon boundary conditions because a �nite ratio between two probabilities isnot the same thing as a ratio close to � hence the argument has to be re�nedby considerations that show that it also implies actual boundary conditionindependence I �nd it easier to just give analytic details on the aboveargument �see Appendix ��A� below� rather than indulging on heuristicdiscussions� The argument clearly shows a mechanism responsible for the�loss� of memory of the boundary conditions as one proceeds in from theboundary down to the center of a �nite interval �� L��Hence the larger the box the smaller is the in$uence of the external particles

on the bulk of the particles in the box hence no inequivalence between theensembles can arise� i�e� no phase transitions in that sense� One also saysthat no long�range order can be established in such systems� in the sensethat one loses memory of the boundary conditions as the boundaries recedeto in�nity in the process of taking the thermodynamic limit�Note that the argument above fails if the space dimension is � in this

case even if the interaction is short ranged the energy of interaction betweentwo regions of space separated by a boundary is of the order of the boundaryarea� Hence one cannot bound above and below the probability of any twocon�gurations in two half�spaces by the product of the probabilities of thetwo con�gurations� each computed as if the other was not there �because thebound would be proportional to the exponential of the surface of separation�which tends to� when the surface grows large�� This means that we cannotconsider� at least not in general� the con�gurations in the two half spacesas independently distributed�Analytically a condition su%cient to imply that the energy between a con�

�guration to the left and one to the right of the origin is bounded above� ifthe dimesion d is d � �� is simply expressed �as it is easy to check� byZ �

�r�

r j�r� j dr � �� #��

One usually says� therefore� that in order to have phase transitions in d � �systems one needs a potential that is �so long range� that it has divergent�rst moment� It can be shown by counterexamples that if the condition��#�� fails there can be phase transitions even in one�dimensional systems�at least in further simpli�ed models� �Dy�� In fact very recently the �rstphase transition in a continuous system and in the absence of symmetrybreaking has been proved to occur precisely in a system violating ��#��see �Jo��The arguments of this section apply also to discrete models like lattice gases

or lattice spin models� see x�� below and Chap�VI� Chap�VII� and x��

�B� Symmetries� By symmetry one means a group of transformations actingon the con�gurations of a system subject to some boundary condition �e�g�periodic or open� and transforming each of them into con�gurations withthe same energy and with the same boundary condition�


Systems with �too much symmetry� sometimes cannot show phase transi�tions� This is best discussed if one uses as de�nition of phase transition theexistence of long�range order�

The latter is de�ned by considering a localized observable F �see �� which has zero average hF i in all Gibbs states obtained as thermodynamiclimits� with suitable boundary conditions �e�g� periodic or open�� becauseof their symmetry properties�

Suppose that the average value hF � ��F i of the product of the observabletimes a translate� by � of itself� a quantity called the spatial autocorrelationof the observable� does not approach � as � � �� Then one says that thesystem shows long�range order for the order parameter F �

The symmetry is continuous if the group of transformations is a continuousgroup� For instance continuous systems have translational symmetry if con�sidered with periodic boundary conditions� so that the number n� of parti�

cles in a small box � is a local observable and such is also ��def� �n�� n��

where n� is the average over translations� For symmetry reasons this quan�tity has zero average in the Gibbs states associated with the Hamiltoniandescribing the system with the symmetric boundary condition �periodic inthis case�� Denote by � � � the box �� obtained by translating � by a vec�tor � and let h�i denote the average in one element of an orthodic ensemble�i�e� an average with respect to a Gibbs state�� The system is said to showlong�range order if the autocorrelation function at distance � is a functionh��i which does not tend to zero as � ��

Note that failure of convergence to zero of h��i� see also x�� footnote#� is precisely what we expect should happen if the system had a crystallinephase �in which case the h��i should show an oscillatory behavior�in �� of the correlation function�� One can also prove that long range or�der of some observable implies that the derivative of the pressure �or ofother thermodynamic functions� with respect to suitable perturbations ofthe energy function has a discontinuity� so there is an intimate connectionbetween phase transitions de�ned in terms of long�range order and in termsof singularities of thermodynamic functions�

As an example of an application of a general theorem� the Mermin�Wagnertheorem� �MW�� Me��Ru�� one can state that if the dimension of theambient space is d � then a system which in periodic boundary conditionsshows a continuous symmetry cannot have any local observable whose aver�age vanishes and whose autocorrelations at distance � do not tend to zero asthe distance � �� This theorem is the �rst of a series of similar theoremsbased on an important kind of inequality called the infrared inequality andit has led to developments that solved several long�standing problems� seefor instance �Fr#�� DLS�#�� Here we choose not to enter into more detailsin spite of the great importance of the technique and the reader is referred

� A translate by the vector � of an observable F is de�ned as the observable ��F suchthat ��F �x� � F �x� �� where x� � is the con�guration obtained from x by translating

by � all particles positions� leaving the velocities unchanged�


to the literature�The limitation to dimension d � is� however� a strong limitation to

the generality of the theorem and very seldom does it apply to higher�dimensional systems� More precisely systems can be divided into classeseach of which has a �critical dimension� below which too much symme�try implies the absence of phase transitions �or of certain kinds of phasetransitions�� see �WF� �� Fr#�� Fr#�� Fi�#��

x�� Absence of Phase Transitions� High Temperature and the

KS Equations

There is another class of systems in which no phase transitions take place�These are the systems so far considered �with stable and tempered interac�tions� see x � � in states with high temperature and low density�We use here as de�nition of phase transition that of a singularity in the

equation of state� although in the cases below one could show that phasetransitions do not occur even in other senses �like persisting sensitivity toboundary conditions as the boundaries recede to ��One can easily show the absence of phase transitions for �� and v large

by showing that the equation of state is analytic� In fact in such regionsthe virial series� �� is convergent and we have analyticity in v�� and� of the equation of state�There are two ways of attacking the problem one is rather direct and

looks for an algorithm that constructs the coe%cients of the virial series�The algorithm can be found quite easily but the k�th order term results asa sum of very many terms �a number growing more than exponentially fastin the order k� and it is not so easy �although it can be done� to show bycombinatorial arguments that their sum is bounded by c��k if � is smallenough� �Gr� ��Pe�� see also equation �� p� �� in �GMM� �� dealingwith a case only apparently di�erent and in fact more general�The other approach is somewhat less natural but it leads quite easily to

the desired solution� It attempts to solve a much more general question�Namely the problem of computing the functions fV � of x�� and ��i�e� �all the properties� of the system�We consider a gas in a cubic container V and with an interaction potential

satisfying �� The state of the system� in the grand canonical ensemble�can be de�ned in terms of the local distributions discussed in x�� or in terms of the more convenient ��spatial or con�gurational�� correlationfunctions

�V �q�� qn� �

��

0V ��

�Xm�

znmZe��q��qn�y��ym� dy� � � � dym

m�

��

where z � e��p

�m��h�� is called the activity it has the dimensionof a density i�e� of a length�� as we included in it also the factor h��m which


in �� was included in fV� � The correlation functions are� therefore� theprobability densities for �nding n particles at the positions q�� qn withany momenta and irrespective of where the other particles are� The squareroot comes from the integration over the momenta variables �which drop outof the scene� with no regret as they play a trivial role in classical statisticalmechanics�� The integral over the y s is over the volume V �The energy ��q�� qn� y�� ym� can be decomposed as

��q�� q�� qn� �

mXj�

�q� � yj� � ��q�� qn� y�� ym� ��

where ��q�� q�� qn� �Pn

i� �q� � qi� is the energy of interaction ofparticle q� with the group of �q�� qn��We can imagine that q� is the �most interacting particle� among the

�q�� qn�� i�e� it is one that maximizes the potential energy of interac�tion with the group of the other particles� Since ��q�� qn� � �Bn bystability� this implies that

��q�� q�� qn� � � B � ��

In some special cases the selection of q� among the n particles q�� qn maybe ambiguous� the choice then can be made arbitrarily �for the purposes ofthe following argument��Then from the de�nition �� and by using the decomposition ��

and if 0V denotes the grand canonical partition function in the volume V �we see that we have the following simple algebraic identities

�V �q�� qn� ��

0Vze��q��q��qn�

�Xm�

zn��m� ��

�Ze��P

j��q��yj�e��q��qn�y��ym� dy� � � � dym

m��

��


�Xm�

zn��m�

�Z h mY

j�

�� e��q��yj� � ��

�ie��q��qn�y��ym� dy� � � � dym

m��

��


�Xm�

mXs�

zn��mX

j��j��js� ��jk�m�

�Z

dy� � � � dymm�

sYk�

�e��q��yjk � � �

�e��q��qn�y��ym�

having developed the product in the second line� By using the symmetryin the yj variables we can suppose that the j�� js are in fact �� s


and rewrite �� as

�V �q�� qn� �

��


�Xs�

zsZ

dy� � � � dyss�

�sY

k�

�e��q��yk� � �

� �Xt�

zn��tZ

dy�� dy�t

t�e��q��y��ys�y

��y

�t�

�� having in the last step called m � s � t and replaced� with the ap�propriate combinatorial factors required by the change� �y�� ym� with�y�� ys� y

�� y

�t�� Hence we see that the integrals reconstruct the cor�

relation functions and �� becomes

�V �q�� qn� � ze��q��q��qn��Xs�

Zdy� � � � dys

s��

�sY

k�

�e��q��yk� � �

��V �q�� qn� y�� ys�

��

in which the term with s � � has to be interpreted as

z e��q��q��qn� �V �q�� qn�� if n � �

z if n � ��

and all the variables q� y are considered to be in V �Relations �� are called the Kirkwood�Salsburg equations they are im�

portant because we can use them to show that the virial series convergesfor � and v small� And in fact they allow us to obtain a complete theory ofthe gases in such regimes of �� v�We can regard �V as a sequence of functions �of one� two�� particle po�

sitions� �V � f�V �q�� qn�gn� q�� vanishing for qj �� V � If we de�nethe sequence �V of functions of one� two�� particle positions by setting�V �q�� if q� � V and �V �q�� qn� � � if n � � or n � �� qj �� V � thenwe can write �� as

�V � z�V � zK�V ��#��

where� if �n� � � for n � � and �n� � � for n � ��

K �V �q�� qn� � e��q��q��qn��V �q�� qn� �n��

�

�Xs�

Zdy� � � � dys

s�

sYk�

�e��q��yk� � �

��V �q�� qn� y�� ys�

�

which shows that the Kirkwood�Salsburg equations can be regarded as linearinhomogeneous �integral� equations for the family of correlation functions


that describe a given system in the box V � The kernel K of these equationsis independent of V �We should note that the quantities in �V have di�erent physical dimensions�

In fact �V �q�� qn� has the dimension of a length to the power ��n�This sounds bad enough to wish to write �� in dimensionless form�

For this we need a length scale� and a natural choice is the �range� of the

potential that could be de�ned as

Rj��x�jjxjdxRj��x�jdx � But a more convenient length

that we can associate with our system is the quantity r��

r�� ZR�

je��q� � �jd�q ��

which can be called the e�ective range at inverse temperature �� Notethat r��

�� if the potential has no hard core� if the potential has a

hard core with radius a� and it is smooth and bounded otherwise� thenr��

�� a

��

The length r�� can be used to de�ne the dimensionless correlations�V �q�� qn� as

�V �q�� qn� � r��n�V �q�� qn� ��

and setting � � zr�� the above equations can be written in dimensionlessform

�V � ��V � � K �V � ��

with

K �V �q�� qn� � e��q��q��qn��V �q�� qn� �

�Xs�

�

�Z

dy� � � � dyss� r��s

sYk�

�e��q��yk� � �

��V �q�� qn� y�� ys�

��

��

Then we can write the recursive formula

�V � ��V � ��K�V � ��K��V � ��K

��V � � � � ��

which gives us an expression for the correlation functions� provided the seriesconverges� of course�The convergence of the series is easily discussed if one notes that

jKp�V �q�� qn�j � e��B

�Xs�

ZR�s

dy� � � � dyss� r��s

�

�sY

k�

je��q��yk� � �j jKp��V �q�� qn� y�� ys�j

��


so that if we call M�p� � maxn� q��qn jKp�V �q�� qn�j we see that

M�p� � e��BM�p� ��

�Xs�

s�� e��B�M�p� ��

and M�� has to be set equal to �� This implies that M�p� � e��B��p sothat the series �� converges if jzj � e��B��r��The convergence is uniform �as V �� and �K

p��V �q�� qn� tends to a

limit as V �� at �xed q�� qn and the limit is simply �Kp��q�� qn�

if ��q�� qn� � � unless n � �� and ��q�� This is because the kernelK contains the factors �e��y�q�� which will tend to zero for y ��not slower than j�y�j� i�e� summably by the temperedness condition� It isalso clear that �K

p��q�� qn� is translation invariant�

Hence the limits as V � � of the correlation functions do exist and theycan be computed by a convergent power series in z� and the correlation func�tions will be translation invariant in the thermodynamic limit and the lackof translation symmetry� due to the con�nement in the box V � disappearswhen the box recedes to ��In particular the one�point correlation function � � ��q� is simply � �z�� O�zr��

�� which to lowest order in z just shows that the activity

can be identi�ed with the density� Activity and density essentially coincidewhen they are small�Furthermore � pV � �

V log 0V �� has the property that �zz�pV �� V

R�V �q�dq� as is immediately checked �by using the de�nition of �V in

�� Therefore the above remarks imply

�pV �� z� � limV��

�

Vlog 0��

Z z

�

dz�

z�� z��

hence the grand canonical pressure p�� z� is analytic in �� z� The density� is analytic in z as well and � z for z small� It follows that the pressureis analytic in the density and � p � � �� O�� at small density� In otherwords the equation of state is� to lowest order� essentially the equation of aperfect gas� and all the quantities that we may want to study are analyticfunctions of temperature and density�The system is essentially a free gas and it has no phase transitions in the

sense of a discontinuity or a singularity in the dependence of a thermody�namic function in terms of others�However the system also cannot show phase transitions in the sense of

sensitive dependence on the boundary conditions this is essentially clearfrom the above analysis �i�e� from the remarked short range nature of thekernel K� which shows that the dependence on the boundary conditiondisappears as the boundary recedes to in�nity while translation invarianceis recovered�One could� nevertheless� think that by taking other boundary conditions

the argument may fail� It can however be shown that this is not the case�


simply by pushing the above analysis a little further� The key remark is infact that any in�nite volume state� obtained by any sequence of boundaryconditions with �xed external particles� will obey the DLR equations� seex�� and the latter can be shown to imply the �in�nite volume Kirkwood�Salsburg equations� �La�� The latter are simply �� with V replacedby R�� and make sense also for in�nite volume as soon as the correlationfunctions satisfy a bound like ��q�� qn� � �n for some � no matter howlarge�The limits of �nite volume states with �xed external particle bound�

ary conditions do satisfy a bound of this type and also the DLR equa�tions and� therefore� the Kirkwood�Salsburg equations in in�nite volume�The uniqueness of the solutions of such equations proves� in the regionjzj e��B�r�� the boundary condition independence �hence the trans�lation invariance� of the Gibbs states� �Do�#c��LR��Finally one can also see that the state of the system can be regarded as

describing a distribution of particles in which particles occupying regionsthat are far apart are �independently distributed�� There are several waysto express this property� The simplest is to say that the correlations have acluster property� see footnote #� x�� This means that

lima�� q�� qn� q

�� a� � � � � q�n� � a� � ��q�� qn��q�� q

�n�� #��

and this property is an immediate consequence of the above analysis in thesmall �� small � regions�In fact� restricting ourselves for simplicity to the case in which the potential

has �nite range r� we easily check that

z�zK�p��q�� qn� q�� a� � � � � q�n� � a� �

�X

p�p�p

z�zK�p��q�� qn�z�zK�p��q�� qn� ��

for all p and provided the distance between the cluster q�� qn and thecluster q�� a� � � � � q�n� � a is greater than p r��This is satis�ed by induction and implies that the power series expansion

for the di�erence between the expression under the limit sign in ��#�and the right�hand side starts at p � O�a�r�� because all the coe%cientswith p � a

r�must vanish due to the fact that the kernel of the operator

K vanishes when its arguments contain points that are too far away� Infact if the argument contains s � � points q�� y�� ys and the maximumdistance between them is greater than �s� ��r� then at least one is furtheraway than r� from q�� see �� Then the above proved convergenceshows that the limit is approached exponentially at a speed that is at least�zr��e��B��a�r� �this being the rate of approach to zero of the remainderof a geometric series with ratio zr��e��B� and starting at order jaj�r��Hence if one wants to look for phase transitions one must forget the regions

of low density and high temperature� The Kirkwood�Salsburg equations


are only one example of equations leading to convergent expansions forthe correlation functions there are many recent developments based onsimilar equations that are derived for other models or even for the sameones considered above� The most interesting concern lattice models� Seex�� below see �Ca#�� KP#�� Br#�� for some examples�

x�� Phase Transitions and Models

As already mentioned the problem of showing the existence of phase tran�sitions in models of homogeneous gases� which we have been considering sofar� is in fact still open�Therefore it makes sense to study the phase transitions problem in simpler

models� tractable to some extent but nontrivial� In fact such an investigationcan give a very detailed and deep understanding of the phase transitionphenomenon�The simplest models are the so�called lattice models� They are models in

which the particles are constrained to occupy points of a lattice in space�In such models particles cannot move in the ordinary sense of the word�because they are on a lattice and the motion would have to take place byjumps� and therefore their con�gurations do not contain momenta variables�The energy of interaction is just a potential energy and the ensembles

are de�ned as probability distributions on the position coordinates of theparticle con�gurations� Usually the potential is a pair potential decaying fastat � and� often� with a hard core forbidding double or higher occupancy ofthe same lattice site�Often the models allow at most one particle to occupy each lattice site� For

instance the nearest neighbor lattice gas� on a square lattice with mesh a � ��is de�ned by the potential energy that is attributed to the con�guration Xof occupied sites

H�X� �Xx�yX

�x� y�� x� �nJ if jxj � a� otherwise �

��

One can de�ne the canonical ensemble� with parameters ��N � in a box ,simply as the probability distribution of the subsets of , with N points

p�X� �e��H�X�P

jX�jN

X�

e��H�X��jX j � N ��

where jX j is the number of points in the set X � and likewise the grandcanonical ensemble with parameters �� in the box , by

p�X� �e��jXje��H�X�P

X�� e��jX�je��H�X��

� ��

Finally we can remark that a lattice gas in which in each site there can beat most one particle can be regarded as a model for the distribution of afamily of spins on a lattice�


Such models are quite common and useful for instance they arise in study�ing systems with magnetic properties� One simply identi�es as �occupied�a site with a �spin up� or � and as an �empty� site a site with a �spindown� or � �of course one could make the opposite choice�� If � � f�xg�is a spin con�guration� the energy of the con�guration will usually take theform

H�� Xx�y�

�x� y��x�y � hXx

�x ��

and one calls canonical and grand canonical ensembles in the box , withrespective parameters ��M or �� h the probability distributions on the spincon�gurations � � f�xg� with

Px� �x � M or without constraint on M �

respectively� de�ned by

p��M �� e��P

x�y��x�y��x�y

Pe��P

x�y��x�y��x��y

p��h�� e��h

P�x��

Px�y

��x�y��x�y

Pe��h

P��x��

Px�y

��x�y��x��y

��

where the sums in the denominators run over the �� withP

x ��x � M in

the �rst case and over all �� s in the second case�As in the study of the previous continuous systems one can de�ne the

canonical and grand canonical ensembles with �external �xed particle con��gurations� and the corresponding ensembles with �external �xed spin con��gurations��For each con�guration X � , of a lattice gas we de�ne fnxg to be nx � �

if x � X and nx � � if x �� X � Then the transformation

�x � nx � � ��

establishes a correspondence between lattice gas and spin distributions� Inthis correspondence lattice gases with canonical �or grand canonical� dis�tributions and given boundary conditions with external �xed particles aremapped into canonical �or grand canonical� spin distributions with suitablycorrespondent boundary conditions of external �xed spins�In the correspondence the potential �x � y� of the lattice gas generates

a potential ��x � y� for the corresponding spin system� The chemical

potential � for the lattice gas becomes the magnetic �eld h for the spinsystem with h � �

� �� P

x�� x��

�x� � � ��

��x�� h �

�

��

Xx��

�x��

The correspondence between boundary conditions is also easy for instance aboundary condition for the lattice gas in which all external sites are occupied


becomes a boundary condition in which all the external sites contain a spin�� The correspondence between lattice gas and spin systems is so completethat one often switches from one to the other with little discussion�The thermodynamic limits for the partition functions of lattice gas models�

de�ned by

� f�� v� � � limj jN v��

�

Nlog

XjXjN�X��

e��H�X� ��#��

and

� p�� lim��

�

,log

XX��

e��H�X��jXj ��

can be shown to exist� by an argument similar to that discussed in Chap�IV��and by far easier�� They have the same convexity and continuity propertiesof the corresponding quantities in the case of the continuous models andthey will be given the same names �free energy and pressure�� They areboundary condition independent� as was the case in the continuum modelswith hard core interactions�Likewise the thermodynamic limits exist also for the spin models partition

functions and they are denoted by f� p

� f��m� � � lim�� M

j j�m

�

,log

XP

x�xM

e��H��

and

� p�� h� � lim��

�

,log

X�

e��H��hP

x�x � ��

however the physical interpretations of f� p are of course di�erent� To �ndthe meaning of the above quantities in the Thermodynamics of a spin systemone would have to go through the discussion of the orthodicity again� in thecase of such systems� One would �nd� as it is easy to check� that p�� h� hasthe interpretation of magnetic free energy while f��m� is a quantity thatdoes not have a special name in the Thermodynamics of magnetic systems�In the next chapter we shall consider some special cases they are the

simplest and they are quite remarkable as in some particular instances theyare even amenable to more or less exact solution �i�e� calculation of thethermodynamic limit of various quantities� like for instance the free energy��The interest� as it will appear� of such models will be the wealth of infor�

mation that they provide about the phenomena related to phase transitions�

One of the developments of the late ��s and early ��s is that natural�extensions� to lattice spin systems of the formalism discussed in x��x��arise in rather unexpected contexts� see Chap�IX�Such �extended� lattice system models are spin systems more general than

the model �� For instance they allow the spins �x at each site x to be


an �arbitrary� �nite set of symbols or �spin values� �rather than necessarily�x � �� The number of values of such �spins� divided by is called thetotal spin so that the case � � �� is the �spin �

�� case�It is convenient� for later reference purposes� to introduce here such ex�

tensions they will be one�dimensional models without phase transitions atleast in the cases that we shall later consider in Chap�IX� they can be ex�tended also to higher dimension and� as such� they will appear in Chap�VII�The model energy H has the form

H �

�Xn�

Xx��xn

x��xn��x� � � � � � �xn� ��

where x��xn�� n� � x�a��xna�� n� for all a � Z��translation invariance�� and

�Xn�

X�x��xn

wn max��n

jx��xn�� n�j ��

for some weights wn � �� If wn � e�jxn�x�j one says that the model �� is a short�range Ising model with many body interactions� The quantity

��

�Xn�

X�x��xn

x��xn��x� � � � � � �xn� ��

will be called the �energy per site�� a few properties of � should be noted�Namely � is �H�older continuous� i�e� if �� are two spin con�gurationsagreeing for jij � k �i�e� �i � ��i for all jij � k�� and if the interaction hasshort range in the above sense� then for some � � ��

j�� j � const e��k � ��

This means that �� depends �exponentially little� on the spins located farfrom the origin�The partition function of the model with �open� boundary conditions will

be simply

Z �X�

e��H��

More generally one can consider the model in the presence of ��xed spinboundary conditions�� This means that for each con�guration �� in thebox , � ��L�L� we consider the biin�nite con�guration � � ��L� �� R�obtained by putting �� on the lattice and then continuing it outside ,with �L to the left and �R to the right� where �L � �� L�� L� � and�R � ��R� � �

R� � � � �� The probability �� of a con�guration in the model

�with boundary conditions �L� �R� will be

�� e�H��L� �R�P�� e

�H��L�� R��


where� if denotes the shift operation on the bilateral sequences and we set�L��R � �� the energy H is

H��L��R� � H��

LXk�L

�� k�� #��

so that one can remark that the exponential is simply written in terms ofthe energy per site �� can also be expressed in a similar way�In fact let � � �o�� be the biin�nite con�guration obtained by extending ��

periodically outside the region ,� Then the energy of interaction betweenthe spins in , and between them and the ones outside ,� is

H��

LXk�L

�� k�� corrections ��

where the �corrections� depend �only� on the spins near the boundarypoints �L and outside the interval , � ��L�L�� in the sense that by varyingthe spin at a site at distance � from the boundary the correction changesby a quantity proportional to e�� if � is the exponent in the weight wnintroduced above� see ��A further extension is obtained by considering a matrix T whose entriesT�� are labeled by the spin values and are supposed to be T�� and by restricting the family of spin con�gurations � to the T �compatiblecon�gurations they are de�ned to be those that satisfy T�i��i� � �� whereT is a matrix with entries � or �� called the compatibility matrix� One alsocalls such models �hard core spin systems� and T describes the hard corestructure� Such systems are also called hard core lattice systems�If the matrix T is such that Tn

�� for all �� and for n large enough�one says that the hard core is mixing� For such cases all the above for�mulae and de�nitions �� extend unchanged provided onlycompatible spin con�gurations are considered�

Appendix ��A�� Absence of Phase Transition in non Nearest

Neighbor One�Dimensional Systems

The method discussed in x��# for hard core nearest neighbour one�dimensional models is called the transfer matrix method� We extend ithere to the more general case of �nite range� but not nearest neighbor� Thetheory is very similar to that in x��#�Let �r� � � for r � �n � ��r� for some integer n� Assume for simplicity

that N is a multiple of n� Let q � �q�� qn� with q� � q� � � � � � qn� andlet us de�ne

a�q� � a�q�� qn� ��

nXi�j�

�qj � qj� ��A��A��


b�q j q�� b�q�� qn j q�� q�n� �

nXi�j�

�qi � q�j�

c�q j q�� a�q� � b�q j q�� a�q��

Then the energy of a particle con�guration is

a�q�� qn� � a�qn��N�n�� qN ��

�

N�n��Xk�

c�qnk� � � � qnkn j qn�k�� qn�k��A�� A��

so that one easily �nds

JN �� p� � h a jT Nn �� ja i ��A��A��

where T is the operator acting on the space of the functions of n coordinatesq � �q�� q�� qn� with qj� � qj � r�

Tf�q� �

Z �

�

e��c�q j qnq��e�

�� qnq

��pf�q��dq� ��A��A��

and the vector ja i is the function e��pqn��a�q��

Since the operator T is a Hilbert�Schmidt operator on the space L��dq�

�i�e�RT �q� q��dqdq� � �� and since its kernel is � � it �immediately�

follows �i�e� it follows from well known results on the theory of operators� orbetter of matrices� like the Perron�Frobenius theorem� see p� �� in �Ru��that the largest eigenvalue t�� p� of T is isolated and simple and thereforeit is analytic as a function of �� p� since T itself is analytic in such variables�Therefore �� p� � �

n t�� p� is analytic in �� p for �� p � �� and convexin such variables �see ��#� � showing that JN is a �linear combination� offunctions depending on � as e�c� hence log JN �� p�� is convex in �� and wecan repeat the argument above to see that the equation of state gives p asan analytic function of �� v�The further extension to systems with a potential with in�nite range butsatisfying ��#�� is also possible and it was the main purpose of van Hove stheorem� �VH�� The condition ��#�� comes in to insure that b�q j q��is uniformly bounded this quantity represents the interaction between acon�gurartion q situated to the left of another con�guration q�� hence it isuniformly bounded if ��#�� holds�


�

��

Chapter V I

Coexistence of Phases

�#�

�

V I� Coexistence of Phases �#�

x�� The Ising Model� Inequivalence of Canonical and Grand

Canonical Ensembles�

The Ising model� plays a very special role in statistical mechanics andgenerates the simplest nontrivial example of a system undergoing phasetransitions�Its analysis has provided us with deep insights into the general nature of

phase transitions� which are certainly better understood nowadays� after thepublication of the hundreds of papers which followed the pioneering work ofIsing� Peierls� Onsager� Kaufman and Yang� �Pe�� On�� Ka�� KO��Ya� ��The main reason why so much attention has been given to this very special

model lies in its simplicity and� in spite of it� in the fact that it �rst gave �rmand quantitative indications that a microscopic short�range interaction canproduce phase transitions which� furthermore� deeply di�er in characterfrom the classical van der Waals �or Curie�Weiss or mean �eld� type oftransitions� see x�� and x�� It should also be mentioned that the two�dimensional Ising model in zero

external �eld is exactly solvable �see x�� this fact has been very oftenused to check of the validity of numerical approximations designed for ap�plications to more complicated models� see the review �Fi�� pp� ��

Last but not least� we mention that the Ising model has given rise toa number of interesting developments and reinterpretations of old resultsin the theory of Markov chains� �Do�#��Sp�� information theory� �Ru��Or�� RM�� random walks� �Gr��Fi��b��GH�� La#�� to quote a fewremarkable works� and therefore constitutes a notable example of a subjectwhich has simultaneously been the object of advanced research in Physics�Mathematics and Mathematical Physics�In the rest of this chapter we give a description� certainly not exhaustive�

� This chapter is mostly taken from the paper Instabilities and phase transitions in theIsing model� La Rivista del Nuovo Cimento� ��

� For a history of the Ising model see �Br�� The original solution for the free energy of the Ising model in two dimensions can befound in �On�� It was preceded by the proofs of existence of Peierls� �Pe�� and vander Waerden� �VW�� and by the exact location of the critical temperature by Kramersand Wannier� �KW��The spontaneous magnetization was found by Onsager� �KO�� but the details werenever published� it was subsequently rediscovered by Yang� �Ya�� A modern derivationof the solution is found in the review article by Schultz� Mattis and Lieb� �SML�� thelatter is reproduced in x�� Another interesting older review article is the paper �NM��A combinatorial solution has been found by Kac and Ward and can be found in �LL��p�� Some aspects of this derivation were later clari�ed� and it has been discussed againin several papers� see �Be�� Another approach to the solution �the Kasteleyn s pfa�anmethod� can be found in �Ka��

�# V I� Coexistence of Phases

of the model and of some selected results� They illustrate properties whichthrow some light on the general nature of the phenomenon of phase tran�sitions� mostly far from the critical point� and which� hopefully� should notbe a peculiarity of the simplicity of the model�There exist some very good accounts of the theoretical arguments leading

to the consideration of the Ising model in the context of physical problems��Fi��a�� Ma�� here we shall completely skip this aspect of the matter��

x�� The Model� Grand Canonical and Canonical Ensembles�

Their Inequivalence

We consider a d�dimensional �d � �� square lattice Zd and a �nitesquare , � Zd centered around the origin� containing j,j � Ld lattice sites�On each site x � , is located a classical �spin� �x � �� The �con�gu�rations� of our system will� therefore� consist of a set � � ��x� � � � � � �xj j�

of j,j numbers �x � �� the number of these con�gurations is j�j� Theensemble of the con�gurations will be denoted U�,��To each spin con�guration a certain energy is assigned� see x��

H�� JX�i�j

�xi�xj � hXi

�xi � B��

whereP

�i�j means that the sum is over pairs �xi� xj� of neighboringpoints� h is an �external magnetic �eld� and B�� describes the inter�action of the spins in the box , with the �rest of the world�� This couldbe the contribution to the energy that comes from the �xed spins boundaryconditions that we considered in x��

For simplicity we shall treat only the case J � ��Of course B�� in �� can be rather arbitrary and� in fact� depends on

the particular physical problem under investigation� It is subject� however�to one constraint of physical nature in case we were interested in letting, �� we should impose the condition

lim��

max� jB��jj,j � � ��

i�e� we want the condition that the energy due to B�� should not be of thesame order as the volume of the box� furthermore B� should depend mostlyon the �x with x near the boundary� e�g� B�� c�� satis�es �� butit should also be excluded�� In other words B� should be a �surface term��

� In some cases the Ising model is a good phenomenological model for antiferromagneticmaterials� this is the case of MnCl� � �H�O� see �FS�� Fi��

� This term is usually omitted and in some sense its importance has only recently beenrecognized after the work of Dobrushin� Lanford and Ruelle� see �Do�� LR�� In thischapter the main purpose is to emphasize the role of this term in the theory of phasetransitions�

� A precise condition could be that for any �xed set D� max jB �� B ��j ��

��


The laws of statistical mechanics provide a relationship between the micro�scopic Hamiltonian �� and the macroscopic quantities appearing in thethermodynamical theory of the system� The free energy per unit volume isgiven by

f�� h� ��

j,j logZ�� h�,�B� ��

where � � T�� is the inverse temperature and

Z�� h�,�B� �X

�U��e��H ��

��

is the grand canonical partition function� Furthermore the probability of�nding the system in a con�guration � of the grand canonical ensembleU�,� is given by the Boltzmann factor

e��H ��

Z�� h�,�B�� U�,� � ��

The grandcanonical ensemble formalism based on �� correspondsto the physical situation in which there are no constraints on the system� Ifone could� by some experimental arrangement� regard for example the totalmagnetization M��

Px� �x as �xed M�� M � mj,j� then the

expression �� for the free energy would no longer be appropriate�One should rather consider the canonical ensemble� i�e� the set of the al�

lowed con�gurations would be the set U�,�m� � U�,� consisting of all the� � U�,� such that

Px� �x � mj,j� �jmj � �� and the Thermodynamics

would be described by the function

g�� h�m� ��

j,j logZ�� h�,�B�m� ��

whereZ�� h�,�B�m� �

X�U��m�

e��H ��

and the free energy would be "f�� h�

"f�� h� � hm�h� � g�� m�h�� #��

where m�h� is the solution of the equation�

h � �g�� m�

m��

if the maximum is over all pairs of spin con�gurations � and �� that di er only on D�i�e� such that �x � ��x for x �� D�

� Here we have not been precise about the problem of what ��m means� since g ��m�is only de�ned for certain rational values of m �whose number is �nite�� One could�for instance� extend g ��m� to all m s by considering instead of g ��m� its convexenvelope �or also one could prefer to consider the g obtained by linear interpolationfrom �� This is not very satisfactory but it should not be very important for largesystems� as discussed in Chap�IV�

�#� V I� Coexistence of Phases

There is no reason for having "f� � f� since they correspond to di�erentphysical problems� it is only when� in some sense� the $uctuations becomenegligible �i�e� in the limit , �� that one can expect the identity between"f and f �

Of course in general the di�erence between "f� and f� should vanish asj,j�� times O�j,j�d��d� �this means O�log j,j� for d � �� but� as we shallsee on many occasions� the situation is not so simple for other quantitiessuch as the correlation functions or the average magnetization�

As discussed in x�� the inequivalence� for �nite volume� of the predictionsof the canonical and grand canonical ensembles should not be interpretedas meaning that statistical mechanics is only approximate when applied to�nite systems� it simply means that in dealing with �nite systems atten�tion must be paid to the boundary conditions as a manifestation of thepeculiarities of the actual physical situation from which the problem underconsideration arises� We conclude by remarking that in the canonical en�semble the probability of a spin con�guration will be given by an expressionsimilar to ��

e��H ��

Z�� h�,�B�m�� U�,�m� � ��

x�� Boundary Conditions� Equilibrium States

Formulae �� or �� provide a complete statistical description ofthe properties of the system� An alternative and often more convenient�equally complete� description is provided by the so�called correlation func�tions

h�x��x� � � � �xni��B �

P� �x��x� � � � �xn e

��H ��P� e

��H ��

whereP

� is extended to the appropriate statistical ensemble� For instance

the average magnetization in the grand canonical ensemble U�,� is

m�� h� �f�� h�

h�

Px� h�xi��Bj,j ��

We shall refer to the family of correlation functions �� regarded as awhole� as the �equilibrium state of the system in the box ,��

We call an equilibrium state� see x�� of the in�nite system any familyfh�x� � � � �xnig of functions such that� for a suitable choice of the B��

h�x� � � � �xni � lim��

h�x� � � � �xni��B ��


for all n � � and all x�� x�� xn � Zd� simultaneously��

An equilibrium state for an in�nite system will simply be called an equi�librium state it is speci�ed by a suitable choice of a sequence fB��g ofboundary conditions satisfying the requirement �� Let us list a number of remarkable boundary conditions

�� Open boundary condition �also called �perfect�wall� boundary condi�tions�� This name will be given to the case

B�� for all � � U�,� ��

� � Periodic boundary conditions� This corresponds to allowing spins onopposite faces of the box , to interact through a coupling �J �i�e� as thebulk spins�� Clearly this can be obtained by a suitable choice of B�� weshall refer to this choice as �periodic boundary conditions��

�� boundary conditions� Let �� be the dj,j�d��d latticepoints adjacent to the boundary of ,� Let � � �� i � �� be�xed� We shall call ��boundary condition the choice

B�� JXxi��

�xj ��j ��

where �xi� �i� are nearest neighbors�The physical meaning of this boundary condition is clear we imagine

that the sites neighboring the boundary , of , are occupied by a spincon�guration � and that the latter spins interact with the spins � throughthe same coupling constant of the bulk spins�The cases � � �� or � � �� will be� respec�

tively� referred to as the ��boundary condition or the ��boundary con�dition�

�� In the two�dimensional case we shall be interested in another boundarycondition� Suppose that the spins on the opposite vertical sides of , areallowed to interact through a coupling�J �i�e� we impose periodic boundaryconditions along the rows of , only�� and suppose that a set �u of �xed spinsis located on the lattice sites adjacent to the upper base of , and� similarly�a set �l of �xed spins is adjacent to the lower base of ,� The spins �u� �l areallowed to interact with the nearest spins in , with a coupling �J � We shall

� This de�nition is essentially in �LR�� where the equivalence of the above de�nitionwith a number of other possible de�nitions is shown� For instance the de�nition inquestion is equivalent to that based on the requirement that the correlation functionsshould be a solution of the equations for the correlation functions that can be derivedfor lattice gases or magnetic spin systems in analogy to those we discussed for the gasesin x�� It is also equivalent to the other de�nitions of equilibrium state in terms oftangent planes �i�e� functional derivatives of a suitable functional� see �Ru�� p� ��Ga��

�#� V I� Coexistence of Phases

naturally refer to this choice of B�� as the ��u� �l��cylindrical boundarycondition�

The particular cases

�u �� l � ��

�u �� l � ��

will be referred to� respectively� as ��cylindrical boundary condition or��cylindrical boundary condition�

x�� The Ising Model in One and Two dimensions and zero �eld

To acquire some familiarity with the model we examine some of the simplestcases� Consider the one�dimensional Ising chain with periodic boundaryconditions� Labeling points of , as �� L� the zero �eld Hamiltonian is

H�� JLXi�

�i�i�� L� � ��

�clearly B�� J�L�� The grand canonical partition function can bewritten

Z�� X�

e�JP

L

i��i�i� �

X�

LYi�

e�J�i�i� � ��

Noting that ��i�i�� and therefore

e�J�i�i� � cosh�J � �i�i� sinh�J ��

�� can be rewritten as

Z�� cosh�J�LX�

LYi�

�� tanh�J �i�i��

If one develops the product in �� one gets a sum of terms of the form

�tanh �J�k�i��i��i��i�� ik�ik� � ��

It is clear that� unless k � � or k � L� each of the terms �� containsat least an index ij which appears only once� Therefore� after performingthe sum over the � s� all terms �� give a vanishing contribution toZ�� except the two with k � � and k � L which are� respectively� � and�tanh�J�L� �� L��L�L�� tanh�J�L� This implies

Z�� cosh�L�L L �� tanh�J�L� ��


Hence

�f�� log� cosh�J� ��

Llog�� tanh�J�L� � ��

It has to be remarked that �f�� as well as �f�� limL�� f�� log cosh�J � are analytic in �� this fact is usually referred to as the �absenceof phase transitions� in the one�dimensional Ising model� The reader cancheck� by using the above method� that the partition function in the grandcanonical ensemble and zero �eld but open boundary conditions �see x��is slightly di�erent from �� and� precisely� is equal to �cosh�J�L L�Consider now the two�dimensional Ising model in a zero �eld and with

open boundary conditions

H�� JLXi�

L��Xj�

�i j�i j� � J

L��Xi�

LXj�

�i j�i� j � ��#��

A better form for H�� is the following

H�� JXb

"�b ��

whereP

b denotes the sum over the bonds� i�e� over the segments b ��i� j�� i� j �� or b � ��i� j�� i � �� j�� and "�b is the product of the twospins at the extremes of b �e�g� if b � ��i� j�� i� �� j�� then "�b � �i j�i��j��The partition function can be written� as in the one�dimensional case� as

Z�� cosh�J��L�L��X�

Yb

�� tanh�J�"�b� ��

Developing the product we are led to a sum of terms of the type

�tanh �J�k"�b� "�b� � � � "�bk ��

and we can conveniently describe this term through the geometric set of linesb�� b�� bk� After the

P� is taken� many terms of the form �� give a

vanishing contribution� The ones that give a nonvanishing contribution arethose in which the vertices of the geometric �gure b� � b� � � � � � bk belongto an even number of bj s �two or four��

�a� �b�

The solution can also be found for instance in �NM��

�## V I� Coexistence of Phases

Fig� �� The dashed line is the boundary of !�

These terms are the ones such that "�b� � "�b� � � � "�bk � �� In Fig� ��awe give a typical nonvanishing term and in Fig� ��b an example of avanishing term �k � ��We shall� in the following� consider the geometric �gures built with k seg�

ments b�� bk such that "�b� � "�b� � � � "�bk � � and call it a k�sided multi�polygon on the box , �needless to say� all the b�� bk are pairwise di�er�ent�� Let Pk�,� be the number of such polygons�The partition function is now easily written as�

Z�� cosh�J��L�L�� L�Xk �

Pk�,� �tanh�J�k � ��

x�� Phase Transitions� De�nitionsWe have already seen� in the preceding section� that the one dimensional

Ising model has no phase transitions in zero �eld� since both f�� and f��are analytic in ��We recall brie$y in the concrete context of the Ising model the general

considerations of Chap�V about the de�nition of phase transition as a phe�nomenon of macroscopic instability slight changes of external conditionsshould imply dramatic changes of some macroscopic variables� it is hard toimagine how in such a situation thermodynamic functions� which we haveseen to be boundary�condition independent� like the free energy� the pres�sure� etc� could be analytic functions of the parameters in terms of whichthey are expressed �say� temperature� chemical potential or magnetic �eld�etc��For this reason an analytic singularity in the thermodynamic functions is

usually thought of as a �symptom� of a phase transition and on this idea itwould be possible to base a de�nition and a theory of the phenomenon ofphase transitions�Here� however� we will not base the investigation of the nature of phase

transitions in the Ising model on the search for singularities of the thermo�dynamic functions� we shall rather adopt and make more precise the other�perhaps more immediate and intuitive� approach based on the detection of�macroscopic instabilities�� introduced in Chap�V�This way of proceeding is more convenient for the simple reason that a

number of very clear and rather deep results have been obtained alongthese lines� But it should be understood that this second approach doesnot �brilliantly� avoid the di%culties of the �rst� It is simply an approachto the theory of phase transitions which� so far� has asked for and provideda less re�ned description of the phenomena of interest� as compared to the

� The expansion can be used as a starting point for the combinatorial solution mentionedabove� see �LL��


description which would be expected from the analysis of the singularities ofappropriate analytic functions �an analysis still in a very primitive stage andwhose problems are often not well formulated even in the simplest cases��

For this reason it provides a wealth of remarkable properties of the phasetransition phenomenon�Let us now discuss in a more precise way the concept of macroscopic in�

stability� Consider the Ising model and de�ne the condition that a phasetransition takes place at the values �� h� of the thermodynamic parame�ters if the system is unstable with respect to boundary perturbations� i�e� ifthere are at least two sequences B�� and B�� of boundary terms �see�� such that �say� in the grand canonical ensemble�

lim��

h�x� � � � �xni��B �� lim��

h�x� � � � �xni��B�

��

for a suitable choice of x�� x�� xn� n�We �rst clarify why we say that� if �� holds� we have a macroscopic in�

stability� We remark that a change in boundary conditions does not changethe extensive properties of the system such as the free energy� In fact� fromde�nition ��

Z�� h�,�B��

Z�� h�,�B�� emax�U� � jB ��jjB� ��j ��

and therefore �� implies

lim��

�

j,j logZ�� h�,�B�� lim��

�

j,jZ�� h�,�B��

On the other hand� if �� is true� intensive quantities like the correlationfunctions are sensitive to the boundary conditions� for instance if

lim��

h�x�i��B �� lim��

h�x�i��B� � ��

we realize that the local magnetization changes as a consequence of a changein boundary condition even if the boundary is very remote�Of course once provided with a �de�nition� of what a phase transition is�

one has not gone very far� The real question is whether the de�nition re$ectswhat is physically expected� this implies� in particular� that one should atleast be able to prove the existence of a phase transition� in the above

�� Of course we do not attach a deep physical meaning to the di erence between these twoapproaches� Clearly they should be equivalent if one pretended to extract all possibleinformation from them� What is really important is that the �rst questions raised byboth approaches are very interesting and relevant from a physical point of view� One ofthe goals of the analytic theory of phase transitions is to understand the nature of thesingularity at the critical point and at the �breaks of the isotherms� A lot of interesthas been devoted to this point and a number of enlightening phenomenological resultsare available� However the number of complete results on the matter is rather limited�An idea of the type of problems that are of interest can be obtained by reading thepapers �Ka�� or the more detailed paper �Fi��

�� V I� Coexistence of Phases

sense� in cases in which one expects a transition� Hopefully the de�nitionand its physical interpretation should allow one to do more for instanceto provide the tools for a closer description of typical phenomena �like thephase separation��Here we end this somewhat philosophical but necessary discussion and

in the coming sections we shall describe in the concrete example of theIsing model� some of the results that have been obtained since the early��s� when the above point of view was starting to be developed� quiteindependently� by several people�

x�� Geometric Description of the Spin Con�gurations

Here we introduce a new description of the spin con�gurations� which weshall use to derive in a very elegant way the exact value of the criticaltemperature in the two�dimensional Ising model� In the following sectionsthe geometric representation� introduced below� will be widely used�Consider an Ising model with boundary conditions of the type ��

boundary conditions� or with periodic boundary conditions �see x��Given a con�guration � � U�,� we draw a unit segment perpendicular to

the center of each bond b having opposite spins at its extremes �in three di�mensions we draw a unit square surface element perpendicular to b�� A two�dimensional example of this construction is provided by Fig� �� wherea very special ��boundary condition is considered��The set of segments can be grouped into lines �or surfaces in three di�

mensions� which separate regions where the spins are positive from regionswhere they are negative�It is clear that some of the lines �or surfaces� if d � �� are �closed polygons�

��closed polyhedra�� respectively� while others are not closed� It is perhapsworth stressing that our polygons are not really such in a geometrical sense�since they are not necessarily �self�avoiding� �see Fig� �� however theyare such that they can intersect themselves only on vertices �and not onsides�� From a geometrical point of view a family of disjoint polygons �inthe above sense and in two dimensions� is the same thing as a multi�polygonin the sense discussed in x�� Fig� ��In two dimensions instead of saying that a polygon is �closed� we could

equivalently say that its vertices belong to either two or four sides�We note that the ��boundary conditions� the ��boundary conditions

and the periodic boundary conditions are such that the lines �surfaces�associated with spin con�gurations are all closed polygons �polyhedra�� Inthe periodic case some polygons might wind up around the two holes of thetorus�In the two�dimensional case and if the boundary conditions are the ��

cylindrical or the ��cylindrical ones �see x�� a geometric constructionof the above type can still be performed and� also in this case� the lines areclosed polygons �some of which may �wind around� the cylinder ,��

V I� Coexistence of Phases ��

A O B

Fig� �� The dashed line is the boundary of !� the outer spins are those �xed by the

boundary condition� The points A�B are points where an open line ends�

For a �xed boundary condition let �� h� �� k� be the disjointcomponents of the set of lines �surfaces� associated by the above construc�tion with a spin con�guration � � U�,�� The �� h are closed polygonsand the �� k are not closed� The example in Fig� �� has one noclosed polygon only �due to the special nature of the boundary condition��half � and half ��Clearly the correspondence between �� h� �� k� and � is� fora �xed boundary condition� one�to�one except for the case of the periodicor open boundary conditions� when it is one�to�two� Changing boundaryconditions implies changing the set of lines �surfaces� which describe thesame spin con�guration ��

A very important property of the above geometric description is that� ifj�j� j�j denote the length �area� of the lines �surfaces� � and �� then theenergy of a spin con�guration is� in zero �eld� given by

H�� J � �number of bonds in ,� � J �Xi

j�ij�Xj

j�j j� � ��

This remark easily follows from the fact that each bond b contributing�J tothe energy has equal spins at its extremes� while the bonds contributing �J


have opposite spins at their extremes and� therefore� are cut by a segmentof unit length belonging to some �i or �j �If N�� number of bonds in ,�� the partition function becomes �in zero

�eld and with �xed spin boundary conditions�

Z�� X

��h

X��k

�e��J

Pij�ij � e��J

Pjj�j j�� e�JN ��

��

where the sum runs over the set of lines associated with a spin con�guration� � U�,� and with the boundary condition under consideration�In the case of periodic or open boundary conditions there may be no � s

�this happens in the periodic case� and there is an extra factor �becausein this case the correspondence between � and �� h� is two�to�one��in the periodic case

Z�� X

��h

e��JP

ij�ij � e�JN ��

and N� � L��Form the above considerations we draw two important consequences

�I� If the boundary condition is �xed� the probability of a spin con�guration� described by �� h� �� k is proportional to

e��J

�Pij�ij

Pjj�j j�

��

�II� In the case of �� or �� boundary conditions and two dimensions weremark that

P��k

in �� is a sum over �multi�polygons� lying on a

shifted lattice and in a box ,� containing �L � �� spins �see the de�nitionin x�� and� therefore� if

Pi j�ij � k we have

Z�� e��L�L��J�Xk �

Pk�,�� e��Jk ��

where Pk�,�� is the number of di�erent multi�polygons with perimeter k�see ��

If we now de�ne � through

tanh�J � e��J ��

with , replaced by a volume ,� with side L� � then a comparison between�� and �� yields

Z��

L��cosh�J��L�L��

Z��

e��JL�L��


Here Z�� is computed with open boundary conditions� while Z�� is

computed with ��boundary conditions�If we assume that the bulk free energy f�� lim��

j�j logZ�� has

one and only one singularity as a function of �� for � real� then �� canbe used to locate the singularity� In fact this implies

f�� log �cosh�J�� f�� J ��#��

having used the fact that the free energy is boundary�condition independent�see �� Hence a singularity in �� if unique� can take place only when� � �� i�e� for � � �c�O such that

tanh�c�O � e��c�OJ ��

which� indeed� has been shown by Onsager� �On�� to be the exact valueof the critical temperature de�ned as the value of � where f�� is singular�in the sense that its derivative diverges��

In the next section we outline the theory of phase transitions in the Isingmodel as a macroscopic instability and a spontaneous breakdown of up�down symmetry� We shall concentrate� for geometric reasons� on the two�dimensional Ising model but� unless explicitly stated� the results hold in alldimensions d � �

�� This geometric picture of the spin con�gurations can be traced back at least as faras Peierls paper� �Pe�� and has been used� together with formula �� to derive�� the �KramersWannier duality relation� and �� KW�� A recent interesting generalization of the duality concept has been given in �We�� where somevery interesting applications can be found as well as references to earlier works� Theduality relation between �� or �� boundary conditions and open boundary conditions�which is used here� has been realized by several people� The reader can �nd othersimilar interesting relations in �BJS�� and further applications came in �BGJS�� Duality has found many more applications� see for instance �GHM�� and� for a recent one��BC�� In particular a rigorous proof of the correctness of the OnsagerYang value ofthe spontaneous magnetization is derived in �BGJS��


x�� Phase Transitions� ExistenceIn this section we shall show that the ��boundary conditions and the ��

boundary conditions �see x�� produce� if the temperature is low enough�di�erent equilibrium states �see x�� i�e� for large � the correlation func�tions are di�erent and the di�erence does not vanish in the limit , � ��see ��More precisely we shall prove that if h � � and � is large enough then

lim��

h�xi�� m��

where the index � refers to the boundary conditions�Clearly �� shows that the magnetization is unstable �in zero �eld and at

low temperature� with respect to boundary perturbations� We also remarkthat by using periodic boundary conditions one would obtain still anotherresult

lim��

h�xi�� periodic � �� if h � � ��

because h�xi�� periodic � �� if h � �� for obvious symmetry reasons�After a description of the very simple and instructive proof of �� we

shall go further and discuss more deeply the character of the phase transi�tion�As already remarked� spin con�gurations � � U�,� are described in terms

of closed polygons �� n� if the boundary condition is �� or ��and the probability of a con�guration � described by �� n� is pro�portional to �see ��

e��JP

ij�ij � ��

Below we identify � with �� n� �with the boundary condition�xed��Let us estimate h�xi�� Clearly h�xi�� P�� where P��

is the probability that in the site x the spin is ��We remark that if the site x is occupied by a negative spin then the pointx is inside some contour � associated with the spin con�guration � underconsideration� Hence if �� is the probability that a given contour belongsto the set of contours describing a con�guration �� we deduce

P�� X�ox

��

where �ox means that � �surrounds� x�Let us now estimate �� if ' � �� n� is a spin con�guration and

if the symbol ' comp� means that the contour � is �disjoint� from �or�compatible� with� �� n �i�e� f� � 'g is a new spin con�guration��then

��

P�� e

��JP

��j��j

P� e

��JP

��j��j � e��Jj�j

P� comp� e

��JP

��j��j

P� e

��JP

��j��j

��

V I� Coexistence of Phases �

Before continuing the analysis let us remark that if � � �� n�then �� n� is obtained from � by reversing the sign of thespins inside �� this can be used to build an intuitive picture of the secondequation in �� Clearly the last ratio in �� does not exceed �� hence

�� e��Jj�j ��

Letting p � j�j and observing that there are at most �p di�erent shapes of �with perimeter p and at most p� congruent � s containing �in their interior�x� we deduce from ��

P�� Xp�

p��pe��Jp � ��

Hence if � �� i�e� the temperature T � �� this probability can be madeas small as we like and� therefore� h�xi�� is as close to � as we like provided� is large enough� It is of fundamental importance that the closeness ofh�xi�� to � is both x and , independent�A similar argument for the ��boundary condition� or the remark thath�xi�� h�xi�� allows us to conclude that� at large �� h�xi�� h�xi�� and the di�erence between the two quantities is uniform in ,�Hence we have completed the proof ��Peierls� argument�� of the fact that

there is a strong instability with respect to the boundary conditions of somecorrelation functions��

We can look upon the above phenomenon as a spontaneous breakdown ofup�down symmetry the Hamiltonian of the model is symmetric� in a zero�eld� with respect to spin reversal if one neglects the boundary terms� thephase transition manifests itself in the fact that there are equilibrium statesin which the symmetry is violated �only on the boundary� and which arenot symmetric even in the limit when the boundary recedes to in�nity�

x�� Microscopic Description of the Pure Phases

The description of the phase transition presented in x�� can be made moreprecise from the physical point of view as well as from the mathematicalpoint of view� A deep and physically clear description of the phenomenonis provided by the theorem below� which also makes precise some ideasfamiliar from a model� which we shall not discuss here� but which playsan important role in the development of the theory of phase transitionsnamely the droplet model� �Fi��c��Assume that the boundary condition is the ��boundary condition and

describe a spin con�guration � � U�,� by means of the associated closeddisjoint polygons �� n��

�� The above proof is due to R�B� Gri�ths and� independently� to R�L� Dobrushin and itis a mathematically rigorous version of �Pe��


We regard the ensemble U�,� as equipped with the probability distributionattributing to � � �� n� a probability proportional to ��Then the following theorem holds

Theorem� If � is large enough there exist C � � and �� with �� e��Jj�j and such that a spin con�guration � randomly chosen out of theensemble U�,� will contain� with probability approaching � as , � �� anumber K�� of contours congruent to � such that

jK�� j,jj � Cpj,j e��Jj�j ��#��

and this relation holds simultaneously for all ��s� In three dimensions onewould have j,j�� instead of

pj,j�It is clear that the above theorem means that there are very few contours

�and that the larger they are the smaller is� in absolute and relative value�their number�� The inequality ��#�� also implies that for some C�� thereare no contours with perimeter j�j � C�� log j,j �with probability ap�proaching � as , � �� this happens when ��j,j � � �because K��is an integer and the right�hand side of ��#�� is � �� Hence a typicalspin con�guration in the grand canonical ensemble with ��boundary con�ditions is such that the large majority of the spins is �positive� and� in the�sea� of positive spins� there are a few negative spins distributed in smalland rare regions �in a number� however� still of order of j,j��Another nice result which follows from the results of x�� and from some

improvement� �BS�� of them� concerns the behavior of the equation ofstate near the phase transition region at low �enough� temperatures�

O�j,j��O�j,j��

m��

�m��

h

�m�� h�

Fig� ��#��


If , is �nite the graph of h � m�� h�f will have a rather di�erent be�havior depending on the possible boundary conditions� e�g� if the boundarycondition is �� or �� one gets respectively the results depicted in Fig��#�� and Fig� ��#� � where m�� denotes the spontaneous magnetizationlimh�� lim��m�� h��


m��

�m��

h

�m�� h�

Fig� ��#�

With periodic boundary conditions the diagram changes as in Fig� ��#��


m��

�m��

h

�m�� h�

Fig� ��#��

The thermodynamic limit m�� h� � lim��m�� h� exists for all h �� and the resulting graph is as shown in Fig� ��#��

m��

�m��

h

�Fig� ��#��

m�� h�

At h � � the limit is not well de�ned and it depends on the boundarycondition �as it must�� It can be proven� if � is large enough� that

limh��

m�� h�

h� �� #� ��

��# V I� Coexistence of Phases

is a �nite number �i�e� the angle between the vertical part of the graph andthe rest is sharp� �BS��

The above considerations and results also provide a clear idea of what aphase transition for a �nite system means�

It is often stated that a �nite system �does not� show �sharp� phase tran�sitions� however this statement is always made when considering a �xedboundary condition� usually of periodic or perfect�wall type� By taking intoaccount the importance of the boundary terms we see which kind of phe�nomena occur in a �nite system� if the corresponding in�nite system has asharp phase transition�

The next section is devoted to the discussion of a number of problems con�cerning the generality of the de�nition of a phase transition as an instabilitywith respect to the boundary perturbations� and other related problems� inthe special case of the Ising model that we are discussing�

x�� Results on Phase Transitions in a Wider Range of Temper�

ature

An unpleasant limitation of the results discussed above is the conditionof low temperature �� large enough�� The results of the preceding sec�tions show that� at a low enough temperature� the Ising model is unstablewith respect to changes in the boundary conditions� A natural question iswhether one can go beyond the low�temperature region and fully describethe phenomena in the region where the instability takes place and �rst de�velops� In the particular case of two dimensions it would also be natural toask whether the maximum value of � to which an instability is associatedis the one given by �� which corresponds to the value of � where thein�nite volume free energy f�� has a singularity� the critical point�

The above types of questions are very di%cult and are essentially related tothe already mentioned theory of the phase transitions based on the searchand study of analytic singularities of the thermodynamic functions �whichis a theory� however� that has still to be really developed��

Nevertheless a number of interesting partial results are known� which con�siderably improve the picture of the phenomenon of the phase transitionsemerging from the previous sections� A list of such results follows

�� It can be shown that the zeros of the polynomial in z � e�h givenby the product of zj�j times the partition function �� with periodicor perfect�wall boundary conditions lie on the unit circle jzj � � ��Lee�Yang�s theorem�� It is easy to deduce� with the aid of Vitali s convergencetheorem for equibounded analytic functions� that this implies that the onlysingularities of f�� h� in the region � � � � �� h � �� can befound at h � ��

A singularity appears if and only if the point z � � is an accumulationpoint of the limiting distribution �as , �� of the zeros on the unit circle�


In fact� if the zeros in question are z�� z�j j then

�

j,j log zj�jZ�� h�,� periodic� � �J � �h ��

j,j�j�jXi�

log�z � zi� ��

and if j,j�� number of zeros of the form zj � ei�j with � j � �

d �� d�� in a suitable sense� we get from ��

�f�� h� � �J � �h ��

�

Z

�log�z � ei�� d ��

where the second term comes from the jzjj�j appearing in ��The existence of the measure �� d�� such that �� is true fol�

lows� after some thought� from the existence of the thermodynamic limitlim�� f�� h� � f�� h��

� � It can be shown that the zeros of the partition function do not movetoo much under small perturbations of the spin�spin potential even if oneallows �many spin� interactions� i�e� even if one perturbs the Hamiltonian�� with perfect�wall boundary conditions into

H �� H�� H��

��H�� Xk �

Xx��xk�

�

k�J ��x�� xk��x� � � � �xk

��

where J ��X� is a function of the set X � �x�� xk� such that

jjJ �jj � supyZd

XyX

jJ ��X�j ��

is small enough�More precisely� suppose that one knows that� when J � � �� the zeros of

the partition function in the variable z � e�h lie in a certain closed set Nof the z�plane� Then if J � �� they lie in a closed set N� contained in aneighborhood of N which can be made as small as we please when jjJ �jj � ��This result� �Ru��b�� allows us to make a connection between the analyt�

icity properties and the boundary condition instability as described in ��below�

�� Here the symbol ��d�� has not to be taken too seriously� it really denotes ameasure on the circle and this measure is not necessarily d�continuous� Also the �convergence statement really means the existence of a measure such that �� holds forall real z� The original proof of this theorem is in �LY�� A much stronger and generalstatement� the Ruelle�s theorem� leading in particular to the Lee�Yang�s theorem is in�Ru��a�� it has been one of the most remarkable among a series of improvements andgeneralizations of LeeYang s theorem �among which I quote �As�� Ru��b��


�� There can be a boundary condition instability only in zero �eld and� inthis case� if and only if the spectrum �� does not vanish around � �one says that there is a gap around � if �� near � ��

The proof of this result relies upon � and the remark that the correlationfunctions are functional derivatives with respect to J ��x�� xk� of the freeenergy de�ned by the Hamiltonian �� Ru��b��

�� Another question is whether the boundary condition instability is al�ways revealed by the one�spin correlation function �as in x�� or whetherit might be shown only by some correlation functions of higher order� Thisquestion is answered by the following result�

There can be a boundary condition instability �at h � � and � �xed� ifand only if

limh��

m�� h� �� limh��

m�� h� ��

Note that� in view of what was said above �point �� m�� h� �lim��m�� h� is boundary condition independent as long as h ��

In other words there is a boundary condition instability if and only ifthere is spontaneous magnetization� This rules out the possibility that thephase transition could manifest itself through an instability of some higher�order correlation function which� practically� might be unobservable froman experimental point of view �ML� ��

�� Point �� implies that a natural de�nition of the critical temperatureTc is to say that it is the least upper bound of the T s such that �� istrue �T � �� It is clear that� at this temperature� the gap around � �closes and the function f�� h� has a singularity at h � � for � � �c � T��c �It can in fact be proven that if �� is true for a given �� then it is truefor all � � �� Gr�� Fi��

�� The location of the singularities of f�� as a function of � remainsan open question for d � �� see however �Gr��Fi�� In particular thequestion of whether there is a singularity of f�� at � � �c is open� Theidentity �c � �c�� for the two�dimensional Ising model has been proved in�BGJS�� and� independently� in �AM��

�� Finally another interesting question can be raised� For � � �c we haveinstability with respect to the boundary conditions �see �� above� howstrong is this instability� In other words� how many �pure� phases canexist�

Our intuition� in the case of the Ising model� suggests that there should beonly two phases the positively magnetized and the negatively magnetizedones�

To answer the above question in a precise way it is necessary to agreeon what a pure phase is� �Ru�� p� �� We shall call �pure phase� anequilibrium state �see footnote #� x�� and ��#�� if it is translationally


invariant and if the correlation functions have a cluster property of the form

h�x� � � � �xn�y�a � � � �ymai��a�� h�x� � � � �xnih�y� � � � �ymi ��

where convergence is understood in a very weak sense� i�e� the weakestsense which still allows us to deduce that the $uctuations of the extensivequantities are o�j,j�� Fi�� which is

�

j,jXa�

h�x� � � � �xn�y�a � � � �ymai�� h�x� � � � �xnih�y� � � � �ymi

�� i�e� the convergence in �� takes place in the �Cesaro�s limit� sense�It can be proved that� in the case of the Ising model� the two states obtained

as limits for , �� of �nite volume states �see x�� corresponding to ��boundary conditions or ��boundary conditions are di�erent for � � �cand are pure phases in the sense of �� above��

Actually it can be proved that� in this case� the limits �� exist in theordinary sense� �GMM� �� rather than in the Cesaro sense� and that at lowtemperature they are approached exponentially fast� see �MS��Furthermore� if � is large enough �e�g� in two dimension �� larger than�c�� these two pure phases exhaust the set of pure phases �GM� a�� Ma� ��For � close to �c� however� the question is much more di%cult neverthelessit has been completely solved in a remarkable series of papers based on thekey work �Ru��b�� see �Hi#�� Ai#�� The work �Ru��b� did provide a realbreakthrough and a lot of new ideas for the theory of the Ising model andpercolation theory� �Ru#�� Hi�� The solution of this problem has led tothe introduction of many new ideas and techniques in statistical mechanicsand probability theory�Another approach� very rich in results� to the theory of correlation func�

tions originates from the combination of the Gri%ths� FKG and other in�equalities� see �Gr�� FKG�� Le�� with the infrared bounds introducedin the work of Mermin and Wagner� �MW�� see x��#� I only quote herethe work �Fr#�� mentioned in x��# of Chap�V� where the reader can �nda very interesting analysis of the behavior at the critical point of variouscorrelations and a clear discussion of the relevance of the dimension of thelattice �if the dimension is � � the correlations are �trivial��Having discussed some exact results about the structure of the phase tran�

sition and the nature of pure phases� we shall turn in the next section tothe phenomenon of the coexistence of two pure phases�

x�� Separation and Coexistence of Pure Phases� Phenomeno�

logical Considerations

Our intuition about the phenomena connected with the classical phasetransitions is usually based on the properties of the liquid�gas phase transi�tion� this transition is experimentally investigated in situations in which the

�� This is an unpublished result of R�B� Gri�ths� His proof is reported in �GMM��

� V I� Coexistence of Phases

total number of particles is �xed �canonical ensemble� and in the presenceof an external �eld �gravity��

The importance of such experimental conditions is obvious� the external�eld produces a nontranslationally invariant situation and the correspondingseparation of the two phases� The fact that the number of particles is �xeddetermines� on the other hand� the fraction of volume occupied by each ofthe two phases� The phenomenon of phase transitions in the absence of anexternal �eld will be brie$y discussed in x��

In the framework of the Ising model it will be convenient to discuss thephenomenon of phase coexistence in the analog of the canonical ensembleU�,�m�� introduced and discussed in x�� where the total magnetizationM � mj,j is held �xed�

To put ourselves in the phase transition region we shall take � large enoughand� for a �xed ��

m � �m�� m�� m��

i�e� we put ourselves in the vertical �plateau� of the diagram �m�h�� seeFig� ��#�� above��

Fixing m as in �� does not yet determine the separation of the phasesin two di�erent regions� to obtain this e�ect it will be necessary to intro�duce some external cause favoring the occupation of a part of the volumeby a single phase� Such an asymmetry can be obtained in at least twoways through a weak uniform external �eld �in complete analogy with thegravitational �eld in the liquid�vapor transition� or through an asymmetric�eld acting only on the boundary spins� This second way should have thesame qualitative e�ect as the former� because in a phase transition regiona boundary perturbation produces volume e�ects �this last phenomenon�which has been investigated in the previous sections� is often also referredto as the �long�range order� of the correlations��

From a mathematical point of view it is simple to use a boundary asym�metry to produce phase separations�

To obtain a further� but not really essential� simpli�cation of the problemconsider the two�dimensional Ising model with ��cylindrical or ��cylindrical boundary conditions�

The spins adjacent to the bases of , act as symmetry�breaking external�elds� The ��cylindrical boundary condition should� clearly� favor theformation inside , of the positively magnetized phase� therefore it will benatural to consider� in the canonical ensemble� this boundary condition onlywhen the total magnetization is �xed to be �m�� see Fig� ��#��

On the other hand� the boundary condition �� favors the separationof phases �positively magnetized phase near the top of , and negativelymagnetized phase near the bottom��

Therefore it will be natural to consider this boundary condition in thecase of a canonical ensemble with magnetization m � �� m�� with� � � � ��


In this last case one expects� as already mentioned� the positive phase toadhere to the top of , and to extend� in some sense to be discussed� up toa distance O�L� from it� and then to change into the negatively magnetizedpure phase�To make precise the above phenomenological description we shall describe

the spin con�gurations � � U�,�m� through the associated sets of disjointpolygons �cf� x��Fix the boundary conditions to be �� or ��cylindrical boundary

conditions and note that the polygons associated with a spin con�guration� � U�,�m� are all closed and of two types the ones of the �rst type�denoted �� n� are polygons which do not encircle ,� the second typeof polygons� denoted by the symbols �� are the ones which wind up� atleast once� around ,�So a spin con�guration � will be described by a set of polygons �� n� �� h�� It is� perhaps� useful to remark once more that thecon�guration � will be described by di�erent sets of polygons accordingto which boundary condition is used �among the ones we are considering�i�e� �� or ��boundary conditions�� However� for a �xed boundarycondition� the correspondence between spin con�guration and sets of disjointclosed contours is one�to�one and the statistical weight of a con�guration� � �� n� �� h� is �cf� ��

e��J�

Pij�ij

Pjj�j j� � ��

It should also be remarked that the above notation is not coherent withthe notation of x�� where the symbol � is used for open polygons �absenthere�� but this will not cause any confusion� The reason why we call �the contours that go around the cylinder , is that they �look like� opencontours if one forgets that the opposite sides of , have to be identi�ed�It is very important to remark that if we consider the ��boundary

conditions then the number of polygons of ��type must be odd �hence �� while if we consider the ��boundary condition then the number of ��type polygons must be even �hence it could be ��

x�� Separation and Coexistence of Phases� Results

Bearing in mind the geometric description of the spin con�guration inthe canonical ensembles considered with the ��cylindrical or the��cylindrical boundary conditions �which we shall denote brie$y asU�,�m�� U��,�m�� we can formulate the following theorem� �GM� b��essentially developed by Minlos and Sinai to whom the very foundations ofthe microscopic theory of coexistence is due

Theorem� For � � � � � �xed let m � �� m�� then for � largeenough a spin con�guration � � �� n� �� h�� randomly chosenout of U��,�m� enjoys the properties �� below with a probability �inU��,�m�� approaching � as , ��


�� contains only one contour of ��type and

j j�j � �� Lj � o�L� ��

where �� is a suitable ��independent� function of � tending to zeroexponentially fast as � ��

�� If ,��,�� denote the regions above and below � we have

j j,�j � � j,j j � �� j,j��

j j,��j � �� j,j j � �� j,j��

where �� exponentially fast�

�� If M� �P

x�� x� we have

jM� � �m�� j,jj � ��j,j��

and a similar inequality holds for M ��

Px��

��x � mj,j �M��

�� If K�� denotes the number of contours congruent to a given � and

lying in ,� then� simultaneously for all the shapes of �

jK�� j,j j � Ce��Jj�j

pj,j C � � ��

where �� e��Jj�j is the same quantity already mentioned in the text ofthe theorem of x�� A similar result holds for the contours below � �cf� thecomments on ��

It is clear that the above theorem not only provides a detailed and rathersatisfactory description of the phenomenon of phase separation� but alsofurnishes a precise microscopic de�nition of the line of separation betweenthe two phases� which should be naturally identi�ed with the �random� line��A very similar result holds in the ensemble U�,�m�� in this case ��

is replaced by

�� no ��type polygon is present

while � �� become super$uous and �� is modi�ed in the obvious way� Inother words a typical con�guration in the ensemble U�,�m�� has thesame appearance as a typical con�guration of the grand canonical ensembleU�,� with ��boundary condition �which is described by the theorem ofx��#��We conclude this section with a remark about the condition that � � � � �

has to be �xed beforehand in formulating the above theorem� Actually theresults of the theorem hold at �xed � �small enough� for all the � s such


that �� min�� i�e� such that the line � cannot touch the basesof , �in which case there would be additional physical phenomena andcorrespondingly di�erent results��

x�� Surface Tension in Two Dimensions� Alternative Descrip�

tion of the Separation Phenomena

A remarkable application of the above theorem is the possibility of givinga microscopic de�nition of surface tension between the two pure phases��GM� b�� We have seen that the partition functions

Z�,� �� X

�U��m��e��J�

Pij�ij

Pjj�j j� ��

and �if m � �� m��

Z��,� �� X

�U��m�

e��J�

Pij�ij

Pjj�j j� ��

will essentially di�er� at low temperature� only because of the line � �presentin U��,�m� and absent in U�,�m�� see the preceding section��A natural de�nition �in two dimensions� of surface tension between the

phases� based on obvious physical considerations� can therefore be givenin terms of the di�erent asymptotic behavior of Z�,�m�� or of thegrand canonical Z�,� �� and Z��,�m�

� �� lim��

�

Llog

Z��,�m�

Z�,�m��

The above limit �which should be ��independent for �� min�� cf�the concluding remarks of the preceding section� can be exactly computedat low enough temperature and is given by

� �� J � log tanh�J ��

which is the value computed by Onsager� �On�� by using a di�erent de�ni�tion� not based on the above detailed microscopic description of the phasesand of the line of separation for a comparison of various old de�nitions ofsurface tension� new ones and a proof of their equivalence see �AGM��We conclude this section with a brief discussion of one particular but very

convenient alternative way of investigating the phenomenon of coexistenceof two phases� Another still di�erent way of investigating the phenomenonwill be discussed in x��Consider the grand canonical ensemble� but impose the following boundary

conditions the spins adjacent to the upper half of the boundary of , are�xed to be �� while the ones adjacent to the lower half are �� and noperiodicity condition�� This is an ��type boundary condition �see x�� and


Fig� �� or also cover �gure� generating an ensemble that we shall denoteby U�

� �,��It is clear that a con�guration � � U�

� �,� is described� under the aboveboundary condition� by one single open polygon � �surface in three dimen�sions� going from one side of , to the opposite side� and by a set of disjointclosed polygons �polyhedra in three dimensions� �� n��The surface � now plays the role of the polygons encircling , in the case

of cylindrical boundary conditions �and two dimensions� and it is also clearthat a theorem very similar to those already discussed should hold in thiscase� The above point of view is more relevant in the three�dimensional casewhere a �cylindrical� boundary condition would have a less clear physicalmeaning� and it would look rather a mathematical device�In the three�dimensional case � is a �surface� with a boundary formed by

the square in the �middle� of , where the �break� between the spins �xedto be �� and the ones �xed to be �� is located�In the next section we investigate in more detail the structure of such a

line or surface of separation between the phases�

x�� The Structure of the Line of Separation� What a Straight

Line Really is

The theorem of x�� tells us that� if � is large enough� then the line � isalmost straight �since �� is small�� It is a natural question to ask whetherthe line � is straight in the following sense suppose that �� regarded asa polygon belonging to a con�guration � � U��,�m� �cf� x�� passesthrough a point q � ,� then we shall say that � is �straight� or �rigid� if the�conditional� probability P� that � passes also through the site q�� opposite�

to q on the cylinder ,� does not tend to zero as , �� otherwise we shallsay that � is not rigid or �uctuates� Of course the above probabilities mustbe computed in the ensemble U��,�m��Alternatively �and essentially equivalently� we can consider the ensembleU�� ,� �see x�� i�e� the grand canonical ensemble with the boundary

condition with the boundary spins set to �� in the upper half of ,� verticalsites included� and to �� in the lower half�� We say that � is rigid if theprobability that � passes through the center of the box , �i�e� �� does nottend to � as , �� otherwise it is not rigid�It is rather clear what the above notion of rigidity means the �excess�

length ��L� see �� can be obtained in two ways either the line � isessentially straight �in the geometric sense� with a few �bumps� distributedwith a density of order �� or� otherwise� the line � is bent and� therefore�only locally straight and part of the excess length is gained through thebending�In three dimensions a similar phenomenon is possible� As remarked at the

end of the last section� in the ensemble U�� ,�� in this case � becomes a

� i�e� on the same horizontal line and L�� sites apart�


surface with a square boundary �xed at a certain height �i�e� �� and we askwhether the center of the square belongs to � with non vanishing probabilityin the limit , ��The rigidity or otherwise of � can� in principle� be investigated by optical

means� one can have interference of coherent light scattered by surface el�ements of � separated by a macroscopic distance only if � is rigid in theabove sense�It has been rigorously proved that� at least at low temperature� the line of

separation � is not rigid in two dimensions �and the $uctuation of the middlepoint is of the order O�

pj,j�� a very detailed description of the separationpro�le is available� at low temperature ��Ga� a��GV� ��Ga� b�� and evenall the way to the critical point �AR�� In three dimensions the situation isvery di�erent it has been shown that the surface � is rigid at low enoughtemperature� see �Do� �� VB�� The latter reference provides a very niceand simple argument for the three�dimensional rigidity�An interesting question remains open in the three�dimensional case and

is the following it is conceivable that the surface� although rigid at lowtemperature� might become loose at a temperature "Tc smaller than thecritical temperature Tc �the latter being de�ned as the highest temperaturebelow which there are at least two pure phases�� The temperature "Tc� ifit exists� is called the �roughening transition� temperature� see �KM#��VB�� KM#�� VN#�� It would be interesting to examine the available experimental data on the

structure of the surface of separation to set limits on Tc� "Tc in the case of theliquid�gas phase transition where an analogous phenomenon can conceivablyoccur even though a theory of it is far from being in sight� at least if onerequires a degree of rigor comparable to that in the treatment of the resultsso far given for the Ising model�We conclude by remarking that the rigidity of � is connected with the

existence of translationally noninvariant equilibrium states �see x��The discussed nonrigidity of � in two dimensions provides the intuitive

reason for the absence of nontranslationally invariant states�Note that the existence of translationally noninvariant equilibrium states

is not necessary for the description of coexistence phenomena� The theoryof the two�dimensional Ising model developed in the preceding sections is aclear proof of this statement�

x�� Phase Separation Phenomena and Boundary Conditions�

Further Results

The phenomenon of phase separation described in x�� and x�� is theferromagnetic analogue of the phase separation between a liquid and a vaporin the presence of the gravitational �eld�It is relevant to ask to what extent an external �eld �or some equivalent

boundary condition� is really necessary� for instance one could imagine asituation in which two phases coexist in the absence of any external �eld�


Let us discuss �rst some phenomenological aspects of the liquid�gas phaseseparation in the absence of external �elds� One imagines that� if the den�sity is �xed and corresponds to some value on the �plateau� of the phasediagram� see Fig� �� then the space will be �lled by vapor and drops ofliquid in equilibrium� Note that the drops will move and� from time to time�collide� since the surface tension is negative the drops will tend to clustertogether and� eventually� in an equilibrium situation there will be just onebig drop �and the drop surface will be minimal�� The location of the dropin the box , will depend on how the walls are made and how they interactwith the particles within ,�Let us consider some extreme cases

�� the walls �repel� the drops�

� � the walls �attract� the drops�

�� the wall is perfect and does not distinguish between the vapor and theliquid�

In the �rst case the drop will stay away from the boundary , of ,� In thesecond case the drop will spread on the walls� which will be wet as muchas possible� In the third case it will not matter where the drop is� the dropwill be located in a position that minimizes the �free� part of its boundary�i�e� the part of the boundary of the drop not on ,�� This means that thedrop will prefer to stay near a corner rather than wetting all the wall�Let us translate the above picture into the Ising model case� Assume

that � is large and m � �� m�� see Fig� ��#�� i�e� assume thatthe magnetization is on the vertical plateau of the �m�h�� diagram in Fig��#��Then the conditions �� can be realized as follows

�� The spins adjacent to the boundary are all �xed to be �� This favorsthe adherence to the boundary of the positively magnetized phase�

� � The spins adjacent to the boundary are all �xed to be �� This favorsthe adherence to the boundary of the negatively magnetized phase�

�� There are no spins adjacent to the boundary� i�e� we consider perfectwall or open �or free� boundary conditions �see x��

The rigorous results available in the case of the Ising model con�rm theabove phenomenological analysis of liquid�vapor coexistence �MS��

Theorem� Fix � � � � � and consider ��boundary conditions� Then aspin con�guration � randomly extracted from the canonical ensemble withmagnetization m � �� m�� has� if � is large enough� properties�� below with a probability tending to � as , ��


�� There is only one � such that j�j � �� log j,j and it has the property��

�� j�j � �p

�� j,j �� pj,j ��

��

with �� as � � � �exponentially fast�� The area enclosed by� is �� j,j �� j,j��

��

�� The magnetization M� �� inside � is on the average equal to �m��and� more precisely�

��M� �� m�� j,j �� j,j��

and� therefore� the average magnetization outside �� is �m��

This theorem also holds in three dimensions but� of course� the exponentof j,j in �� changes �from �

� to ��

The above theorem shows that a typical con�guration consists of a pos�itively magnetized pure phase adherent to the boundary and of a �drop�of negatively magnetized phase not adhering to the boundary �since � isclosed�� The size of the drop is p�� j,j �as it should be��Note that the drop is almost square in shape �as follows from ��

�� this should not be astonishing since the space is discrete andthe isoperimetric problem on a square lattice has the square as a solution�rather than a circle��The opposite situation is found if one �xes a ��boundary condition� a

square drop forms in the middle of the box with side p�j,j and averagemagnetization m��Finally if the boundary condition is of perfect wall type �B�� then

the above theorem does not hold and one can prove �say� in two dimen�sions� that a typical spin con�guration has just one open contour � �withends on ,� which separates the space in two parts which are occupied byopposite phases� the line � should be the shortest possible compatible withthe condition that the volume , is divided by it into two regions of vol�ume essentially �j,j and ��j,j �respectively occupied by the positivelymagnetized phase and by the negatively magnetized phase� see �Ku#��If one interprets the spins equal to �� as particles and the spins equal to ��

as empty sites� then one has a lattice gas model which undergoes a liquid�vapor phase transition presenting the phenomenological aspects outlined atthe beginning of this section for these transitions�

�� The number �� is just an arbitrary constant and it is reported here because it appearedin the original literature� �MS�� as a joke referring to the contemporary papers on theKAM theorem ��Moser s constant�� In fact it looks today somewhat confusing andquite strange� the modern generation do not seem to appreciate this kind of humourany more� they became more demanding and would rather ask here for the �bestconstant� this is my case as well�


To conclude we remark that� in the phase separation phenomenon� the�niteness of the box only plays the role of �xing the density� The detailedstructure of the phenomenon depends on the boundary conditions which�in experimental situations� turn out to be something intermediate betweenthe three extreme cases discussed above�Note that �� does not provide a satisfactory estimate of j�j since the

allowed error is still of the order ofpj,j� but better estimates can be ob�

tained to determine exactly �i�e� with an error much smaller than O�pj,j��

the size of the boundary and its macroscopic shape� Such remarkable resultsalso provide a rigorous microscopic theory of the ancient Wul��s construc�tion of the shape of a droplet in the absence of gravity when the surface ten�sion is not spherically symmetric as a function of the normal to the dropletsurface� see �DKS� �� PV�� MR�� Mi�� The results hold in the twodimensional case and for low enough temperature their literal extension tothe whole coexistence region or to the three�dimensional case �even at verylow temperature� seem out of reach� if at all possible� of present day tech�niques in this respect� unexpected� remarkable progress has been achievedvery recently� �PV�� with new techniques that cover� at least in dimen�sions the whole phase.coexistence region �showing that despair is out ofplace�� However one can get surprisingly detailed informations by generalconsiderations based on inequalities and convexity properties of the surfacetension� see �MMR� ��Another problem is the investigation of the dependence of the correlation

functions on the distance from the surface of the drop�The analogs of the �rst two questions just raised were previously satis�

factorily answered in the two�dimensional Ising model with the �easier�cylindrical boundary conditions �see x�� i�e� in the case of an �in�nite�drop with a $at surface� This problem has been approximately studied evenin the case of a $at drop� �BF��

x�� Further Results� Some Comments and Some Open Prob�

lems

In x�� we dealt with the case of a nearest neighbor Ising model� Ithas become customary� in the literature� to apply the name of Ising modelto more general models in which the �bulk� Hamiltonian �i�e� without theboundary interactions and conditions� has the form� see x��

�hXxi

�xi �Xi�j

J��xi� xj��xi�xj �Xi�j�k

J��xi� xj � xk��xi�xj�xk � � � �

��where the potentials Jn�x�� xn� are translationally invariant functionsof �x�� xn� and satisfy certain restrictions of the type

Xx

jJ�� x�j�Xx�y

jJ�� x� y�j� � � � � ��


If only pair potentials are present� i�e� if the bulk Hamiltonian has the form

�hXx

�x �Xi�j

J�xi � xj��xi�xj ��

and if J�r� � �� then most of the results described in this chapter and ap�propriately reformulated can be extended or become very reasonable con�jectures� see �Ru�� p� � �� MS�� BS�� Do�#c� and the review �Gi��Many results remain true for more general pair potentials and for other

models �like continuous gases� at least from the qualitative point of view�in fact it is reasonable that the results selected here for discussion shouldhave� at least qualitatively� an analog in the �general� case of a classical �asopposed to quantum� phase transition�Results such as analyticity and absence of phase transitions at high tem�

perature� or exact solutions� are a peculiarity of lattice models and havebeen partly discussed in Chap�V and they will be analyzed again later�Below I list a number of rather randomly chosen and interesting problems

suggested by the topics of this chapter�

�� The solution of the two�dimensional Ising model is based on the socalled �transfer matrix�� The investigation of the transfer matrix has beenpursued in some detail in the case of periodic or open boundary conditionsin two or three dimensions� �MS�� CF�� see also �On��Ab��The transfer matrix with non�symmetric boundary conditions has also beenstudied in the two�dimensional case��Ab#�� i�e� the transfer matrix betweentwo rows �or planes� where the line �or surface� of separation should passif straight� A qualitative di�erence should arise between two and threedimensions �see� for more details� x��

� � In Fig� ��#�� we see that the isotherm m�� h� as a function of h � �abruptly ends at h � �� It is a natural question whether h � � is an analyticsingularity of m�� h� or whether m�� h� can be analytically continued toh � �� There has always been strong evidence for a singularity� �LR�� andit has been shown� rigorously� that at h � � there is an essential singularity�at least at large �� although the function m�� h� is in�nitely di�erentiableas a function of h for h � �� Is#��

�� The answer to � � makes clear that one has to give up the theory of�metastability� based on the possibility of an analytic continuation of themagnetization as a function of h through h � �� The latter idea was foundedin fact on the absence of an analytic singularity at h � � in the equationof state deduced from mean �eld theory �that is in the van der Waals the�ory of phase transitions� whose version for spin systems is called the CurieWeiss� theory� for an interesting mathematically complete treatment of themetastability phenomenon in the case of very weak and very long rangedforces see �LP�� The question of how one can explain metastability phe�nomena in systems with short range forces has been investigated in great

� V I� Coexistence of Phases

detail as a dynamical phenomenon and the results are very many� detailedand varied� see for instance �CCO�� KO��MOS��

�� There is a great number of other lattice models for which phase transi�tions are proven to take place� the basic techniques originated by the workof Minlos and Sinai� �MS�� have evolved and have been highly enriched�one can compare the situation in the early ��s� before the �rst proof ofthe existence of a phase transition in a continuous system� still based onsymmetry breaking �Ru��b�� see the review �Gi�� with the present dayvery re�ned results �see �BLPO�� BC�� In this respect one must men�tion the very recent progress in nonlattice models� �Jo�� and �MLP�#��The latter works should be considered major breakthroughs on the problemof phase transitions occurring in continuous systems or� more generally� insystems without special symmetries which are spontaneously broken at thetransition�

�� The question of whether� once a phase transition is known to occur�one can count how many pure phases exist is often a very intricate questionas various examples show �see �BC��

�� A detailed description of the correlation functions near the line orsurface of separation has still to be discussed� see �AR��

�� The microscopic de�nition of surface tension in the particular case ofthe three�dimensional Ising model has been studied but there are many openproblems� see �Mi�� particularly concerning the cases when the boundaryconditions would impose an ideal surface of separation� between the twophases� which is not parallel to the lattice planes� Furthermore it is awell founded conjecture that there is a temperature lower than the criticaltemperature for the appearance of spontaneous magnetization� above whichthe separation surface shows large $uctuations �possibly of order

plogL��

see �VB�� In this regime there would probably be no more translationallynoninvariant states� and it is likely that the surface tension �� is notanalytic as a function of � �while at low temperature it is known thatthe surface tension relative to an ideal surface of separation parallel to thelattice planes is such that �� J is analytic in e��J�� This wouldidentify a second type of phase transition which has been called in x�� theroughening transition� see �KM#�� KM#�� and� for a review� �VN#��

�#� The problem of the existence of phase transitions in models close tosymmetric models but asymmetric was expected to give interesting results��Me�� Substantial progress towards the understanding of phase transitionsnot directly associated with spontaneous breakdown of symmetry has beenachieved by the understanding of the model in �� with J� �� PS��Although the models in �PS�� are �close� to symmetric models the absenceof a rigorous symmetry was a major obstacle and the solution proposedin �PS�� has generated a large number of investigations the theory is


generally known as the Pirogov�Sinai theory of phase coexistence� see forinstance the applications to Potts models� �LMMRS�� MMRS��

�� In connection with �#� an interesting problem arises on the correctde�nition of approximate symmetry� The analysis has been attempted in�EG�� but the results are still very partial� There is also the possibilitythat in some cases symmetries that are apparently already broken in theHamiltonian are in fact dynamically restored �and then� possibly� sponta�neously broken�� An example in which this happens has been proposed in�BG�� see �BGG�� and �Ga�#b� for some applications� in a totally dif�ferent context� but its relevance for the theory of phase transitions mightcreate surprises�

�� Last but not least� the phase transitions problem in quantum statis�tical mechanics will not be discussed in this book� but it is� of course� veryimportant� The conceptual frame in which it is developed in the literatureis the same as the one we described in the classical cadre� However the phe�nomenology becomes even richer and full of surprises see �BCS��Br��Phase transitions of quantum systems can be studied in �lattice systems�at a rather sophisticated mathematical level� �Gi��b�� DLS�#�� and we shallmeet some in Chap�VII where they appear because of their relation withtransitions in classical spin models� However the theory is� generally speak�ing� less developed on a mathematical level but it is enormously developedon a phenomenological level� see �An#��


�

��

Chapter V II

Exactly Soluble Models

��

�

V II� Exactly Soluble Models ��

x�� Transfer Matrix in the Ising Model� Results in d � ��

Many well�known exact solutions of statistical mechanical models are basedon the transfer matrix method� In fact the summation over the states of thesystem� e�g� the sum over the values of the spin at each site� can be quiteeasily interpreted as an operation of summing over the labels of a productof matrices in order to compute the trace of the product� Thus the problemof computing� say� a partition function is �reduced� to that of diagonalizingcertain matrices with the purpose of computing their eigenvalues and eigen�vectors� The latter sometimes also provide informations on the correlationfunctions�The di%culty is that the matrices that one obtains are �large dimension

matrices� and they are di%cult to diagonalize� This can be done� at thecost of remarkable e�ort� in a few cases� We shall discuss a few samples ofthem� The simplest case is the d � � Ising model already studied in x��with a di�erent method and at zero �eld h�Consider the one dimensional Ising model with periodic boundary condi�

tions� see �� Ch�IV� If �L� � �� the partition function Z�,� �� h� canbe written as

X��L

LYi�

e�J�i�i��h�i �X

��L

LYi�

e�� h�ie�J�i�i�e

�� h�i� �

�X

��L

V��V�� V�L�� TrV L

��

where V is a two�by�two matrix such that ��

V�� e�� h�e�J��

�

e�� h�

�

� V �

�e��hJ� e��J

e��J e��h�J� �

��

If � � �� are the two eigenvalues of V � we �nd

Z�,� �� h� � �L � �L� ��

so that

�f�� h� � limL��

�

LlogZ � log� � ��

It is easy to check that �� h� is analytic in � and h for � � � � �and �� h � �� i�e� there are no phase transitions �as singularities off�� h�� In fact

�f�� h� � log�e�J cosh�h �

�e��J�sinh�h�� e��J

� ��

��

�# V II� Exactly Soluble Models

as the elementary calculation of � shows� from �� This is manifestlyan analytic function of �� h in the region of physical interest �� h real� sothat the model has no phase transitions in the sense of no singularities ofthe thermodynamic functions or of their derivatives� as already discussed�A similar method can be applied to the two�dimensional Ising model �, is

now an M �N box�� Suppose� for simplicity� h � �� then Z�� h� is

X�

MYi�

NYj�

e�J�i�j�i��j�J�i�j�i�j� �

�X��

� � �X�M

MYi�

n NYj�

e�J� �i�j�i�j�e�J�i�j�i��j

�J� �i��j�i��j�

o ��

where in the second line we denote by �i � ��i�� i�N � all the spins onthe i�th row of ,� the periodic boundary conditions are imposed by setting�� M� and �i�� i�N�� Clearly� if we de�ne the N � N matrix

V��

NYj�

e�J� �j�j�e�J�j�

�j

�J� ��j�

�j� �

� exp

NXj�

��J �j�j� � �J�j�

�j �

�J

��j�

�j�

� ��

where �� N�� N�� we realize that

Z�,� �� TrV M ��#��

We have dealt so far only with periodic boundary conditions� We couldintroduce transfer matrices also in the case of other boundary conditions�For instance� assume� for simplicity� that there are periodic boundary con�

ditions along the columns� we shall consider the three cases below

�� Perfect wall� or �open� boundary conditions� see x�� along the rows�

� � Boundary conditions on the rows corresponding to the existence of �xedspins �i � �� or �i � �� for all the i s on the lattice sites adjacent to theend points of the rows�

�� Boundary conditions which are of the same type as in � but half therows end in positive spins �say the upper half� and half in a negative spin�

We shall now write down a transfer matrix expression for Z�,� �� in theabove cases� In case �� Z�,� �� TrV ��M where

V�� e

PN��

j�� J� ��j�j��

�j��j��

PN

j��J�j�

�j � � ��


In case � � Z�,� �� TrV ��M where

V�� e��J��

��N��N �V

��

In case �� assuming here that the height of , is M with M even� we have

that Z�,� �� Tr �V ��M��V ��V ��

��M��V �� with

V�� e��J��

��

�N��N �V

��

The transfer matrix V in �� is the matrix that was diagonalized in thefamous paper of Onsager� �On�� The matrix V �� has also been diagonal�ized exactly in �Ab��The matrices V �� have been studied and diagonalized in �AM�� Many

more exact calculations of interesting quantities have been performed� seefor instance�Ab�� MW�� Ab�#a�� Ab�#b�� Ab#��The problem of computing the partition function can be formulated sim�

ilarly in the three dimensional case� Some very interesting results on thespectral properties of the generalization to three dimension of the matrix V�periodic boundary conditions� have been obtained in �MS��CF��In three dimensions one expects that the analogue of V �� in contrast

to V �� V �� has spectral properties which di�er radically from those ofV � In two dimensions the phenomenon should not occur and all the abovematrices should have the same spectrum �asymptotically as , � �� Asmentioned in x�� problem �� this should be related to the fact thatV �� should contain some information about the rigidity of the line orsurface of phase separation �which is �rigidly sitting� right near the twolines between which V �� transfer�� see Ch�VI��A very interesting heuristic analysis of the spin correlation functions in

terms of the transfer matrix has been done in �CF��

x�� Meaning of Exact Solubility and the Two�Dimensional Ising

Model

Before proceeding to study more interesting cases it is necessary to saythat usually by �exactly soluble� one means that the free energy or someother thermodynamic function can be computed in terms of one or morequadratures� i�e� in terms of a �nite�dimensional integral� whose dimensionis independent of the system size�In some cases one can even compute a few correlation functions but there

remain� as a rule� quite a few physically interesting quantities that onecannot compute �in the above sense of computing��Another characteristic problem of the �solutions� is that sometimes their

evaluation �in terms of quadratures� involve a few exchanges of limits thatare not always easy to justify� For instance the value of the spontaneousmagnetization in the Ising model was derived in an unknown way by On�sager �who just wrote the �nal formula� �� below� on a blackboard at a

� V II� Exactly Soluble Models

meeting in Firenze� �KO�� and later by Yang� �Ya� �� but both derivationsrelied on some hidden assumptions �in modern language the �rst assumedthat there could be only two pure phases and the second that an external�eld of order O� �

N � would be strong enough to force the system into a puremagnetized phase��In the case of the Ising model spontaneous magnetization the mathematical

problems were solved much later� independently in �BGJS�� and �AM��But in several other cases there are still open problems in establishing thevalidity of the �exact solutions� with full rigor �in the current sense of theword��The two�dimensional model is soluble only in zero external �eld �h � �� in

the sense that in this case the free energy f�� the magnetization m�h�and a few correlation functions can be expressed quite simply� For instanceif �J � J�� is de�ned by tanh J�� e��J � see �� anddenoting cosh�� as the inverse function to cosh

�f��

log sinh� �J��

�

Z

�

d

��j cosh��

�cosh �J cosh J � sinh �J sinh J cos

�j � ��

A simple analysis of the � dependence of this function shows that it issingular at the value � � �c for which it is J � J �i�e� sinh �J � �� andthe singularity appears as a logarithmic divergence of the derivative of f�with respect to �� i�e� as a divergence of the speci�c heat�The exact value of the spontaneous magnetization� i�e� of the �right� h�

derivatives at h � � of the free energy� can also be computed as said aboveand the result is

limh��

��f�h

�� h� � m��

�� if sinh �J � �� sinh �J��

� �� otherwise �

�� The importance of the above formulae� besides their obvious beauty� canhardly be overestimated� For instance they proved that in statistical me�chanics there are phase transitions with critical exponents di�erent fromthose of mean �eld theory e�g� from �� one realizes that m��

��c�

as ��c�� rather than the ��c�

�� foreseen by mean �eld theory� seeCh�V� x�� and x�� Many other quantities have been computed exactly some already in the

original Onsager papers and others in successive works� Among them wequote

�� The correlation function h�O�xi where O denotes the origin and xis a lattice point on one of the two lattice axes or� alternatively� it is alattice point on the main diagonal� the symbol h�i denotes the average value�of the quantity inside the brackets with respect to the Gibbs equilibriumdistribution�

V II� Exactly Soluble Models �

One shows that� if �� jJ � Jj and if jxj is large� then the functionh�O�xi is proportional to

h�O�xi �

��

e��jxjpjxj for � � �c

jxj�� for � � �c

e��jxj

jxj� for � � �c

� ��

One should not view �� as a discontinuity in the asymptotic behaviorof the pair correlation function when the temperature passes through thecritical temperature Tc� A more detailed analysis shows in fact that if T isclose to Tc the correlation starts depending on x as if T � Tc this howeverproceeds only until jxj becomes so large to be comparable to the correlationlength �� which is longer and longer the closer T is to Tc� Afterwardsthe exponential decay sets in �with a di�erent power correction dependingon whether � � �c or � � �c��In fact one can compute the asymptotic behavior of all correlations func�

tions �i�e� of the average value of products of spin values in an arbitrarynumber of sites� in various regimes for instance for � �� c when the spinsites separate from each other homothetically� �WMTB�� MTW��In the same situation a beautiful asymptotic formula for the value of the n

spins correlation function for n spins aligned along a line has been derivedby Kadano�� Ka��

� � The surface tension between coexisting phases� de�ned as� see �� Ch�VI�

�� lim��

�

Llog

Z��/�

Z��/��

�� if � � �c ��J � J� if � � �c

��

where Z� Z� denote respectively the partition functions of the modelsobtained by �xing the boundary spins all equal to �� in the �rst case andequal to �� in the upper half and to �� in the lower half in the secondcase� Here L is the perimeter of the container /� which is assumed to be asquare� See �On�� GMM� �� GM� b�� AM��

�� Quite a lot is known about correlation functions of spins associatedwith boundary sites� In the case of a box with open boundary conditions�i�e� no interaction with the spins located at external sites� it is� for in�stance� remarkable that for � � �c the spontaneous magnetization on theboundary� does not tend to � as � � �c with the same critical exponent �

�characteristic of the bulk magnetization �i�e� of the magnetization at a site

� Whose square is de�ned here as the limit as L of the correlation h�x�yi with x� y

being two points on �! and at distance O�L��

V II� Exactly Soluble Models

at a �nite distance from the origin� hence in�nitely far from the boundary�the new critical exponent is in fact �

� �in the limit as / � �� of course��Ab#��

Even more remarkable is that one can �nd the critical exponent of themagnetization computed at a corner� where it is �� more generally oneconsiders the Ising model in a wedge�shaped planar lattice with openingangle �so that � � is the case of a point in the middle of a side of thecontainer� whose shape does not matter as long as the point is at a distancetending to in�tity from the points where the boundary starts bending �half�plane case�� and �

� is the case of a �square corner�� The conjecture isthat the critical exponent for the spontaneous magnetization at a �cornerwith opening � is

�� This has been proved �together with various otherresults about the magnetization at points a �nite distance away from acorner� for � � �exponent �� as mentioned above� and �

� to whichthese corresponds an exponent �

� � see �AL��

x�� Vertex Models�

Consider a rectangular region / � Z� with opposite sites identi�ed �peri�odic boundary conditions�� We imagine that the microscopic states of thesystem are obtained by �xing an orientation on each lattice bond linkingnearest neighbors of /�

Given a microscopic con�guration � of the system� at every lattice site weshall see one among the �� possibilities shown in Fig� �� below�

�A� �B�

�C� �D�

�E�

�F �

Fig� ��

The eight�vertex models or #V �models are characterized by allowing onlythe con�gurations � which in every lattice site the bonds orientations lookas in A�B�C�D� see p� � #� p� �� in �Ba# �� Furthermore the energyassociated with a con�guration is� in the general eight�vertex model� a sumof contributions �j coming from each lattice site j� Allowing only vertices


A�B�C generates the six�vertex models� or �V �models�

The vertices A�B are called polar vertices while the C�D are called nonpolar the A vertices are usually labeled as � and � the B vertices arelabeled � and �� the C are � and �� the D are � and #� We denote by cone of the latter eight arrow con�gurations an #V con�guration will be acollection of arrows put on the lattice bonds that at each lattice node formone among the eight arrows con�gurations� called �allowed vertices��

Each lattice site x gives to the total energy of the con�guration an additivecontribution ��c� depending on which of the eight possible vertices c areformed by the arrows entering or exiting the lattice site x�Not all the eight�vertex models are �exactly soluble� one gets a soluble

model if the energies of the vertices �� i�e� A� are equal and so are those ofthe vertices �� i�e� B� of �� i�e� C� and of �� #� i�e� D� Thus the family ofeight vertex soluble models has three parameters �because one of the fourcan be eliminated by recalling that the total energy is de�ned up to anadditive constant��

We call �Adef� �� the common value of the energies corresponding

to the vertices �� and likewise we set �Bdef� �� C

def� �� and

�Ddef� ��

The eight vertex models can be interpreted as spin models �hence as latticegas models�� A trivial way to do this is to interpret an arrow con�gurationas a spin con�guration with the lattice of the spins being the lattice of thebonds� see Fig� �� the up and down arrows can be identi�ed with � and� spins located at the center of the arrows and� likewise� the right and leftarrows can be identi�ed with � and � spins� This naive procedure howeverrelates the #V models to spin models with constraints or hard cores becausenot all �� con�gurations of spins on the arrows relative to a vertex are goingto be possible�A much more interesting representation of the eight vertex models is ob�

tained by considering Ising models in which interactions between next near�est neighbor spins and many spins interactions occur �see x�� betweenquadruples of spins involving the four spins of a unit lattice cell� see p� ��in �Ba# �� An excellent introduction to the vertex models and their rela�tionship with other models can be found in �LW� ��Ka��

We call the lattice of the centers of the vertices the �#V lattice� and weconsider a con�guration of arrows which at each point of the #V lattice isone of the eight allowed con�gurations� We de�ne a con�guration of signs� or � located at midpoints of the bonds of the #V lattice � represents aup or right arrow while � represents a down or left arrow�

The product of the signs of the four bonds of the #V lattice that merge intoa vertex must be � this is the condition that all vertices are of the above

� In fact the restriction �� is not really such because the total energy depends onlyon �� Furthermore if �� the models with �� and �� are alsosoluble so that the class of soluble �V models has four parameters� one more than the�V model� see below�


�rst eight types� Therefore we have a one�to�one correspondence betweenthe eight vertex con�gurations on the #V lattice and the sign con�gurationsat the centers of the bonds which multiply to � on each of the #V latticebonds that merge into a vertex�We imagine putting a spin � � �� at the center of each square� also called

a �plaquette�� formed by four of the #V lattice bonds� the lattice of thecenters of the #V lattice plaquettes will be called the Ising lattice and it willcarry in this way a spin con�guration ��The product of the spins on nearest neighbor sites v� w of the Ising lattice is

a sign �� we use this name rather than spin to stress the di�erence betweenthese auxiliary variables and the spins introduced above�� This sign can benaturally associated with the bond of the #V lattice that separates v� w�and by construction the product of the signs on the sides that merge into avertex is necessarily � so that the sign con�guration can be interpreted asan #V con�guration and there is a one�to�two correspondence between #Vcon�gurations and Ising spin con�gurations� In fact a spin con�guration �and �� give the same sign con�guration� hence the same #V con�guration�Setting a � exp��A�� b � exp��B�� c � exp��C�� d � exp��D�

and de�ning J� J �� J ��

a � exp��J � J � � J �� b � exp��J � J � � J ��c � exp��J � J � � J �� d � exp��J � J � � J ��

��

it is now immediate to check that any con�guration of the #V model hasthe same energy as the corresponding spins con�guration �� in the Isingmodel with energy

H�� X

J�i�i� �X

J ��i�i�� X

J ��i�j�j��j��

��

where the sum runs over the sites i � / and i� denotes the nearest neighborof i along the diagonal of the second and fourth quadrant� and i�� the onealong the �rst and third quadrant diagonal� j� j�� j�� are three sites that�together with i� form a unit square �with i in the lower left corner�� see p� �� in �Ba# ��One can consider also the sixteen�vertex models� obtained by consideringall arrow con�gurations in Fig� �� above� including the E�F ones� Thismodel is also equivalent to a suitable Ising model� with also three�spin in�teractions� see �LW� �� p� �� for the discussion of the general cases�This model has many interesting special cases� some of which were rec�

ognized to be soluble before Baxter s work� In fact the breakthrough inthe whole theory was the solution by Lieb of the six�vertex Pauling s �Icemodel� discovering the method of solution for the other soluble �V �models�Among the latter there are the six�vertex models� whose con�gurations

only allow for the vertices A�B�C each of which gives a contribution to theenergy �A� �B � �C �

�� The just mentioned Pauling s ice model �xes �A � �B � �C � �� itcorresponds to J � � J�� J � �J�� J �� J� and J� � ��


� � The KDP model �xes the B�C vertices energies to � � � and to � thoseof the vertices A the occurrence of �non polar� vertices and of two of thefour polar ones is energetically less favored�

�� The F model �xes the energies of A�B to � � � and those of C to �the non polar vertices are favored�

Such models were solved �i�e� their free energy was calculated� in rapidsuccession after the solution of the �rst� �Li��a�� Li��b�� Li��c�� Su��The analyis and solution of the most general �V model is in �LW� ��

The �V models above are limiting cases of the #V model in which thecouplings J� J �� J �� in �� tend �suitably� to �� Because of this limitingprocedure they have been sometimes regarded as �pathological�� see below�

The �V models have a physical meaning within the theory of the hydrogenbond and of similar chemical bonds� In the ice model the lattice sitesrepresent the sites occupied by O� and the bond orientations tell where thetwo H atoms are located if an arrow emerges from a lattice site this meansthat an H atom is located on the bond and near the site� The associationof the arrows with the bond provides a two�dimensional version of the icerule �which states that on any bond there is one H atom� and not more�located closer to one of the two oxygens�� a rule deduced by Pauling fromthe observation that the ice entropy is lower than the one it would have ifin an ice crystal the H atoms could be found� unconstrained� near every Oatom �so that one could �nd con�gurations like E�F or even with oppositearrows on each bond�� Of course the model should be� to be realistic� threedimensional� but the appropriate three�dimensional version is not exactlysoluble�

The KDP�model has been proposed as a model for the ferroelectric prop�erties of KH�PO� a substance that crystallizes in tetrahedra with KPO�

at the center and the two H atoms on the lines between the KPO� onlyone H can be located on each such line and it can be there in two positions�i�e� near one or the other extreme�� KH�PO� is a polar molecule withoutspherical symmetry so that not all dipoles give equal contribution to thetotal energy of a con�guration� In the two�dimensional version of the modelthe two nonpolar vertices C and two of the polar ones �e�g� B� are unfa�vored and contribute energy � � � while the others contribute � � � atlow temperature a spontaneous polarization� or ferroelectricity� is expectedto occur�

The F�model� instead� is a model for an antiferroelectric polar materialresisting �at low temperature at least� polarization by a �eld�

A deeper discussion of the physical interpretation of the �V models and oftheir relation with other remarkable combinatorial problems and statisticalmechanics models can be found in �LW� �� The vertex models are equivalentin various senses to several other models� see for instance� �VB�� Ba# ��

The problem of the existence of the thermodynamic limit for the #V modelsis a special case of the general theory� see Ch�IV� with the minor modi�ca�


tion required to study lattice spin systems rather than continuous particlesystems �see �� an easier problem� in fact�� However the situationis di�erent in the case of the �V models� because the constraints imposedby arrow con�gurations can propagate the boundary conditions on a largebox all the way inside it� The theory has been satisfactorily developed in�LW� ��In the ice model case one �nds� �Li��a�

lim��

�

j/j logZ�/� ��

log

�

��

which is really beautiful�In the F�model case the free energy is given by� if � � ��

�e��

� �fF ��

�

��

R�� log cosh��cos ��

cosh��d�

cosh�� if cos� � j�j � �

��

P�n�

e�n� tanhn�n if cosh� � ��

��

and in the KDP�model case� setting � � ��e

�� the free energy is

� �fKDP ��

�

��

R��

cosh��cos�cosh��cos ��

d�cosh�� if � � � cos� � �

� otherwise

��

where the above solutions for the KDP�model is due to Lieb� �Li��b�� andfor the F�model to Lieb and Sutherland� �Li��c�� Su�� See x�� below fora technical introduction�Furthermore the F�model and the KDP�model� unlike the soluble eight

vertex models� can be solved even in the presence of an �electric �eld� E� ifsuch �eld is modeled by assuming that the energy contribution of a vertexincreases by �Ep with p being the number of arrows pointing up minus thenumber of those pointing down� The solution in the presence of such anelectric �eld is fairly simple but we do not report it here� It is a model withone more free parameter in which the energies �� are di�erent and theenergies �� are also di�erent by the same amount�The even more general model in which the electric �eld has also a horizontal

component so that the energy of a vertex increases also by�E�q with q beingthe number of arrows pointing right minus the number of those pointingleft is also soluble although the result is not as simple to discuss� �LW� ��Taking this into account amounts to a freedom of taking �j � j � �� asindependent parameters� Hence the number of free parameters in the mostgeneral �V soluble model is � �they are �� and one can be�xed to be ��The elementary analysis of the above formulae� and of their extensions

to cases with E �� leads to the following results that we describe by


denoting by fF ��E� and fKDP ��E� the free energies of the two modelsat temperature �kB�� and in the presence of a �vertical� electric �eld E�

The function fF �� is in�nitely di�erentiable and analytic in � exceptfor an essential singularity� at the value � � �c where � � �� one has�therefore� in zero �eld a phase transition of in�nite order �the order of atransition as a function of a parameter can be de�ned to be the order of thelowest derivative of the free energy� with respect to the parameter� which issingular� see Ch�VI� x��

The polarization� de�ned as the average number of arrows pointing upminus those pointing down� is proportional to the derivative of fF withrespect to E� it vanishes for E � � for all values of �� but if � � �c �lowtemperature� it remains � even if E �� for a while� and it becomes nonzeroonly if E grows beyond a critical value Ec�� and in this sense the modelhas an antiferromagnetic behavior� If one keeps E �xed� then by varying� one �nds a second�order phase transition with a speci�c heat singularityproportional to �� c�

��The free energy of the KDP�model� fKDP ��E�� is essentially di�erent�

Also in this case there is a critical temperature � � �c at zero �eld E�de�ned by � � �� in zero �eld and if � � �c the polarization has value �identically� and the free energy is constant� if � � ��c the speci�c heat tendsto � as ��c � �� but the internal energy does not tend to zero althoughthe value of the internal energy for � � �c is �� therefore there is a phasetransition of �rst order with latent heat and at low temperature there isspontaneous polarization �maximal� p � �� so that the system is �frozen�and it has trivial thermodynamic functions��

The above properties� selected among many that can be derived by simplyexamining the expressions for the exact solutions� show the richness of thephenomenology and their interest for the theory of phase transitions� inparticular as examples of phase transitions with properties deeply di�erentfrom those found in the Ising model case� �LW� ��

It is important to stress that the �V models form a four�parameters classof models and some of the soluble �V models are limiting cases of the three�parameter family of soluble #V models� By varying the parameters one can�nd a continuous path linking the F�model critical point �in zero �eld� tothe KDP�model critical point �also in zero �eld��

As remarked above the #V models are genuine short�range Ising models�� with �nite couplings� Hence one can study how the critical point sin�gularity changes in passing from the F to the KDP�model� The remarkableresult found via the solution of the #V models is that the critical exponents�the ones that can be computed� change continuously from the F values tothe KDP values�

This fact has great importance at the time of the solution of the ice modeland of the consequent solution of the F and KDP models the universalitytheory of critical point singularity was not yet developed in its �nal form�So when the renormalization group approach arose around �� see �BG��for a review and references� the �V model appeared as a counterexample to

# V II� Exactly Soluble Models

the universality that the renormalization group was supposed to predict�

This was stressed by Lieb on several occasions regrettably his commentswere either not understood or they were dismissed on the grounds that the�V models regarded as Ising models were �models with constraints�� in thesense discussed above in the derivation of ��

Baxter s solution of the #V model made clear to everybody Lieb s point ofview� Even genuine �nite�range �unconstrained� Ising models could show acritical singularity that was neither that of the F�model� neither that of theKDP�model nor that of the Ising model �to which the #V models reduce ifJ �� The three behaviors were the limiting cases of a three�parametercontinuum of possibilities� so that the phenomena shown by the �V modelswere not examples of pathological properties dubiously attributable to thehard core features of such models�� but they were the rule� This led to amuch better understanding of the theories that were put forward to explainthe universality phenomena� �rst among which the renormalization groupitself�

It would take too long to discuss the eight�vertex model properties� �Ba# �it is not surprising that it o�ers a varied and interesting phenomenology�besides the enormous theoretical interest of the sophisticated analysis nec�essary to obtain the solutions� As mentioned it can be solved only in zero�eld �i�e� for �� Some results in nonzero�eld can be obtained by �perturbing� the zero �eld models� see �MW#��

It is however important to stress once more that the exactly soluble modelsgive very limited information about the actual thermodynamics of the sys�tem� For instance the informations that can be obtained about correlations�even just pair correlation functions� is very scanty�

There are a few remarkable cases in which one can compute explicitly thepair correlation function� like the two�dimensional Ising model� see �MW��WMTB�� MTW�� AR�� or even higher correlations� like the n�spins correlations in the two�dimensional Ising model when the spins areon the same lattice line and far apart� �Ka�� Correlation inequalities canbe very useful to study more general situations without having recourse toexact solutions� �MM��

Often one is able to evaluate the correlation length because it can be re�lated to the second or third largest eigenvalue of the transfer matrix �oftenthe highest is almost degenerate and what counts is the third� which canbe studied quite explicitly �for instance in the #V models�� see �Ba# � p� �� #��

Even the latter results provide little information about the �critical� caseswhen there is no gap isolating the top of the transfer matrix spectrum fromthe rest� The renormalizations group methods� which received so muchclari�cation from the exact solution of the �V and #V models� do provideat least in some regions of the coupling parameters a� b� c� d �� a rather

� Because as mentioned above their statistical mechanics properties are quite normal� asshown in �LW�� p� ��


detailed analysis of the properties of the correlation decay� For a recentdevelopment see �Ma�#��There are many other exactly soluble models� e�g� one could just men�

tion the spherical model� �BK� �� the dimer model� �Ka�� and �TF��the XY �model� the ground state of the one�dimensional Heisenberg model�whose solution can be related to that of the six and eight vertex models�YY��Ba# �� see also �Li��a��Su��Ka�� besides the Ising model onsome lattices other than the square lattice and the hard hexagon model ona triangular lattice� �Ba# �� The reader should consult the monographs onthe subject �MW�� Ba# � to which a good introduction is still the reviewpaper �SML�� and the book �LM��

x�� A Nontrivial Example of Exact Solution� the Two�Dimen�

sional Ising Model

The actual computations for the exact solutions are always quite involvedbut their elegance surpasses Jacobi s theory of the action angle variablesfor the pendulum� To give an idea of the procedures involved we describebelow the classical solution of the Ising model� as presented in the paper�SML��

One starts from the remark in x�� that the free energy of the Ising modelin an N �M box with periodic boundary conditions is given by the trace��#� of the M �th power of the matrix V in ��It is easy to construct a convenient representation for the matrix V � Con�

sider the three Pauli matrices

�x �

��

�� y �

�� ii �

�� z �

��

��

��

and consider the tensor product H of N bidimensional linear spaces E andthe operators ��j de�ned by

H �

NYj�

E � E �E � � � ��E

��j �I � � � �� I � � � �� I� � � x� y� z

��

where I is the identity operator on E and �� is located at the jth�place�The operators �zj are pairwise commuting and one can easily diagonalize

them� If j� i � E is a vector such that �z j� i � � j� i �with � � �� thenthe most general eigenvector of the operators �zk on H can be written as atensor product of vectors

� This was important progress as it provided a simple and easily understandable entirelynew approach to the solution of the Ising model� at a time when rested on the originalworks �On�� Ka�� Ya�� which were still considered very hard to follow in the ��s�

�� V II� Exactly Soluble Models

j� i �

NYj�

� j�j i � � � �� N � ��

and the N such vectors �each corresponding to a string � � �� N �of digits �j � �� form an orthonormal basis in H�Consider the operator w on E such that

h� jw j�� i � e�J��

��

which can be written as a matrix as

w �

�e�J e��J

e��J e�J

�� e�JI � e��J�x � AeJ

�x ��

with A � � sinh �J�� and tanh J � e��J � We see from �� thatthe tensor product

V� �

NYj�

�w � w � � � �� w � ANeJP

j�xj ��

has matrix elements

h� jV� j�� i �NYj�

e�J�j��j � ��

��

Hence if we de�ne the operator V� on H� diagonal on the basis consistingin the vectors j� i � which on j� i acts as

V� � e��J

Pj�zj �

zj� ��#��

then we immediately check that the matrix elements of the operator V �V�V�V� between j�� i and h� j are exactly the transfer matrix elementsV�� in ��This means that the problem of the evaluation of the partition function

��#� is solved once we know the eigenvalues of V � V�V�V�� or of anyoperator unitarily equivalent to it�We shall� therefore� perform the unitary transformation U on H such that

U�xj U�� zj U�zjU

�� xj ��

which transforms the matrix V into "V � UV U��

"V � AN�e��J

Pj�xj �

xj�

��eJP

j�zj��

e��J

Pj�xj �

xj�

��

which we can write� de�ning the matrices "Vj � as "V� "V� "V��


Here it will be very usefule to consider the following Pauli�Jordan trans�formation

��j ��

��xj � i�yj �� a�j � ��j

j��Ys�

��zs � ��

so that ��j � a�jQ

s�j�� as a�s � and �j �

�j � aj a

�j � The usefulness of

�� is due to the remark that

�aj � aj� � � � � �a�j � a

�j� �� aj � a

�j� � � �jj� ��

if �� denotes the anticommutator� In other words the above transforma�tions changes the operators ��j which have mixed commutation relations�i�e� commuting for j �� j� while anticommuting for j � j�� into operatorsa�j with purely fermionic commutations relations� We set N �

Pj a

j a

�j �

This makes it easy to complete the calculations one �rst remarks that the

parity operator ��Ndef� ��

Pj�j �

�j � ��

Pjaj a

�j commutes with the

factors de�ning "V � hence it commutes with "V itself�Hence the space H is a direct sum of two orthogonal subspaces H and H�

on which the parity operator has values respectively �� and �� i�e� H

contains an �even number of fermions� and H� an �odd number��This means that if j/ i is a vector in H such that a�j j/ i � � for j �

�� N � and one can see that there is always one and only one such vector�the H is the space spanned by the vectors aj� � � � a

j�nj/ i � n � � while H�

is spanned by the vectors with aj� � � � aj�n�

j/ i � n � ��

Assuming from now on that ��N is even� for simplicity� one can make the

following key remark� if we de�ne the new operators

A�q �e�i��p

N

NXj�

e�ijqa�j � q � � �

N��

�

N� � � �� N � ��

�

N��

where the q s are de�ned so that eiqN � �� then on H the followingalgebraic identities hold

a�j �e��p

N

Xq

e�ijqA�q

"V� �A��Ne

JP

q��A

q A�q ��

"V� �e��J

Pq

��A

q A�q A

�qA

��q� cos q�A

q A�qA

��qA

�q � sin q

��

where� in deriving the second and third relations� careful account hasbeen taken of the facts that while �xj �

xj� � ��j � ��j ��j� � ��j� �

�aj � a�j ��aj� � a�j�� for J � N it is� instead� ��N � ��N ��

��N ��N � a�N ��a� � a�� and that on H it is ��N � ��


The anticommutation relations �� imply similar anticommutation re�lations for the A� so that the expressions in �� containing di�erent�q s commute and therefore we can write� only on� H�

"V � ANYq�

Pq ��

with A de�ned after �� and

Pq � e�J��A

q A�q A

�q

A��q

� cos q�Aq A

�q

A��q

A�q � sin q��

e�J�A

q A�q A

�q

A��q�� e�J

��A

q A�q A

�q

A��q

� cos q�Aq A

�q

A��q

A�q ��

and the operators Pq commute� Hence they can be considered as � � �matrices acting on the space Dq spanned by the vectors j/ i � A

q j/ i �

A�q j/ i � A

q A�q j/ i �

Furthermore it is clear that the operator Pq has Aq j/ i as eigenvectors

and A�q j/ i with eigenvalues e��J cos q � Hence the N

� operators Pq canbe considered as operators on the space D

q spanned by the two vectors

j/ i � Aq A

�q j/ i � Note that such space is invariant under the action of Pq �

Diagonalization of �� will account only for �� N eigenvalues and eigen�

vectors because the operator "V coincides with the transfer matrix only oonH� The remaining �

� N are in the space H� and they can be found inthe same way de�ning new operators A�q as in �� with q such that

eiqN � � one obtains a representation like �� which is now only correctif restricted to the space H��Therefore the problem has been reduced to the diagonalization of N

� � matrices obtained by restricting each Pq � q � �� to the space spanned bythe two vectors A

q A�q j/ i � j/ i � The four matrix elements of Pq on such

vectors can be evaluated easlily starting from the remark that

Pq � e��J cos qe�J��z cos q�x sin q�e�J

�ze�J��z cos q�x sin q� ��

where

�z ��Aq A

�q � A

�qA��q � �� x � �A

q A�q � A��qA

�q �

�y ��

i�A

q A�q �A��qA

�q �

��#��

and the matrices �x� �y � �z have the same matrix elements as �x� �y� �z in

�� respectively� if the vectors Aq A

�q j/ i � j/ i are identi�ed with

��

�

and

��

�� hence they satisfy the same commutation and multiplication

relations and have the same spectrum�


By using the properties of the Pauli matrices it is possible� with someobvious algebra� to rewrite the product �� as

Pq � e��J cos q�

cosh ��q� � ��z cos q � �x sin q� sinh ��q��

��

where ��q� is the positive solution of

cosh ��q� � �cosh J cosh �J � sinh J sinh �J cos q� ��

and the angle q is de�ned� setting t � tanh�J and t � tanh J� by

e�i�q �� eiq t

t �� tteiq�� e�iq t

t �� tte�iq��

Coming back to Pq one sees that on Dq

Pq � e��J cos qe��q��z cos�q�

x sin�q� ��

but since the latter expression has Aq j/ i and A

�q j/ i as eigenvectors

with the �correct� eigenvalues e��J cos q we see that Pq is given by ��on the entire four�dimensional space Dq� Hence we must diagonalize

"Pq � e��J cos qe��q�

��A

q A�q A

�q

A��q�� cos�q�A

q A��q

A��q

A�q � sin�q

��

�� Note that if one sets �for q � q � ��

Aq �B

q cosq � B��q sinq

A��q ��Bq sinq � B�

�q cosq��

which is an example of a well�known transformation� called the �Bogoliu�bov�Valatin transformation�� and q � � q one �nds

"Pq � e��J cos qe��q��Bq B

�q B

�qB��q��

so that the transfer matrix on H can be written as AN exp�P

q��J cos q�

��q��Bq B

�q � �

� �� see �� noting that ��q� � ��q� the �� N eigen�

values of the transfer matrix on H are� therefore� simply given by

ANe

Pq

�� eq � eq � ��q� where the signs can be arbitrarily chosen as long

as the is an even number of � signs �because the only relevant eigenvaluesare those associated with an even number of fermions�� The terms �J cos qdisappear because of the trigonometric identity

Pq cos q � ��

There is a unique vector j/� i such that B�q j/� i � � for all q � �� and it

is a linear combination of the vectors obtained by applyingQ

k AqkA�qk to

j/ i this means that the �new vacuum� j/� i is in the even space H�


Clearly the largest eigenvalue is A�Pq�

��q� and it corresponds to

the eigenvectorQ

q Bq j/� i which� being in H� really corresponds to an

eigenvector of the transfer matrix�

A similar calculation can be performed to �nd the eigenvalues of the trans�fer matrix on H� this time one takes q such that eiqN � � and proceedsin the same way getting an analogous result� The analysis of the relationbetween the maximal eigenvalues in H and H� is a simple matter how�ever it is very illuminating as it reveals that if � � �c� with �c de�nedby �cJ � J��c�� there is a gap between the two eigenvalues �which doesnot shrink to � as N � �� while for � � �c �i�e� at low temperature�the di�erence between the two eigenvalues tends to � exponentially fastas N � �� In the end this can be seen as the reason for the di�erentasymptotic behavior of the correlations� in ��

In the analysis on H� one must pay attention to the fact that now thevalues q � �� are permitted but they are not paired as the other valuesof q� This implies that the modes q � �� must be treated di�erently� andthe key di�erence that a careful discussion yields is that the value of e��is not necessarily � � it becomes negative for T � Tc and this accounts forthe di�erence in the spectrum of the transfer matrix above and below Tc�

With the above explicit expressions for the eigenvalues it is not di%cultto see that in the limit as N � � the only eigenvalues that count arethe two with maximum modulus� which are almost degenerate if � � �cand separated by a gap of order O�� for � � �c� This means that thelimit as N � � of ��#� is dominated by the largest eigenvalue � andthe free energy �f�� is always �for all �� given by logA plus the limit of�N

Pq� ��q� which is the integral in �� as a consequence of ��

The calculation of the spontaneous magnetization is more involved �see�Ya� �� it is� however� quite simple in the method of �SML�� and it requiresextra assumptions� which can be removed by using further arguments asmentioned in Ch�VI� �BGJS� �� AM��

We see that the above calculation requires some wit but its real di%culty isto realize that the transfer matrix can be written essentially as a quadraticform in certain fermionic operators� This reduces the problem to a � � �matrix diagonalization problem� after that it is clear that the problem is�solved�� although some computations are still necessary to get a reallyexplicit expression for the free energy�

The above calculation should be performed in all its details by those inter�ested in statistical mechanics� as it is one of the high points of the theory�in spite of its apparent technicality� There are alternative ways to computethe free energy of the two�dimensional Ising model� but all of them havea key idea and a lot of obvious technicalities that accompany it� The oneabove is particularly interesting because of its connection with the quantumtheory of fermionic systems �and because of its simplicity��

x�� The Six Vertex Model and Bethe�s Ansatz


We consider an N �M periodic lattice �N�M even� and on each bond weput an arrow so that the arrows entering or leaving each lattice node forma con�guration c among the A�B�C in the Figure in x�� Let ��c� dependon c but suppose that it takes the same values in each pair in A or in B orin C� the value of ��c� will be the contribution to the energy of an arrowcon�guration� The partition function will be

Z�� XC

e��E�C�� E�C� �Xc

��c� ��

where the sum runs over all vertices� i�e� over all the arrow con�gurationscompatible with the six�vertex constraint and which can be put on the NMvertices�Consider �rst the case ��c� � � �the ice model�� Given a con�guration C

consider the M rows of vertical arrows� We associate a number �mj � ��to the j�th bond �j � �� N� on the m�th row �m � �� M� indicatingwhether the arrow points up �� or down �� Let �m � ��m�� mN ��Given �� M � i�e� the collection of vertical arrow con�gurations there

will be in general several horizontal arrows settings that will be compati�ble with �� M � i�e� which together with the vertical arrows form anallowed con�guration around each vertex� Of course the set of horizontalcon�gurations that can be between two rows �r and �r� depends solelyupon �r� �r�� Hence we can de�ne

T �� fnumber of horizontal configurations allowed

between two rows of vertical arrows � and ��g � ��

and for instance in this case with ��c� � � it is

Z �X

��

M

T �� T �� T ��M � ��

Therefore the free energy �which in the latter example is the so�calledresidual entropy� i�e� a number that measures how many con�gurations arepossible for the system� is

I � limN��

limM��

�

MNlog TrTM � lim

N��

Nlog�max � ��

One checks� see Fig� �� that

T �� T �� if � ��

If � � �� N � and �� N � and if � � x� � x� � � � � � xn �

N are the n�labels for which �xj � �� and � � x�� x�� x�n� � Nthe n��labels for which ��x�

j� �� then T �� only when n � n� and

one of the following two chains of inequalities holds


� � x� � x�� xn � x�n � N

� � x�� x� � � � � � x�n � xn � N ��

Hence we see that T is divided into blocks which do not mix spaces withdi�erent �total spin�� i�e� one must have

Pi �i � N� n � N� n� �

Pi �

�i

to have T �� Fixed n �i�e� the total spin� we can consider the matrix T �n� obtained by

restricting T to the space generated by the unit vectors that we can labelj� i with

Pi �i � N � n� This can therefore be regarded as an operator

on the functions f�x�� xn� with � � x� � x� � � � � xn � N

�T �n�f��x�� xn� �

x�Xy��

x�Xy�x�

� � �

xnXynxn��

f�y�� yn��

�

x�Xy�x�

x�Xy�x�

� � �

NXynxn

f�y�� yn�

��

where the � denotes that the terms in which there are at least two equal y smust be considered absent� see the following footnote ��

If ��n�max is the largest eigenvalue of T �n� then �max � max��n�N �

�n�max� One

can� however� show that in the most interesting cases �max � ��N� �max�

Bethe�s ansatz is that the largest eigenvector of the matrix T �n� is a linearcombination of plane waves

f�x�� xn� �XP

AP eiP

jxjKPj ��#�

��

where the sum runs over the n� permutations P � �p�� pn� of the nindices and K�� Kn are n distinct �wave numbers��The idea of trying to �nd eigenvectors of the form ��#� appeared �rst in

the work of Bethe� �Be�� who found that the eigenvectors of the matrix Hde�ning the one�dimensional Heisenberg model could be expressed in thatform��

� The Heisenberg model on the lattice �� N with periodic boundary conditions isan operator written in terms of the matrices de�ned in �� and it is�

H �

NXj��

Jx�xj �

xj�� Jy�

yj �

yj�� Jz�

zj �

zj�� N��

which can be regarded as a matrix acting on the vectors in �� which can be denoted

j� idef� jx�� xn i if x�� xn are the lattice pints where �xj � �� Thus if the

generic vector is writtenP

x��xnf�x�� xn� jx�� xn i the operator H becomes

a matrix acting on the same space as the transfer matrix �� If J � Jx � Jy akey remark is that given T the matrix H commutes with T if the coe�cient Jz�J � #is suitably chosen as a function of the parameters a� b� c of the sixvertex model� # �a��b��c�

�ab� Then� see �� and �� # � �

�for the ice model� # � � � �

�e�� in

the case of the Fmodel and # � ��e� for the KDP model�


Bethe s eigenvectors were not immediately useful� not even to compute theactual value of the lowest eigenvalue of H �ground state energy� because thecoe%cients AP were di%cult to treat and their evaluation required solvinga linear integral equation which� at the time� could not be studied�

Wave functions of the form ��#� turned out to be very useful also in otherproblems the �new era� started when it was shown� in �LL�� that theground state energy of a simple one�dimensional Bose gas with a nontrivialinteraction corresponded to an eigenvector of the form ��#��

In the papers �YY�� a complete study of the integral equation necessary forthe evaluation of the ground state of the Heisenberg model was presented�and in �Li��a� it was discovered that one could �nd eigenvectors of �� ofthe form ��#�� among which one with f�x�� xn� � � corresponding ton � N� when N is even� as usually assumed� From this Lieb was able toshow� in the ice model case� that the largest eigenvalue of T was preciselythe same as that which gave the ground state energy of a correspondingHeisenberg model �with suitably chosen couplings� see footnote �� and whoseform had become known by the �YY�� key work�

The commutation property� cited in footnote �� between the ice modeltransfer matrix and the Heisenberg model Hamiltonian with � � �

� wasalso a consequence of the results on the ice model� This becomes clear assoon as one realizes that the two matrices have the same eigenvectors �allof the Bethe ansatz form� ��#��

The knowledge of the ground state eigenvalue for the ice model transfermatrix led to the explicit evaluation of the ice model residual entropy and�shortly afterwards� the works �Li��b��Li��c�� and �Su�� determined thelargest eigenvalue of the matrix T in the F and KDP�models�

It would be fairly easy to reproduce the work of �Be�� to see that the ma�trix H � written in the basis of the last footnote �� admits eigenvectors of theform ��#�� And remark that the Heisenberg model matrix H and trans�fer matrix T for the above considered �V models commute �if� of course� �is suitably chosen� implies that T has eigenvectors of the form ��#� andprovides a way for �nding the solution oof the model this� however� wasnot the path followed in the discovery of the ice model solution�

A direct check that one can adjust AP �KP in ��#� to make them eigen�vectors of the matrix �� Li��a�� is possible and very instructive� al�though it is surprisingly di%cult to write in words� The procedure suggestedin �Ba# � is perhaps the simplest�

Having noted that T decomposes into blocks that do not mix vectors in thespaces generated by the basis elements jx�� xn i �i�e� the functions ofx�� x�m vanishing unless m � n and x�i � xi� when their value is �� seefootnote �� with di�erent n s one studies �rst the case n � � �trivial�� thenn � � �also very easy�� then n � which is easy but requires attention as thealgebra is already quite involved� The calculation is strongly recommended�and one should �rst attempt it in the ice model case � � �

� �

For the ice model �of the F and KDP�models which are not harder� one

�# V II� Exactly Soluble Models

�ndsA��

A�� z� � z�z�

�� z� � z�z�� zj � eiKj

zN� �A��

A�� zN� �

A��

A��

��

If one has really performed the calculation at least in the � � �� case� with

the necessary patience� then �the solution of the eigenvalue problem forarbitrary n is a straightforward generalization of the n � case�� p��# of�Ba# �� If P � �p�� pn� is the generic permutation of �� n� then

Ap��pn � ��PYi�j

spi�pj � sp�qdef� �� zq � zpzq�

zNj � ��n��Y��j

s��jsj��

��

where the last relations must be interpreted as equations for zj hence� by�� for the Kj s� The calculations also provide the value of the eigen�value � corresponding to a sequence K�� Kn of n values that are pairwisedistinct and satisfy the last of �� For instance in the ice model case

� �

nYj�

�

�� zj�

nYj�

zj�� zj

��

It is remarkable that if j�j � � and p� q are real then

sp�qsq�p

� �� eip � ei�pq�

�� eiq � ei�pq�� ei��p�q� � ��

where the 2 s are real and are given by

2�p� q� � arctg� sin �

� �p� q�

cos �� p � q�� cos �

� �p� q��

which is real if p� q are real� The conditions on Kj become

�Ij � NKj �nX��

2�Kj �K�� Ij � j � �

�n � ��

which has been discussed in �YY�� and shown to admit� for all j�j � ��a unique real solution K�� Kn� Therefore� modulo mathematical rigorproblems� we expect that K�� Kn become dense when N � � andn�N � � stays constant� the number of Kj s in an interval dK shouldbe described by a density ��K� such that

R � ��K�dK � � � n

N � Thedistribution ��K� will be nonzero inside an interval ��Q�Q� � �� because j in �� varies between � and �

� �n � ��


The form of �� means that the number of Kj s which are � K is suchthat

N

Z K

�Q��K

��dK � � ��n��NK�N

Z Q

�Q2�K�K ��K ��dK ��

hence di�erentiating with respect to K

��K� � � �

Z Q

�QK2�K�K ��K �� dK � ��

which is an equation whose solution for � �� is still an open problem�

The problem for � � �� was solved in ��# �reference to Hulth3en� �Hu�#��

in �Ba# �� for the case � � �� and in �Wa�� for the cases � � �� Thej�j � � cases were completely solved by Yang and Yang� �YY�� for � � ��The case � � � is trivial� as noted by Lieb� because one can see that the

maximum eigenvalue is the one corresponding to n � � and it is aN � bN �see �Ba# �� see �� for the de�nition of a� b��The reason why the case n � �

�N is so special and exactly computablelies in the existence of a change of variables transforming �� into a

convolution equation� in the cases �� cos�def� � � �� The change of

variables is K��

eiK �ei� � e�

ei��

ordK

d��

sin�

cosh�� cos��#��

which maps the interval ��Q�Q� into �� and �� K��R�� with

R�� satisfying

R�� sin�

cosh�� cos��

�

Z �

��

sin �

cosh�� cos �R��d� ��

which can be solved by Fourier transform� and it even leads to the simplesolution

)R�x� ��

�

cosh�x��

for the Fourier transform )R of R��The �rst instance in which a transformation of the type �� is used

to transform the integral equation �� into a simple equation is in theremarkable brief paper �Wa�� which introduced the change of variablescorresponding to �� in the case � � �� The latter paper is basedupon another remarkable paper� �Or�#�� which studies the same equation inan approximate way�It is now a matter of simple algebra to obtain the formulae of x��

�� see �LW� ��Ba# ��


The #V model is not soluble by Bethe s ansatz� I cannot give here even asketchy account of its solution� The reader should look at Baxter s book��Ba# �� detailing what is one the main achievements of mathematical physicsin the ��s� The book also illustrates several other exactly soluble models�

��

Chapter V III

Brownian Motion

�

�

V III� Brownian Motion ��

x�� Brownian Motion and Einstein�s Theory�

Brownian motion was �rst observed by Brown ��# #� he recognized thatthe motion of �molecules�� of size of �� cm� of a pollen in a solution��colloid�� was not due to internal causes� as it was believed at the time�but it had a mechanical �unknown� origin� It was in fact observable bylooking at particles of comparable size and of any material� both organicand inorganic�

Very soon the movements were attributed to collisions with the microscopicconstituents of matter� Among the �rst to recognize this was Cantoni��Ca�� in a remarkable paper he says

�in fact� I think that the dancing movement of the extremely minute solidparticles in a liquid� can be attributed to the di�erent velocities that mustbe proper at a given temperature of both such solid particles and of themolecules of the liquid that hit them from every side� I do not know whetherothers did already attempt this way of explaining Brownian motions� � �

In this paper an impressive number of experiments� performed by Cantonihimself� are reported in which he �nds evidence for the equipartition of en�ergy between the suspended particles and the solvent molecules to conclude

�In this way Brownian motion provides us with one of the most beautifuland direct experimental demonstrations of the fundamental principles of themechanical theory of heat� making manifest the assiduous vibrational statethat must exist both in liquids and solids even when one does not alter theirtemperature��

This work is most remarkable also in view of the fact that it is contemporaryof the �rst papers of Boltzmann on the heat theorem and equipartition�

Brownian motion attracted the interest of many leading scientists� amongwhich was Poincar3e� Brownian motion theory was worked out by Einsteinand� independently� by Smoluchowski� �� soon followed by theexperimental con�rmation of Perrin� ��#�� see �Ei��VS��Pe��

The main critique �N�ageli� �VN�� to the microscopic kinetic nature ofBrownian motion was the remark that experimental data and kinetic theo�ries permitted one to estimate that �the particles in suspension being hun�dreds of millions times larger than the molecules of the liquid� the velocityvariations at each collision had a random sign� so that it seemed inconceiv�able that one could see a nonvanishing average e�ect� A fallacious argument�as it was stressed �for instance� by Poincar3e �� Po�� He also noted�with others� that the hypothesis that the colloidal particles motion had akinetic nature could contradict thermodynamics �see below��

The fallacy of the reasoning was� in any event� well known as it appears

�� V III� Brownian Motion

from an esoteric article by Bachelier �� Ba�� on the austere Annalesde l�Ecole Normale of Paris �a few pages away from the French translationof Hilbert s Grundlagen der Geometrie�� it is a paper on stock market spec�ulations where� precisely� the problem is posed of how can many randomsmall variations produce visible e�ects �see below��Einstein s approach starts with the remark that the suspended particles�

although of huge size� can still be considered as large molecules and� hence�one can apply statisical mechanics to them� so that they will exercise osmoticpressure� just as with ordinary solutions� satisfying �therefore� Raoult�vant4Ho� s law at least at small concentrations� Hence Van t4Ho� s law holdsnot only for solutions of microscopic particles but also for calculating thepartial pressure due to particles of arbitrary sizes �e�g� glass balls��The idea was �revolutionary� and� as Einstein realized� possibly in contrast

with classical thermodynamics� but not with statisical mechanics and theatomic hypothesis� Hence he immediately posed the question of how to �ndobservable macroscopic consequences�Nonrectilinear motion of particles is thus attributed to their random colli�

sions with molecules� Hence it is a random motion� at least when observedon time scales � large compared to the time necessary to dissipate the ve�locity v acquired in a single collision with a molecule �by friction� due� alsoto microscopic collisions between $uid molecules�� The dissipation of suchvelocity can be estimated� in the case of macroscopic particles� by remarkingthat in a single collision with a molecule the acquired speed v is dissipatedinto heat by the action of a force F which� by Stokes� law� is

mv

t� F � ��Rv �#��

where � is the $uid viscosity coe�cient� R is the radius of the suspendedparticles� and v the speed� hence the characteristic time scale for the lossof the velocity acquired in a single collision is t� � ��m��R�� Thisis a very short time �for instance if R � �� and if m is evaluated byassuming that the density of the material constituting the large particles isthe same as that of the liquid in which they are suspended �i�e� water� sothat � � �� cgs�units�� one realizes that the time scale is t� � �� sec��Therefore on the time scale � � t� motion will be di�usive� In such motionsthere is transport of matter only when there is a density gradient�The logic of Einstein s analysis is quite fascinating� Using an ideal experi�

ment �a method characteristic of his thinking� he links microscopic quanti�ties to macroscopic ones� The background solvent �xes the temperature andthe time scale over which a particle undergoes a di�usive motion to �nd thedi�usion coe%cient for a single particle one considers a gas of particles ofarbitrary density �� but so small that the ideal osmotic pressure law holds�We recall Raoult�van t�Ho��s law if p is the osmotic pressure �i�e� the par�tial pressure due to the particles� and � is their numerical density one hasp � kBT�� with T being the $uid temperature and kB Boltzmann s con�stant� This is done in spite of the fact that in the classical experiments the


suspended particles are so few that they can be considered isolated fromeach other�Hence in the �rst step of the ideal analysis one replaces a single colloidal

particle with a gas of such particles� with density ��One then imagines that some external force F acts on the gas �also a �ction�

in the ideal experiment� which only acts upon the colloidal particles �andnot on the solvent $uid�� it generates� in a stationary state� a gradient F�of pressure� denoted xp� The pressure gradient also equals kBTx� by theRaoult.van t Ho� law �or more precisely by the assumption that the osmoticpressure law holds for macroscopic particles�� The pressure gradient whichexactly balances the force action is xp � F� � kBTx� by the Raoult.vant Ho� law�Supposing that the solvent obeys the Navier.Stokes equations one can

then compute� via Stokes law� the particle velocity in terms of the viscosity�strictly speaking here it is necessary that particles be in fact macroscopic�so that one can compute the $ux generated by the pressure gradient

� � � v � � � F

��R� � kBT

��Rx� � �#��

Finally the assumption that the individual particles undergo a di�usivemotion implies that the $ux has to be proportional to the density gradientgiving

� � �Dx� �#��

where the proportionality constant D is the di�usion coe�cient�Equating the two expressions for the $ux of particles all auxiliary quanti�

ties� used to mount the ideal experiment� have disappeared and one infersthat assuming kinetic theory then a macroscopic particle �even just one� ina $uid and in thermal equilibrium �i�e� in a stationary state� must have adi�usive motion with a di�usion constant related to the viscosity by

D �kBT

��R� �#��

which is called the Einstein�Smoluchowski relation that one should at�tribute entirely to Einstein� see the following x#� �The quantity D is also directly related to the average value �over many

trajectories� hr�t��i of the squared displacement r�t�� of the colloidal par�ticle in a time interval t� we shall see that hr�t��i � �Dt� Since the valueof hr�t��i is directly measurable in a microscope� this is a �rst theoreticalrelation that can be checked experimentally�Conceiving of macroscopic particles as behaving like microscopic molecules

�and generating an osmotic pressure obeying Raoult.van t Ho� law� much astrue chemical solutions do� is an important idea that was� in itself� a noveltybrought by Einstein s work �heralded by Cantoni s experiments� �Ca�� Itallowed everybody who had not yet accepted the atomic hypothesis to see


that thermodynamics laws must have a statistical nature� This in fact hap�pened at least after Perrin showed� experimentally� that Brownian motiontheory was correct� �Pe��

In fact� since it seems possible to build semi�permeable walls for macro�scopic objects� it becomes possible to perform thermodynamic cycles at con�stant temperature which� by using osmotic pressure� convert heat into workthe walls create an entity similar to Maxwell s daemon� One can think of acylinder �lled with liquid and divided in two by a semi�permeable movablewall� on the left side there is also a colloidal solution seeing the impermeableside of the wall� while the right side contains no colloid� The wall can thenbe pushed to the right by using the osmotic pressure to extract work fromthe process� at the end of the run the wall is taken out �performing a workas small as we wish� in principle� because we only need to displace the wallhorizontally� and reinserted back at the original position but with the twofaces inverted�

Then comes a long waiting time while the observer does nothing but witnessthe colloidal particles randomly hit the wall on the permeable side and getcaught in the left part of the cylinder� again� After the last colloidal particlecrosses the wall the initial conditions are restored and Carnot s principle hasbeen violated�

The in�nitely sharp eye of Maxwell s daemon can thus be replaced byour microscope� as Poincar3e stressed �having in mind a somewhat di�erentapparatus based on the same ideas�� Perrin highlighted this same aspect ofthe Brownian motion phenomenon� and he also noted that a machine like theabove would have required unimaginably long times to extract appreciableamounts of energy� see x�� of �Pe��

It is important to keep in mind that here we are somewhat stretchingthe validity of thermodynamic laws the above machines are very idealizedobjects� like the daemon� They cannot be realized in any practical wayone can arrange them to perform one cycle� perhaps� �and even that willtake forever if we want to get an appreciable amount of energy� see x�� of�Pe�� but what one needs to violate the second law is the possibility ofperforming as many energy producing cycles as required �taking heat out ofa single reservoir�� Otherwise their existence �only� proves that the secondlaw has only a statistical validity� a fact that had been well established sincethe work of Boltzmann�

In fact an accurate analysis of the actual possibility of building walls semi�permeable to colloids and of exhibiting macroscopic violations of the secondprinciple runs into grave di%culties it is not possible to realize a perpetualmotion of the second kind by using the properties of Brownian motion� It isin fact possible to obtain a single violation of Carnot s law �or a few of them��of the type described by Perrin� but as time elapses and the machine is leftrunning� isolated and subject to physical laws with no daemon or other idealextraterrestrial being intervening �or performing work unaccounted for��the violations �i�e� the energy produced per cycle� vanish because the cyclewill be necessarily performed as many times in one direction �apparently


violating Carnot s principle and producing work� as in the opposite direction�using it��This is explained in an analysis of Feynman� see �Fe�� vol� �� x�� where

the semi�permeable wall is replaced by a wheel with an anchor mechanism�a �ratchet and a pawl�� allowing it to rotate only in one direction under theimpulses communicated by the colloidal particle collisions with the valvesof a second wheel rigidly bound to the same axis� Feynman s analysis isreally beautiful� and remarkable as an example of how one can still saysomething nonboring about perpetual motion� It also brings importantinsights into the related so�called �reversibility paradox� �that microscopicdynamics generates an irreversible macroscopic world��Di�usive motion produces a displacement r�t� over a time t whose squared

average is hr�i�t� � �Dt� because the probability f�x� t�d�x for �nding aparticle� initially located at the origin� in the little cube dx around x is thesolution of the di�usion equation tf�x� t� � D�f�x� t�� i�e�

f�x� t� �e�x

��Dtd�x

��Dt��#��

�the equation that Einstein derives by imitating Boltzmann s method to ob�tain the Boltzmann s equation also �nding� at the same time� a microscopicexpression for the di�usion coe%cient D�� The squared average value of thedisplacement is then simply

hr�t��i �

Zx�f�x� t� d�x � �D t �#��

We see that� although each collision produces a very small velocity varia�tion� immediately followed by variations of similar size and of either sign�nevertheless the particle undergoes a motion that over a long time �com�pared to the frequency of the collisions� leads to a change of each coordinateof the order of

p Dt �or

p�Dt if one looks at the three�dimensional vari�

ation� which not only is nonvanishing� but can also be considerably largeand observable�As an application Einstein deduced �� the value of Boltzmann s con�

stant kB � hence of Avogadro s number NA� from the measured di�usion ofsugar suspended in water� �nding NA � �� the error being mainlydue to a computational mistake� On the basis of accurate experiments�by using the theory of Einstein� Perrin and collaborators obtained a valueessentially equal to the recently accepted value of NA� see x�� of �Pe��Brownian motion theory was derived by Einstein without him being really

familiar with the details of the experiments that had been performed forabout #� years� He proceeded deductively� relying on ideal experiments�starting from the remark that particles� even if of macroscopic size� had toobey the laws of statisical mechanics� In particular they had to show energyequipartition and their osmotic pressure had to obey the perfect gas law�Raoult.van t Ho� s law��

�# V III� Brownian Motion

His theory leads manifestly to motions that� if observed over time scaleslong compared to t� �cf� lines following �#�� must be motions for whichthe velocity would depend on the time interval over which it is measured andit would diverge in the limit t � � or� better� it would become extremelylarge and $uctuating as t approaches the time scale t� beyond which thetheory becomes inapplicable�It provided� therefore� an example of an actual physical realization of cer�

tain objects that had until then been just mathematical curiosities� likecontinuous but nondi�erentiable curves� discovered in the #��s by mathe�maticians in their quest for a rigorous formulation of calculus� Perrin himselfstressed this point very appropriately� see x�# of �Pe��The assumption that $uid resistance to macroscopic particle motion follows

Stoke s law is by no means essential but it is a characteristic aspect distin�guishing Einstein s theory from Smoluchowski s� as we shall see below� Infact if the assumption was changed to v � CRF with CR suitably dependingon R� then �#�� would be replaced by D � kTCR� If� for instance� theparticle was suspended in a rare�ed gas� rather than in an incompressibleliquid� then CR would be di�erent�More precisely if the colloidal particle proceeds with velocity v in a gas

with density �� then the number of gas particles colliding with an averagevelocity �vm is �R��v � vm�� while �R��v � vm�� is the number ofparticles colliding with an average velocity �vm� The former undergo amomentum variation� per unit time� m�vm�v� and the latter m�vm�v��Hence� instead of Stokes law� the force of the $uid on the particle is

�

�R��v � vm�� vm � v�� m � c�R�vmm�v �#��

with c� � ��In the above calculation we supposed that half of the particles had velocity

equal the absolute velocity average and half an opposite velocity� further�more the particle has been treated as a disk perpendicular to the directionof motion� A more correct treatment should assume a Maxwellian veloc�ity distribution and a spherical shape for the particle� The evaluation ofthe corrections is without special di%culties if one assumes that the gasis su%ciently rare�ed so that one can neglect the recollision phenomena�i�e� repeated collisions between the particle and the same gas molecule�and it leads to a �nal result identical to �#�� but with a di�erent factorreplacing �� One would eventually �nd� following the argument leading to�#��

D �kBT

c�R�mvm��

kBT

cR��p

mkBT�

pkBT

cR��p

m�#��#��

and the constant c is in fact p�� For obvious reasons the regime in which

the expression �#�� for the friction holds is called Doppler�s regime and

� i�e� long compared to ��sec as is necessarily the case because of our human size


it is relevant for rare�ed gases� while the Stokes regime� in which �#�� isacceptable� pertains to friction in liquids�

x�� Smoluchowski�s Theory

Smoluchowski s theory� shortly following that of Einstein� shines light onEinstein s hypotheses� The latter lead to �#�� at the price of tacit assump�tions similar to the molecular chaos hypothesis familiar from Boltzmann sattempts at deriving Boltzmann s equation in fact according to �#��colloidal particles show a di�usive motion� with mean square displacementproportional to time t�

Smoluchowski� to disprove N�ageli s argument� considers a concrete micro�scopic model for the collisions a particle of mass M is subject to a largenumber of collisions with the molecules� of mass m� of the $uid �� sec��

in many cases�� If vk is the particle velocity after k collisions and if the k�thcollision is with a molecule with velocity v before colliding� one infers fromthe elastic collision laws that

vk� � vk �m

M�R� �� v if jvj � jvkj� M � m �#� ��

where R is a random rotation �depending on the impact parameter� alsorandom�� The above equation only deals with a single collision between twoparticles and it does not take into account that the heavy particle moves in agas of light particles with positive density this causes a cumulative frictione�ect� hence when the velocity vk grows the Doppler friction in �#�� willstart to damp it by a force �c�R�vm�mvk� hence with acceleration ��vkwith � � c�R�vm�m�M � Thus the velocity variation should rather be as

vk� � vke��

m

M�R� �� v �#� � ��

which includes� empirically� the damping e�ect�We can consider �R� ��v as a random vector� at each collision� with zero

average and square width �m�M��h��R� ��v��i � �m�M��v�m� Further�more collisions take place� in average� every time interval � such that

��R��vm � �� c�m

M�#� ��

hence the space run in a time t during which n � t�� collisions take place is�if the initial positon and velocity of the particle are both �� for simplicity�

r �

n��Xk�

�

kXh�

e��k�h�wk �#� ��

with wk independent random vectors with hw�ki � v�m�m�M�� Hence we


can immediately compute the average square dislacement as�

hr�

ti � v�m

�mM

��

�

��

��

v�mc��

�

��R�vm� �a

kBT

c�R�m�vm�#� ��

where a � ��c�� is a numerical constant of O�� see below� and c� wasintroduced in �#�� we see that we �nd again� essentially� Einstein s for�mula in the case in which Stokes law is replaced by Doppler s resistanceof a rare�ed gas� just as we would expect to �nd given the nature of themodel �apart from the factor a�� One shall realize� see x#�� that Uhlenbeck.Ornstein s theory is a macroscopic version of Smoluchowski s theory�This led Smoluchowski to say that if� instead� the $uid is an incompress�

ible liquid one has �simply� to replace� in the denominator of �#� �� theDoppler regime viscosity with that in the Stokes regime� with a somewhataudacious logical leap� he thus �nds

Dsmol� � aDeinst� �#� ��

Hence Smoluchowski s theory is in a sense more ambitious than the Ein�steinian theory because it attempts at proving that the colloid motion is adi�usive one without neglecting completely the time correlations betweenconsecutive collisions �which Einstein� as already mentioned� implicitly ne�glects�� The model proposed is to think of the $uid as a rare�ed gas which�therefore� does not obey the Stokes viscosity law� Strictly speaking Smolu�chowski s model deals with a colloid realized in a rare�ed gas� a situation notvery relevant for the experiments at the time� because it is not applicable toa colloid realized in a $uid� Einstein s method is more general and appliesto both cases� although it does not really provide a microscopic justi�cationof the di�usive nature of the motions�Conceptually Smoluchowski could not possibly obtain Einstein s formula

because he was not able to produce a reasonable microscopic model of a$uid in the Stokes regime �which even today does not have a satisfactorytheory�� His method in fact is not very �objective� even in the rare�ed gascase since it leads to a result for D a�ected by an error of a factor a withrespect to Einstein s�This factor can be attributed to the roughness of the approximations�

mainly to the not very transparent distinction between velocity and averagevelocity in the course of the derivation of �#� �� which does not allow us tocompute an unambiguously correct value for a� Nevertheless Smoluchowski�without the support of the macroscopic viewpoint on which Einstein wasbasing his theory� is forced to take seriously the factor a that he �nds andto transfer it �with the logical jump noted above� to an incorrect result inthe case of a liquid motion�

� We use that ��kBTm

� ��v�m� The factor a changes if one makes a less rough theory of the

Doppler friction taking into account that there are di erences between various quantities

identi�ed in the discussion� likeph�#vk�

�i and hj#vkji �which� by the Maxwellian

distribution of the velocities� modi�es c� hence a��


As in the case of the factor c in �#��#� a more precise theory of the col�lisions between molecules and colloid is possible� in which one replaces theaverage values of the velocity with $uctuating values �distributed accordingto the appropriate Maxwellian� in this way the value a � � arises� HadSmoluchowski proceeded in this manner� although �nding the correct resultin Doppler regime� he would have still needed a logical jump to treat thecase of a colloid in a liquid solution�

It thus appears that Smoluchowski s theory was not comparable with theexperimental data available at the time� this was so for intrinsic reasonsand it� perhaps� explains why he did not publish his results before Ein�stein s papers �results that� as he says� he had obtained years earlier�� It isnot impossible that by reading Einstein s memoir he could make the aboveanalyzed logical jump which was necessary in order to make a comparisonof the theory with the experiments �which occupies few lines in his longmemoir��

Later Smoluchowski abandoned the factor a and he adopted �Einstein s value� �a � ��

It remains true� however� that Smoluchowski s work is a milestone in kinetictheory and his was among the �rst of a series of attempts aimed at obtainingequations for macroscopic continua� The continua are regarded as describ�ing microscopic motions observed over time scales �and space scales� verylarge compared to microscopic times �and distances�� so that the number ofmicroscopic events involved in an observation made on a macroscopic scalewas so large that it could be treated by using probability theory techniques�or equivalent methods��

The use of probability theory is the innovative feature of such theoriesalready Lagrange� in his theory of the vibrating string� imagined the stringas composed of many small coupled oscillators but his theory was entirely�deterministic�� so much as to appear arti�cial�

In �� six years before Smoluchowski s work� Bachelier published theabove mentioned research� �Ba�� with the rather unappealing title ofTh�eorie de la sp�eculation which� as is maintained by some historians� wouldhave been left unappreciated because it was superseded or shadowed byEinstein s �� paper� Bachelier s work� it is claimed� did in fact presentthe �rst theory of Brownian motion�

It is in fact only a posteriori possible to see a connection between thetheory of $uctuations of erratic �� stock market indicators and Brownianmotion� nevertheless Bachelier s memoir can perhaps be considered to bethe �rst paper in which dissipative macroscopic equations are rigorouslyderived from underlying microscopic models�

In his work Brownian motion is not mentioned and his model for the evo�lution of list prices is that of a random increase or decrease by an amount�x in a time �t with equal probability� The novelty with respect to theclassical error analysis is that one considers the limit in which �x and �ttend to � while studying the list price variations at various di�erent timest under the assumption that they are given by partial sums of the price

� V III� Brownian Motion

variations� In classical error theory one studies only the total error� i�e� thesum of all the variations� so that the summation index does not have theinterpretation of time value� but is an index enumerating the various errorvalues�One deduces that the probability distribution of the list price values at

time t satis�es a di�usion equation� furthermore the probability distribu�tion of successive increments of value is a product of independent Gaussiandistributions and one arrives at a kind of preliminary version of the stochas�tic process that was later studied by Wiener �essentially in �Ba�� only�the analysis of the continuity of the sample paths as functions of time ismissing��Thus we can consider Bachelier s work as similar to Smoluchowski s theory

and� therefore� rather loosely related to Einstein s theory and� furthermore�in his analysis no mention appears of physics and thermodynamics� But wehave seen in the above discussion that it is precisely here that one of themain di%culties of Brownian motion lies�Smoluchovsky s point of view is by no means superseded in the last few

decades it has been developed into very re�ned theories aiming at under�standing the more general problem of deriving macroscopic continua equa�tions from microscopic dynamics see �Sp�� part II� for a perspective andtechnical details�

x�� The Uhlenbeck�Ornstein Theory

As remarked by Einstein �as well as by Smoluchowski� Brownian motiontheory held for experimental observations taking place at time intervalsspaced by a quantity large compared to the time scale characteristic for theloss of the velocity acquired in a single collision� which is t� � ��m��R��For shorter time intervals it still makes sense to de�ne the velocity of the

particles and motion cannot be described by the di�usive process charac�teristic of Brownian $uctuations proper� The trajectories appear� whenobserved over time scales larger than t�� erratic and irregular so that if onetries to measure the velocity by dividing the space run by the correspond�ing time one �nds a result depending on the time interval size and thatbecomes larger and larger the more the time interval is reduced� This is animmediate consequence of the fact that� on such time scales� the average ofthe absolute value of the displacement is proportional to

pt� rather than

to t� But this �divergence� of the velocity ceases as soon as one examinesmotion on time scales short compared to t�One is then faced with the problem of developing a theory by describing

motions in the �normal� phase at small time intervals� as well as in theBrownian phase� at larger time intervals� Langevin proposed a very simplemathematical model for the complete Brownian motion equations�He imagined that successive collisions with $uid molecules had an e�ect on

the variations of each velocity component that could be described in termsof a random impulsive force F �t� and� hence� the equation of motion of a


coordinate of a colloidal particle would be

m !v � ��v � F �t� �#��

where � is the friction coe%cient for the colloid motion �i�e� ��R in thecase of a $uid in a Stokes regime and cR�vmm� in the case of a Dopplerregime� see �#��The Langevin equation� �#�� can be discussed once a suitable random

force law is assigned for F � The model proposed by Uhlenbeck and Ornsteinfor F �t� was that of a white noise� i�e� that F was such that

�� no correlation existed between the values of F �t� at di�erent timeinstants�

� � the distribution of an n�tuple F �t�� F �t�� F �tn� of values of theforce� observed at n arbitrary instants t� � t� � � � � � tn� was a Gaussiandistribution� and

�� the average value of F �t� vanished identically as a function of t�

This leads to the notion of a �centered� Gaussian stochastic process andto the more general notion of stochastic process and� also� it leads to thepossibility of regarding Brownian motion as an �exactly soluble� stochasticprocess�Consider a stochastic process� i�e� a probability distribution� on a space of

events that can be represented as functions of one �or more� zero averagevariables t � F �t�� see x�� footnote �� It is characterized by giving theprobability of observing an n.tuple F �t�� F �t�� F �tn� of force valueswhen measuring the F �t� at n instants t� � t� � � � � � tn as a Gaussiandistribution on the force values�It can be shown that such a process �i�e� the probability distribution of

the functions t� F �t�� is uniquely determined by the two�point correlationfunction� also called the covariance� or propagator� This function is de�nedas the average value of the product of the function values at two arbitraryinstants t�� t�

C�t�� t�� hF �t��F �t��i �#��

and this means that the Gaussian distribution of the probability of an ar�bitrary n.tuple of values of F at n distinct time instants can be simply


expressed in terms of the covariance C �it is� in fact� simply expressible interms of the inverse matrix of the matrix C�ti� tj�� i� j � �� n��

In this way the �white noise� is de�ned as the Gaussian process with co�variance

C�t� t�� f��t� t�� #��

where f� is a constant and � is Dirac s delta function�We shall write f� as �kBT �thus de�ning T � and it will appear that if T is

identi�ed with the absolute temperature of the solvent then the Uhlenbeck.Ornstein theory and the Einstein theory will agree where they should� seebelow�For C given by �#�� an explicit solution of �#�� is possible and it

follows� as shown by Uhlenbeck and Ornstein� �UO�� that each velocity andposition component which is generated from initial data s�� v� �respectivelyfor the position and the velocity� is a Gaussian process with nonzero average�If � � ��m � t�� see the lines following �#�� their average� at time t�is given by

s�t� � s� �v��

�� e��t�� v�t� � v�e��t �#��

which follow simply by averaging �#�� over the distribution of F �so thatthe term with F disappears because hF i � � by assumption�� and then inte�grating the resulting equations for the average velocity� and the probabilitydistribution of a velocity component v at time t is the Gaussian

G�v� t� �� m

�kBT �� e��t�

��exp

�� m

kBT

�v � v�t��

�� e��t�� #��

��

� Starting from a Gaussian probability distribution over n variables x�� xn of the form

��dx� � const e� ��

Pi�j

Mi�jxixjYh

dxh ��

where M is a positive de�nite symmetric matrix� then�Z��dx�xixj � �M��i�j ��

de�ning the covariance of the Gaussian process � and identifying it with the inverse ofthe matrix of the quadratic form de�ning �� Also� for all vectors � � �� n�� we

have� Z��dx� e

Pjjxj

� e��

Pi�j

�M��i�jij� ��

Viceversa �� plus Gaussianity of � implies the other two and also �� implies the othertwo �and gaussianity of �� Finally one should remark that any random variable whichis a linear combination of the Gaussian variables xi has a probability distribution thatis Gaussian� One now replaces sums with integrals and matrices with operators andone obtains the corresponding notions and relations for a Gaussian stochastic processindexed by a continuum time label�


This is computed also from �#�� by writing it as

v�t� � e��tv� �

Z t

�

d�

me��t��F �� #��

so that squaring both sides and using �#�� one gets

h�v�t� � e��tv��i �

�

Z t

�

d� d� � e��t��e��t�� f

�

m��t� t��

f�

m�

�� e��t� �

�#��

hence �#�� follows because the distribution of v�t� must be Gaussian since�#�� shows that v�t� is a linear combination of Gaussian variables� seefootnote ��By integrating �#�� once more one obtains in the same way also the

distribution of the position component s�t�� It is a Gaussian with center ats�t� and quadratic dispersion

��t� �kBT

m�� t� � � �e��t � e��t��t��

kBT

m�t � Deinst�t �#��#��

or

H�s� t� ��

��t�

��e��s�s�t��

��t� �#��

The formulae just described reduce to the previous ones of Einstein s theoryin the limit t � �� but they hold also if t � t� � m

� and hence they solvethe problem of the colloidal particle motions over time scales of the orderof t� or less�The relation kBT

m� � D �i�e� kBT�� D� connecting viscosity �or �dissi�

pation�� and microscopic force $uctuations due to collisions with the solventwas the �rst example of a series of similar relations called ��uctuation�dissipation theorems��Uhlenbeck and Ornstein also computed the �joint� probability distribu�

tions of the values v�t�� s�t�� v�tn�� s�tn� for arbitrary t�� tn� hencethe resulting Gaussian process �i�e� the probability distribution of thetwo component functions t � �v�t�� s�t�� is therefore called a Ornstein.Uhlenbeck process�A modern discussion of the theory can be found in �Sp�� part I� Chap�#

here the motion of a Brownian particle is discussed by treating it is a �trac�er� revealing the underlying microscopic motions� This is perhaps the mainrole of Brownian motion in macroscopic physics�

x�� Wiener�s Theory�

From a mathematical viewpoint one can consider an idealized random mo�tion with the property that the position r at a time t� t relative to that at


time t has� even for an extremely small time t� the probability given by

P �r� t� �e�r

��Dt

��Dt��#��

i�e� given� even for small time� by the asymptotic distribution �as t�� ort� t�� of the Brownian motion�Clearly the proportionality of r� to t �rather than to t�� for small t means

that one shall �nd a motion with the remarkable property of not having awell�de�ned velocity at any time just as the Brownian motion observed overtime scales longer than the previously introduced t��The very possibility of rigorously de�ning such an object is the remarkable

contribution of Wiener �� who showed that the Gaussian process withtransition probability �#�� already introduced by Bachelier in the abovequoted article� �Ba�� is well de�ned from a mathematical viewpoint andthat with probability � the paths described by the particles are continuous�and in fact H�older continuous with exponent � �with any � � �� see�#�� below� �Wiener theorem� �Ne��IM��The Gaussian process describing the probability of trajectories t� r�t� in

which the increments of r are distributed independently and with Gaussiandistribution �#�� is� in probability theory� called a Wiener process or�simply� a �mathematical� Brownian motion� From the Physics viewpointit corresponds to the description of the asymptotic behavior of a colloidalparticle in a $uid� for times t large compared to the characteristic relax�ation time t� �while for generic times� including the short times� it is ratherdescribed by the Ornstein.Uhlenbeck process��More technically we can translate into a rather simple formula the state�

ment that the increments of one �of the three� coordinate ��t� � ��t�� ofa Brownian motion are indpendent and distributed according to �#��This means that if � � t� � t� � � � � � tn then the probability p that��t�� dx�� t�� dx�� tn� � dxn is given by

p �

nYj�

dxjp��D�tj � tj��

e� ��

�xj�xj��

D�tj�tj�� x� � �� t� � � � �#��

Although Wiener s process is� as we have seen� a �mathematical abstrac�tion� it has� nevertheless� great theoretical interest and it appears in themost diverse �elds of Physics and Mathematics�Its �rst application was to provide several quadrature formulae that express

solution of various partial di�erential equations in an �explicit form�� asintegrals over families of curves randomly distributed with a Wiener processlaw� �Ne��Obviously the calculation of such integrals is� usually� not simpler than the

solution of the same equations with more traditional methods� Neverthe�less the explicit nature of the formulae provides an intuitive representationof the solutions of certain partial di�erential equations and often leads tosurprisingly simple and strong a priori estimates of their properties�


A classical example is the theory of the heat equation

tu � D�u� ujt� � u��x� �#��

whose solution can be written as

u�x� t� �

Zdy

ZP �d��tx�yu��y� �#��

where the integral is extended over all continuous curves � � �� thatat � � � start from the position y arriving at time t at x� The integration��sum�� over the paths is performed by using the distribution of the Wienerprocess� subject to the condition of reaching x at time t�Equation �#�� has the following interpretation heat undergoes a Brow�nian motion� i�e� it is transferred from point to point following a randommotion with distribution given by Wiener process� Therefore the amount ofheat u�x� t� which at time t is in x can be obtained by imagining that theamount of heat initially in a generic point y is equitably distributed amongall trajectories of the Wiener process that leave y� so that the amount ofheat that one �nds in x at time t �i�e� u�x� t�� is the sum over all Brow�nian paths that arrive at x each carrying an amount of heat proportionalto the amount u��y�dy initially around the point y where they originated�and the proportionality factor is precisely equal to the fraction of Browniantrajectories that start in the volume element dy and arrive at x in the timeinterval t�Formula �#�� is the simplest instance of a class of formulae that solve

partial di�erential equations� a further classical example is provided by theequation

tu � D�u � V �x�u� ujt� � u��x� �#��

which can be explicitly solved by the quadrature

u�x� t� �

Zdy

ZP tx�y�d��e

�R t�V ��d�

u��y� �#��

which is called the Feynman�Kac quadrature formula�As an example of a simple application of �#�� one can derive a comparisontheorem for solutions of the equation tu � D�u � Vj�x�u� u�� u� � �where Vj�x�� j � �� are two functions �not necessarily positive� such thatV��x� � V��x�� Then �#�� immediately implies that u��x� t� � u��x� t�for all x and t � �� a property that is not so easy to prove otherwise�Equation �#�� admits various extensions� relevant both in mathematics

and in physics in very diverse �elds ranging from probability theory or par�tial di�erential equations to statisical mechanics and relativistic Quantum�eld theory and even to the foundations of quantum mechanics �see theanalysis of hidden variables in Nelson s or in B�ohm s quantum mechanicsformulations� �Ne�� BH�� One can say that in these applications theformulae of explicit solution really play a role similar to that played by theclassical quadrature formulae in classical mechanics�

�# V III� Brownian Motion

Wiener s process had� and still has� a particular importance in probabilitytheory� where it introduced a wealth of ideas and problems �and providedsolutions to several preexisting problems�� We quote the following as exam�ples among the mathematical properties of the process�

�� P� Levy�s regularity law� This gives the behavior of the variation ��t��t�� of a position component ��t� of a Wiener process trajectory t� ��t�observed at two nearby instants t� t� within a pre�xed time interval �� t��It has already been said that Wiener proves that the trajectory is H�oldercontinuous with an exponent � that can be �xed arbitrarily provided it issmaller than �

� � This means that� for arbitrarily �xed � � �� we shallhave with probability �

limt�t��

j��t�� t��jjt� t�j� � � �#��

if � � t� t� � &t� The arbitrariness of � � �� makes it interesting to askwhat is the �optimal� value for �� if any� Levy s law says that the is nooptimal �� but at the same time it provides an answer to what is the actualregularity of a trajectory � because it states

lim supt�t��

��t�t��t

j��t�� t��j��Djt� t�j log �

jt�t�j ��

� � �#��#��

with probability �� IM��

� � But the Levy s regularity law does not provide us with informationsabout the properties of the trajectory in the vicinity of a given instant infact �#��#� only gives the worst behavior� i�e� it only measures the maximallack of regularity within a speci�ed time interval �� t�� If we concentrateon a given instant "t then� in general� the trajectory will not be as irregular�This is in fact the content of the iterated logarithm law of Kintchin the lawgives the regularity property of a trajectory at a pre�xed instant "t� Fixing"t � �� and supposing �� the law is

lim supt��

j��t�j��Dt log

�log �

t �� #��

with probability �� Equation �#�� is not incompatible with �#��#�� In factit only says that the worst possible behavior described by �#��#� is in factnot true with probability � at a pre�xed instant i�e� it happens certainly� bythe previous law� but certainly as well it does not happen at the time "t atwhich one has decided to look at the motion� �IM��

�� The above two laws deal with the behavior of the trajectories at �nitetimes� one can ask what is the long�time behavior of a sampled trajectory�The Einstein and Smoluchowski theories foresee that the motion goes away


from the vicinity of the origin �i�e� of the starting point� by a distance thatgrows proportionally to

pt�

An analysis of these theories indicates that by average one means averageover a statistical ensemble� If at time t one measures the square of a co�ordinate of the particle which at time � was at the origin and if one doesthis for many Brownian particles �i�e� if one repeats the measurement manytimes� one �nds Dt� in the average�However this does not mean that if one �xes attention on a single motion

and one observes it as t grows� every coordinate ��t� squared grows atmost as Dt in the sense that the maximum limit as t � � of ��t��t is D� In fact such growth is really given by the global iterated logarithm law�Kintchin�

lim supt��

j��t�j��Dt log�log t�

�� #��

with probability �� IM��Although �#�� deals with Wiener process properties it expresses a prop�

erty of relevance for experiments on Brownian motion unlike the two previ�ous laws� which look at properties characteristic of the Wiener process andnot of real Brownian motions �which are rather described by the Ornstein.Uhlenbeck process�� this is a property that refers to the large time behavior�which is the same for the Wiener process and for the Ornstein.Uhlenbeckprocess�� However it is very di%cult to perform experiments so accurateas to reveal a correction to the displacement which is proportional to thesquare root of an iterated logarithm�

�� Equation �#�� does not invalidate the measurability of D based onthe observation of a single trajectory� Such measurements are performed byfollowing the displacement ��t� of a coordinate as t varies between � and t�One then sets

X�t� � t��Z t

�

��d� �#��

and a �t is attempted by comparing the data X�t� with the function Dt�The procedure is correct� at least asymptotically as t � �� because oneshows that

limt��

X�t�

Dt� � �#��

with probability �� This is the ergodic theorem for the Wiener process� Likethe comparison between laws �� above� law �� tells us that our particlewill be �far� ahead of where it should in�nitely many times �i�e� a factor

�log log t�� ahead� �#�� although in the average it will be at a distance

proportional to t ��#�� IM��

�� The trajectories run by the Wiener process are rather irregular� as theLevy and Kintchin laws quantitatively show� One can ask which is thefractal dimension of the set described by a Wiener process trajectory� If


the dimension of the space in which the motion take place is � then thefractal dimension is � in the sense of Hausdor�� This essentially means thatif one wants to cover the trajectory with spheres of radius ��n one needsO�n�� spheres in the sense that given � � � then O�n�� are �eventually asn � �� not su%cient while O�n�� are su%cient �even if n � �� Notethat a smooth curve� with �nite length� can be covered by just O�n� spheresof radius n��The property of showing dimension can be expressed also in several other

ways which are perhaps intuitively equivalent� at a super�cial level of under�standing� but strictly speaking di�erent� and the analysis of the alternativeways illustrates subtle aspects of the Wiener s process trajectories�For instance if one considers two distinct points in Rd and from each of

them one starts a Wiener path� then one �nds that the two paths willeventually �cross� �i�e� they reach the same point at some later time� withprobability � if d � � �� as expected intuitively on the basis that dimension�ally they are �surfaces�� But they do not cross if d � � �with probability ��if they were really .dimensional geometric objects we would expect thatthey would intersect not only for d � � � but for d � � as well� This isLawler�s theorem� �La#�� see also �HS� ��

�� A further celebrated property of the Wiener process� due to Wienerhimself� exhibits interesting connections with harmonic analysis and Fourierseries theory� Consider a sequence g�� g�� of Gaussian independent andequidistributed random variables and suppose that the distribution of eachof them is � �� exp�g�� Set

��t� �t

��g� �

�

�

��Xk �

sin kt

kgk � �#��

Then the random function ��t�� for � � t � �� has a probability distribution�induced by the one assumed for the coe%cients gk� identical to that of aWiener process sample path in dimension � �and D � �

� �� IM�� Thisremarkable fact is easy to check� since the covariance of a Gaussian processdetermines the process� it su%ces to check that if t � t� then� see �#��

h��t��t��i � ��p �

��Z

e��x�

t e� ��x�y��

�t�t�� x y dx dy �#��

The left�hand side is immediately computable from �#�� and from theassumed Gaussian distribution of the gk s and the r�h�s� is an elementaryintegral so that the identity is easily checked�To conclude one can say that the Wiener process is a mathematical ab�

straction originated from the physical phenomenon of Brownian motion itdescribes its �large time� behavior �the behavior for all times being caught�more appropriately� by the Ornstein.Uhlenbeck process�� It is neverthelessa mathematical entity of great interest which �nds applications in the mostdi�erent �and unexpected� �elds of mathematics and physics� Further read�ing on both the mathematical and physical aspects of Brownian motion canbe found in �Sp��

��

Chapter IX

Coarse Graining and Nonequilibrium

�

�

IX� Coarse Graining and Nonequilibrium ��

x�� Ergodic Hypothesis RevisitedAn informal overview of the basic ideas of Chap�IX is given in Appendix

�A� below� which is the text of a conference at the S3eminaire de Philosophieet Math3ematiques of 3Ecole Normale Superieure in Paris�Giving up a detailed description of microscopic motion led to a statisti�

cal theory of macroscopic systems and to a deep understanding of theirequilibrium properties which we have discussed in Chap�I�VII�It is clear today� as it was already to Boltzmann and many others� that

some of the assumptions and guiding ideas used in building up the theorywere not really necessary or� at least� could be greatly weakened or justavoided�A typical example is the ergodic hypothesis� Although we have analyzed

it in some detail in Chap�I it is interesting to revisit it from a di�erentperspective� The analysis will not only help clarify aspects of nonequilibriumstatistical mechanics� but it will be important for its very foundations�We have seen the important role played by the heat theorem of Boltzmann�

�Bo#�� We recall that one can de�ne in terms of time averages of totalor kinetic energy� of density� and of average momentum transfer to thecontainer walls� quantities that one could call� respectively� speci�c internalenergy u� temperature T � speci�c volume v� pressure p� and the heat theoremstates that when two of them varied� say the speci�c energy and volume bydu and dv� the relation

du � p dv

T� exact ��

holds�In the beginning� �Bo�� this was discussed in very special cases �like free

gases�� but about �fteen years later Helmholtz� in$uenced by the progress ofBoltzmann on the proof of the heat theorem� wrote a series of four ponder�ous papers on a class of very special systems� which he called monocyclic� inwhich all motions were periodic and in a sense non�degenerate� and he notedthat one could give appropriate names� familiar in macroscopic thermody�namics� to various mechanical averages and then check that they satis�edthe relations that would be expected between the thermodynamic quantitieswith the same name�Helmholtz assumptions about monocyclicity are very strong and seem to

be satis�ed in no system other than in con�ned one�dimensional Hamilto�nian systems� Here are the details of Helmholtz reasoning �as reported byBoltzmann�� in a simple example�Consider a one�dimensional system in a con�ning potential�� There is only

one motion per energy value �up to a shift of the initial datum along itstrajectory� and all motions are periodic so that the system is monocyclic�We suppose that the potential �x� depends on a parameter V �

� A potential ��x� such that j��x�j � � for jxj � �� and ��x��x��

�� IX� Coarse Graining and Nonequilibrium

Suppose that one identi�es a state with a motion with given energy E andgiven V � Then� let

U � total energy of the system � K � �T � time average of the kinetic energy K�V � the parameter on which is supposed to depend�p � � time average of V �

A state is parameterized by U� V and if such parameters change by dU� dVrespectively we de�ne

dL � �pdV� dQ � dU � dL � ��

then

Theorem �Helmholtz� The di�erential dQ�T � �dU � p dV ��T is exact�

Repeating� for convenience� the few lines of proof already discussed in Ap�pendix ��A� to Chap�I� this can be proved by directly exhibiting a functionS whose di�erential is �dU � pdV ��T � In fact let x��U� V � be the extremesof the oscillations of the motion with given U� V and de�ne S as

S�U� V � � log

Z x�U�V �

x��U�V �

pU � �x� dx ��

and ��S is the logarithm of the action since U � �x� is the kinetic energy

K�x�U� V �� so that

dS �

R�dU � V �x�dV � dxp

KRK dxp

K

� ��

Noting that dxpK

�q

�mdt� we see that the time averages are obtained by

integrating with respect to dxpK

and dividing by the integral of �pK

� Hence

dS �dU � p dV

T� ��

completing Helmholtz remark� For a more extended discussion of the the�orem see Appendix ��A� to Chap�I�Boltzmann saw that this was not a simple coincidence his interesting �and

healthy� view of the continuum which� probably� he never really consideredmore than a convenient artifact� useful for computing quantities describinga discrete world where sums and di�erences could be approximated by in�tegrals and derivatives� cf� x�� and �Bo�� p� �� led him to think that insome sense monocyclicity was not a strong assumption�Motions tend to recur �and they do in systems with a discrete phase

space� and in this light monocyclicity would simply mean that� waiting


long enough� the system would come back to its initial state� Thus its mo�tion would be monocyclic and one could try to apply Helmholtz ideas �inturn based on his own previous work� and perhaps deduce the heat theoremin great generality� The nondegeneracy of monocyclic systems becomes thecondition that for each energy there is just one cycle and the motion visitssuccessively all �discrete� phase space points�Taking this viewpoint one had the possibility of checking that in all mechan�

ical systems one could de�ne quantities that one could name with �thermo�dynamic names� and which would satisfy properties coinciding with thosethat thermodynamics would predict for them� see Chap�I� II�He then considered the two�body problem� showing that the thermody�

namic analogies of Helmholtz could be extended to systems which weredegenerate� but still with all motions periodic� This led to somewhat ob�scure considerations that seemed to play an important role for him� giventhe importance he gave them� They certainly do not help in encouragingreading his work the breakthrough paper of �##�� Bo#�� starts with asso�ciating quantities with a thermodynamic name to Saturn s rings �regardedas rigid rotating rings�� and checking that they satisfy the right relations�like the second principle� see ��In general one can call monocyclic a system with the property that there

is a curve � � x�� parameterized by its curvilinear abscissa �� varying inan interval � � � � L�E�� closed and such that x�� covers all the positionscompatible with the given energy E�Let x � x�� be the parametric equations so that energy conservation can

be written� for some m � ��

�

m !�� x�� E � ��

then if we suppose that the potential energy depends on a parameter Vand if T is the average kinetic energy� p � �hV i then� for some S�

dS �dE � p dV

T� p � �hV i� T � hKi ��

where h�i denotes the time average �see Appendix ��A�� Chap�I��A typical case to which the above can be applied is the case in which the

whole energy surface consists of just one periodic orbit� or when at leastonly the phase space points that are on such orbit are observable� Suchsystems provide� therefore� natural models of thermodynamic behavior�A chaotic system like a gas in a container of volume V � which will be re�

garded as an important parameter on which the potential �which includesinteraction with the container walls� depends� will satisfy �for practical pur�poses� the above property� because �Feynman� �if we follow our solution�i�e� motion� for a long enough time it tries everything that it can do� so tospeak� �see p� �� in �Fe�� vol� I�� Hence we see that we should be ableto �nd a quantity p such that dE � p dV admits the average kinetic energyas an integrating factor�


On the other hand if we accept the viewpoint �ergodic hypothesis� thatphase space is discrete and motion on the energy surface is a monocyclicpermutation of its �nitely many cells� the time averages can be computed byintegrals with respect to the uniform distribution� that we shall call Liouvilledistribution�� see x��Hence if � is the Liouville s distribution on the surface of constant energyU � and T is the ��average kinetic energy then there should exist a functionp such that T�� is the integrating factor of dU � pdV �Boltzmann shows that this is the case and� in fact� p is the ��averageh�V i and it is also the average momentum transfer to the walls per unittime and unit surface� i�e� it is the physical pressure� see Appendix ��A��This is not a proof that the equilibria are described by distributions �

of the the microcanonical ensemble� However it shows that for most sys�tems� independently of the number of degrees of freedom� one can de�ne amechanical model of thermodynamics� i�e� one can de�ne various averagesof mechanical quantities and name them with names of thermodynamicalfunctions� and check that they satisfy the relations that would follow fromclassical thermodynamics� see Chap�I�II for more details�

thermodynamic relations are� therefore� very general and simple conse�quences of the structure of the equations of motion� They hold for smalland large systems� from one degree of freedom �the case of Helmoltz� mono�cycles� to �� degrees and more �the case of a gas in a box��

The above arguments� based on a discrete view of phase space� suggest how�ever that� in general� the thermodynamic relations hold in some approximatesense� as we have no idea of the precise nature of the discrete phase space�However� in some cases� they may hold exactly even for small systems� ifsuitably reformulated for instance in the �##� paper� �Bo#�� Boltzmannshows that in the canonical ensemble the relation �� i�e� the secondlaw� holds without corrections even if the system is small� as explained inChap�II�Thus the ergodic hypothesis does help in �nding out why there are mechan�

ical �models� of thermodynamics they are ubiquitous� in small and largesystems alike� but usually such relations are of interest in large systems andnot really in small ones�A critical comment and a warning is important at this point for large

systems any theory claiming to rest on the ergodic hypothesis may seembound to fail� see x�� because if it is true that a system is ergodic� it isalso true that the time the system takes to go through one of its cycles issimply too long to be of any interest and relevance this was pointed outvery clearly by Boltzmann� �Bo�� and earlier by Thomson� �Th��The reason why we observe approach to equilibrium over time scales far

shorter than the recurrence times is due to the property that the micro�

� Which is the only invariant distribution if one accepts the above discrete point of view�probably Boltzmann s�


canonical ensemble is such that on most of phase space the actual valuesof the observables� whose averages yield the pressure and temperature andthe few remaining other thermodynamic quantities� assume the same value��La� �� and p� �� of �Bo�� This implies that such values coincide withthe average and therefore satisfy the heat theorem�

The ergodic hypothesis loses its importance and fundamental nature and itappears simply as a tool used in understanding that some of the relations thatwe call �macroscopic laws� hold in the same form for all systems� whethersmall or large�

x�� Timed Observations and Discrete Time

The question that we shall now investigate is whether there can be anythingsimilar to the above done out of equilibrium but still in a stationary stateare there statistical properties that hold for small and large systems alikeunder the �only� assumption that the systems evolve in a very disordered��chaotic�� way� If so such properties might have physical relevance for largesystems when the system is large they may become observable becausethey may become a property of most of the individual con�gurations of thesystem without need of time averaging� just as it happens in equilibrium�

And they might be checked� and perhaps even be interesting� in small sys�tems which become therefore a natural testing ground� mainly because ofthe availability of fast computer experiments� just as it happens in equilib�rium with the ergodic hypothesis� which is usually tested in systems withvery few degrees of freedom�

The �rst step in the investigation is a dynamical hypothesis on the natureof the motions of complex systems �like a gas in a box�� This hypothesishas developed quite slowly in the past epoch it developed from the theoryof complex motions in $uid mechanics and it was formulated by Ruellein the early ��s �� and written explicitly later� �Ru#�� Ru�� Itin$uenced research strongly� see for instance �ECM��ECM�� and it ledto some concrete results� after being reformulated and put in the context ofnonequilibrium statistical mechanics� �much later� �GC�� The hypothesiswill be stated below and is called the chaotic hypothesis�

To proceed to the formulation of the hypothesis we need to set up a conve�nient kinematic description of disordered motions� convenient for the studyof chaotic evolutions this is necessary because the usual kinematics is wellsuited for orderly motions but is insu%cient for disordered ones�

In Chap�I we have already hit the di%culty of a proper representation ofthe evolution of a system of N particles in a box V as a permutation ofphase space cells� The di%culty came from the hyperbolic nature of theevolution that stretches some coordinates and contracts others� This forcedus to use very small cells and very small time intervals as phase space andtime units�

�# IX� Coarse Graining and Nonequilibrium

The division in cells of phase space is� in this way� extremely �ne �although�as we discussed� there are conceptual limits to the precision and reliabilitythat one can reach in this way�� so �ne that we can regard space as discrete�It is interesting to change viewpoint and� keeping the conception of space

as a continuum� to try using a discrete representation of motion based onphase space cells that are not as small as possible �but that are neverthelessstill compatible with the discrete picture that we follow after Chap�I�� weshall call such a description a �coarse grained� one�In other words it may be more convenient to use larger phase space cells

and to �nd a description of dynamics in terms of them� which explicitlyand �exactly� still takes into account the �often� �hyperbolic� nature of theevolution� i�e� its �usual� high instability or its chaoticity�The use of the �coarse grained� cells brings to mind something �approx�

imate� and not too well de�ned� Here we do not want to convey suchintuition and the representation of motion� that we look for� will be in prin�ciple as exact as wished� and involve no approximation at all� The moderne�orts to clarify the notion of coarse graining can be traced back to Krylov��Kr�� whose work strongly in$uenced Sinai� �Si�� whose work in turnin$uenced Ruelle leading him� eventually� to his hypothesis�We consider a system evolving on a bounded surface ( and with the evo�

lution acting near a given point x by expanding some line elements and bycontracting some others� The evolution will be described by a map S thatcan be thought of as being obtained from the time evolution $ow x � Stxby monitoring it every time some special event happens �for instance a col�lision between some pair of particles�� In this way the surface ( consists ofthe collection of the special events that one monitors� which will be calledtiming events or monitored events�The geometrical meaning of the construction of S from the $ow St is il�

lustrated in Fig� �� where a trajectory x � Stx in the phase space forthe evolution in continuous time ��usual phase space�� is depicted togetherwith the surface ( consisting of the monitored events �which is the phasespace where the motion is described by the map S the map S acts� in theFig� �� on the monitored event � mapping it into S� and S��


S��

S�

�

x�trajectory

(Fig� ��

Line elements on ( are transformed by the time evolution S and some ofthem become longer and others shorter �or keep their length�� Note that inorder to measure the length of a line element we need a metric de�ned on (we shall take the natural metric induced on the surface ( by the metric inthe space in which ( lies �which is usually the Euclidean space of positionand velocity vectors of the system particles�� But of course the metric thatwe use is rather arbitrary and we have to be careful� it is best to try dealingonly with notions that turn out to be metric independent�

In the following we shall always use the above discrete time evolution� butall that we say can be quite easily translated in terms of properties of thecontinuous time $ow St�

�

x�� Chaotic Hypothesis� Anosov Systems

Boltzmann s equation and Brownian motion theory are examples of at�tempts at studying nonequilibrium problems� The �rst has the ambition ofdiscussing the approach to equilibrium� while the second deals with motionsthat take place in equilibrium�

In general we shall say that a system is in a nonequilibrium situation when�ever nonconservative external forces act on it and� usually� sustain macro�scopic motions� Such systems will evolve and reach in due course a station�ary state� which will not be one of the equilibrium states with which we arefamiliar from the preceding chapters�

This is because time evolution will ultimately be described by di�erential

� One should not confuse this map with the map used in Chap�I for the one time stepevolution� Two successive timing events still contain very many time steps in the senseof Chap�I� here we do not discretize time or space� Motion appears evolving discretelysimply because we choose to observe it from time to time� when something that weconsider interesting happens�


equations which will contain dissipative terms and consequently phase spacevolume will be contracting� hence probability distributions correspondingto the statistics of stationary states must be concentrated on sets with zerovolume�

Before attempting a general theory of the approach to an equilibrium stateor� more generally� to a stationary state it seems reasonable to study theproperties of the stationary states themselves� end products of the evolutionunder external driving forces �which may vanish� however� so that equilib�rium theory will still be a �special case��

This will include investigating phenomena like Brownian motion whichcan be regarded as dynamical properties of equilibrium states� as well asgenuinely nonequilibrium phenomena� like thermodynamical relations be�tween quantities that can be de�ned in systems in nonequilibrium stationarystates�

And one can imagine stretching the analysis to stationary states of themacroscopic equations that are supposed to be obeyed out of equilibrium�like the Euler or Navier�Stokes equations for systems macroscopically be�having as $uids�

Approach to equilibrium or to a stationary state is likely to be a moredi%cult problem and it will be set aside in most of what follows� Hence weshall not deal with states of systems evolving in time rather we refer toproperties of states that are already in a stationary state under the in$uenceof external nonconservative forces acting on them� For instance think of anelectric circuit in which a current $ows �stationarily� under the in$uenceof an electromotive �eld� or of a metal bar with two di�erent temperatures�xed at the extremes� and one can even think of a Navier�Stokes $uid in a�stationary� turbulent Couette $ow or a more general $ow� What followsapplies also to such apparently di�erent systems �and in fact the basic ideaswere developed having precisely such systems in mind��

The �rst two systems� regarded as microscopic systems �i�e� as mechanicalsystems of particles�� do certainly have very chaotic microscopic motionseven in the absence of external driving �while macroscopically they are ina stationary state and nothing happens� besides a continuous� sometimesdesired� heat transfer from the system to the surroundings�� The thirdsystem also behaves� as a macroscopic system� very chaotically at leastwhen the Reynolds number is large�

A basic problem is that the situation is quite di�erent from that in whichBoltzmann was when attempting a microscopic proof of the heat theoremthere is no established nonequilibrium thermodynamics to guide us�

The great progress of the theory of stationary nonequilibrium that tookplace in the past century �the XX�th�� at least that which was unanimouslyrecognized as such� only concern properties of incipient nonequilibriumi�e� transport properties at vanishing external �elds �we think here of On�sager�s reciprocity and of its more quantitative form given by the Green�Kubo transport theory�� So it is by no means clear that there is any generalnonequilibrium thermodynamics�


Nevertheless in �� a �rst suggestion that a general theory might bepossible for nonequilibrium systems in stationary and chaotic states wasmade by Ruelle� eventually written down only later in �Ru�#c�� Ru#��This suggestion was made with full understanding of its ambition

�If one is optimistic� one may hope that the asymptotic measures will playfor dissipative systems the sort of role which the Gibbs ensembles played forstatistical mechanics� Even if that is the case� the di�culties encounteredin statistical mechanics in going from Gibbs ensembles to a theory of phasetransitions may serve as a warning that we are� for dissipative systems� notyet close to a real theory of turbulence�� Ru�#c��

The proposal is very ambitious because it suggests a general and essen�tially unrestricted answer to what should be the ensemble that describesstationary states of a system� whether in equilibrium or not� In the recentformulations of Cohen and Gallavotti it reads

Chaotic hypothesis for the purpose of studying macroscopic properties� thetime evolution map S of a many�particle system can be regarded as a mixingAnosov map��

We defer discussing in detail the technical notion of �mixing Anosov map�to the coming sections and the hypothesis is written here only for con�creteness and later reference� For the moment it will su%ce to say thatmixing Anosov maps are the paradigm of chaotic motions they are wellunderstood dynamical systems which show chaotic behavior in the �purest�possible way� They play a role in nonlinear dynamics very similar to thatplayed by harmonic oscillators in the theory of stable motions�

Remark If the evolution is very dissipative and motions tend to an attract�ing set smaller than the whole phase space the hypothesis may be interpretedas meaning that the attracting set can be regarded as a smooth surface andthat the restriction of the evolution to it is a mixing Anosov map� see below�which is a very special case of a wider class of chaotic systems called AxiomA systems�� BGG��BG�� However a less strict interpretation could beto say that the attractor is an �Axiom A attractor�� see below�

The ergodic hypothesis led Boltzmann to the general theory of ensembles�as acknowledged by Gibbs� p� vi in �Gi#�� whose work has been perhapsthe main channel through which the allegedly obscure works of Boltzmannreached us� besides giving the second law� �� it also prescribed themicrocanonical ensemble for describing equilibrium statistics�

The reasoning of Ruelle was that from the theory of simple chaotic systemsone knew that such systems� just by the fact that they are chaotic� will reach

� This is a notion that in the original work �GC�� was called �transitive Anosov map�however it turns out that the established nomenclature is di erent and here I try toadhere to it� as much as possible�

� IX� Coarse Graining and Nonequilibrium

a �unique� stationary state� Therefore simply assuming chaoticity wouldbe tantamount to assuming that there is a uniquely de�ned ensemble whichshould be used to compute the statistical properties of a stationary systemout of equilibrium�This argument is based on the idea that a chaotic system� even when it is

not exactly a mixing Anosov map in the mathematical sense� does share itsmain qualitative features� �Ru�#c�� in a sense similar to the one in which instability theory one infers properties of nonlinear oscillations from those ofharmonic oscillators or of integrable systems� And a key property of mixingAnosov maps is that their motions show a unique statistics �� This meansthat there is a unique probability distribution � on phase space F such thatfor all �smooth� observables F �x�

limT��

�

T

T��Xn�

F �Snx� �

ZFF �y��dy� ��

apart from a set of zero volume of initial data x � F ��

The distribution � is called the SRB distribution or the statistics of mo�tions it was proven to exist by Sinai for Anosov systems and the resultwas extended to the much more general Axiom A attractors by Ruelle andBowen� �BR�� Ru�� Natural distributions were� independently� discussedand shown to exist� �LY�� for other �related and simpler� dynamical sys�tems� although in an apparently less general context and with a less generalvision of the matter� �Si��BR��Therefore one is� at least in a very theoretical way� in a position to inquire

whether such a unique ensemble has universal properties valid for small andlarge systems alike �of course we cannot expect too many of them to hold�but even a single one would be interesting��In fact in equilibrium theory the only universal property is precisely the

heat theorem� besides a few general �related� inequalities �e�g� positivity ofthe speci�c heat or of compressibility�� The theorem leads� indirectly as wehave seen� to the microcanonical ensemble and then� after one century ofwork� to a rather satisfactory theory of phenomena like phase transitions�phase coexistence and universality�From this point of view the criticized �and more and more often dismissed

as� at best� unnecessary� ergodic hypothesis assumes a new status andemerges as greatly enhanced� Ruelle s proposal seems to be its natural�and� perhaps� the unique� extension out of equilibrium�The proposal was formulated in the case of $uid mechanics but it is so

clearly more general that the reason why it was not explicitly referringto statistical systems is� probably� due to the fact that� as a principle� itrequired some �check� if formulated for statistical mechanics� As originally

� In general replacing S by S�� in �� leads� when the new limit exists� to a di�erentprobability distribution on phase space� which we can call the statistics towards the past�while �� de�nes the statistics towards the future� One cannot expect that� whenthey exist� the two statistics coincide�


stated� and without any further check� it would have been analogous to theergodic hypothesis without the heat theorem �or other consequences drawnfrom the theory of statistical ensembles��Of course the chaotic hypothesis will su�er from the same objections that

are continuously raised about the ergodic hypothesis namely �there is thetime scale problem�� see x��To such objections the answer given by Boltzmann should apply unchanged

large systems have the extra property that the interesting observables takethe same value in the whole �or virtually whole� phase space� Thereforetheir values satisfy any relation that is true no matter whether the system islarge or small such relations �whose very existence is� in fact� surprising�might even be of no interest whatsoever in small systems �like in the above�mentioned Boltzmann s rigid Saturn ring� or in his other similar example ofthe Moon regarded as a rigid ring rotating about the Earth��Evidence for the nontrivial applicability of the hypothesis built up and

it was repeatedly hinted at in various papers dealing with numerical ex�periments� mostly on very few particle systems �� to give an indi�cation�� HHP#�� In attempting at understanding one such experiment��ECM�� the above �formal� interpretation of Ruelle s principle was for�mulated� �GC�� for statistical mechanics �as well as for $uid mechanics�replacing �many�particles system� with �turbulent $uid��

The hypothesis was made �rst in the context of reversible systems theywere in fact the subject of much of the experimental work that bloomedonce the importance and relevance of reversibilty was strongly stressed�and supported by experiments� by Hoover and coworkers� for highlightssee �HHP#��EM��ECM��DPH�� Note that saying that reversibilitycan be relevant to� and even facilitate� the analysis of dissipative motionsis highly nontrivial and it required insight and intellectual courage to beintroduced�

The strict interpretation of the chaotic hypothesis given in the aboveremark rules out� when the attractive set is smaller than phase space�attractors� with a fractal closure �i�e� attracting sets which are not smooth

� It is important to distinguish between attracting set and attractor� The �rst is a closedset such that all points close enough to it evolve in time keeping a distance from itthat tends to � as t �� and furthermore are �minimal in the sense that theydo not contain subsets with the same properties� Consider� to de�ne an attractor� anattracting set which admits a statistics � given by �� for all but a zero volume setof nearby points x� Any subset C of such an attracting set with ��C� � � is called anattractor� More generally we can imagine choosing initial data x near an attracting setwith a probability distribution �� that can even be concentrated on sets of zero volume�i�e� that is completely di erent from the volume measure� Supposing that all data x buta set of zero �� measure satisfy �� of course� in general� with a statistics �� di erentfrom � and ��dependent�� The subsets of the attracting set that have probability �with respect to the statistics generated by such initial data x� will be called �attractorsfor the data with distribution �� and they may have � measure with respect to thestatistics � de�ned by �� We see that in general the notion of attracting set isuniquely determined by the dynamics� while the notion of attractor depends also onwhich initial data we are willing to consider� and even once the class of initial data ischosen the notion of attractor is not uniquely de�ned as we can always take out of an


surfaces�� to include them� one should replace� in the formulation of thechaotic hypothesis� the word �Anosov� into �Axiom A� �a similar but muchweaker notion� but it seems appropriate to wait and see if there is real needof such an extension� It is certainly an essential extension for small systems�but it is not clear to me how relevant could fractality be when the systemhas �� particles�� Therefore we shall concentrate our attention mainly onAnosov systems�Finally I add one more comment on the words at the beginning of the

chaotic hypothesis �for the purpose of � � �� an easy critique could be thatthis is �vague� since it is obvious that virtually none of the systems ofinterest in statistical mechanics �or $uid dynamics� are Anosov systemsin the mathematical sense �they are often not smooth� or obviously nothyperbolic and often not even ergodic��Nevertheless I think that the hypothesis is well founded and it has an

illustrious predecessor in the early ergodic hypothesis of Boltzmann he wastrying to prove the heat theorem� he needed the condition that the motionswere periodic� he said that if they were not they could still be consideredso for practical purposes �because a nonperiodic orbit can be regarded asperiodic with in�nite period��Assuming that the motions were periodic �i�e� the systems were �mono�

cyclic�� led him to discover a hitherto unknown property of periodic motions�the heat theorem�� Indeed things �went as if the motions were periodic��Bo�� a �rst rough formulation of the ergodic hypothesis� We have dis�cussed at length the interpretation and the interest of the ergodic hypothesisin Chap�I�II�III� and we have seen that it leads to important relations be�tween averages� one can think that �all systems are Anosov� in the samesense�Mathematically what is being said is that there might be general properties

of Anosov systems that might have been missed in spite of the vast reasearchon the subject�

x�� Kinematics of Chaotic Motions� Anosov Systems

To proceed we need a more precise formulation of the notion of Anosovsystems and an analysis of the kinematics of their motions� We give herean informal de�nition and in the next section we give a detailed discussionof the kinematics of motions�

De�nition A mixing Anosov system is� see p� of �AA�� a smooth mapS �i�e� of class C�� of a smooth manifold M ��phase space�� and aroundevery point x one can set up a local coordinate system with the followingproperties associated with it

�a� depends continuously on x and is covariant �i�e� it follows x in itsevolution� and

attractor one orbit and still have an attractor �unless the attractor consists of �nitelymany points��


�b� is hyperbolic� i�e� transversally to the phase space velocity of any chosenpoint x the motion of nearby points looks� when seen from the coordinateframe covariant with x� like a hyperbolic motion near a �xed point� Thismeans that in a sphere of small radius � around x there will be a connected�local stable coordinate surface�� the stable local manifold W s��

x through x�whose points have trajectories that get close to the trajectory of x at ex�ponential speed as the time tends to �� and a �local unstable coordinatesurface�� the local unstable manifold W u��

x � whose points have trajectoriesthat get close to the trajectory of x at exponential speed as the time tendsto �� Furthermore the exponential speed of approach must admit a boundindependent on x� i�e� it has to be �uniform��

�c� the global stable manifold and the global unstable manifold of everypoint� i�e� the sets ��n�S

nW u��x and ��n�S

�nW s��x are dense in M � This

excludes the possibility that the phase space consists of disconnected invari�ant parts� It also excludes the case that it consists of n disconnected partsM�� Mn cyclically permuted by S so that Sn is a mixing Anosov map asa map of Mj into itself� clearly a case in which one is �improperly de�ning�phase space and evolution map �which should be rather de�ned as M�� say�and Sn respectively��

If only �a��b� hold the system is simply called an �Anosov system�� It isa theorem �Anosov� that the planes tangent to W u

x �Wsx are quite smoothly

dependent on x� they are H�older continuous in x �in general not more�however� because in general they are not di�erentiable in x even if the mapS is analytic�� AA�#��Ru#��

If the system is described in continuous time the direction parallel to thevelocity has to be regarded as an �extra� neutral direction where� on theaverage� no expansion nor contraction occurs� However here we shall adhereto the discrete viewpoint based on timed observations� see x�� For adiscussion of the continuous time point of view see �BR�� Bo�� Ge�#��

The simple but surprising and deep properties of Anosov maps are by andlarge very well understood� �Ru#�� Unfortunately they are not as wellknown among physicists as they should be many seem confused by thelanguage in which the above concepts are usually presented� however it is afact that such remarkable mathematical objects �i�e� Anosov systems� havebeen introduced by mathematicians� and physicists must� therefore� makean e�ort at understanding the new notion and its physical signi�cance�

In particular� as mentioned above� if a system is Anosov� for all observ�ables F �i�e� continuous functions on phase space� and for all initial data x�outside a set of zero volume� the time average of F exists and can be com�puted by a phase space integral with respect to a distribution � uniquelydetermined on phase space F as expressed by ��

Clearly the chaotic hypothesis solves in general �i�e� for systems that canbe regarded as �chaotic�� the problem of determining which is the ensembleto use to study the statistics of stationary systems in or out of equilibrium


�it clearly implies the ergodic hypothesis in equilibrium�� in the same sensein which the ergodic hypothesis solves the equilibrium case�

The chaotic hypothesis might turn out to be false in interesting cases�like the ergodic hypothesis which does not hold for the simplest systemsstudied in statistical mechanics� like the free gas� the harmonic chain andblack body radiation� Worse� it is known to be false for trivial reasons insome systems in equilibrium �like the hard core gas�� simply because theAnosov map de�nition requires smoothness of the evolution and systemswith collisions are not smooth systems �in the sense that the trajectoriesare not di�erentiable as functions of the initial data��

However� interestingly enough� the hard core systems are perhaps the ob�ject closest to an Anosov system that can be thought of� being at the sametime of statistical mechanical relevance� �GG�� ACG�� to the extentthat there seem to be no known �physical� properties that this system doesnot share with an Anosov system� Aside from the trivial fact that it is nota smooth system� the hard core system behaves� for statistical mechanicspurposes� as if it was a �mixing� Anosov system� Hence it is the prototypesystem to study in looking for applications of the chaotic hypothesis�

In fact if the system is smooth one can imagine� and sometimes proverigorously� see �RT�#�� that phase space contains big nonchaotic islandswhere the system looks very di�erent from a Anosov system it is importantto understand the relevance of such regions for the statistical properties of�moderately� large systems because the chaotic hypothesis states that theyshould not be too relevant for nonequilibrium theory�

The problem that �remains� is whether the chaotic hypothesis has anypower to tell us something about nonequilibrium statistical mechanics� Thisis the real� deep� question for anyone who is willing to consider the hypoth�esis� Of course one consequence is the ergodic hypothesis� hence the heattheorem� but this is manifestly too little even though it is a very importantproperty for a theory with the ambition of being a general extension of thetheory of equilibrium ensembles�

A chaotic motion� as discussed in x�� is recognized from the expansionand contraction that the evolution map x � Sx produces on line elementsof phase space ( �the space of the events��

However the notion of expansion and contraction depends on the metricthat we use near x and Sx to measure lengths� hence it is clear that expan�sion and contraction at x are not de�nable in terms of x and of the actionof S near x alone� The latter are� in fact� local notions� but it will makesense to say that a line element � emerging from x �lying on (� �expands�if it does so asymptotically� i�e� if

jSn�jSnx � Ce�nj�jx for all n � � ��

where j � jx denotes the length of � measured with the metric used at thepoint x of ( and C� � � � are suitable constants� Likewise one has to reasonin the case of contraction�


The collection of the expanding line elements generates a plane T ux �tangent

to (� called the expanding plane at x and the collection of the contractingline elements generates another plane T s

x � This can be illustrated in a two�dimensional case by the �rst of Fig� �� where two line elements emergingfrom a point x � ( are directed along the expanding and the contractingdirections�

x

T sx

T ux

W sx

W ux

�

Fig� ��

Here the two tangent planes emerging out of x are drawn as two arrowsone has to imagine that the planes can be drawn at all points of (� in factthe family of planes T s

x �or T ux � can be integrated and generates a family of

smooth manifolds W sx �or W u

x � that are tangent at each of their points yto the plane T s

y �or T uy �� The second drawing in Fig� �� illustrates this

property and shows the beginning of the two manifolds through x� i�e� a partof the manifolds that is enclosed in a sphere of radius �� small compared tothe curvature of the manifolds�More generally we can hope that it will be possible to give a decomposition

of the tangent plane Tx at a point x as a sum of linearly independent planesT jx

T �x � T �

x � � � �� T px ��

of dimensions n�� np and to de�ne p Lyapunov exponents �� p suchthat

limn��

�

nlog jSn�jSnx � �j if � � T j

x � ��

If such a decomposition and such exponents exist we say that the point xadmits a dynamical base for the evolution S �towards the future��If Snx denotes the matrix of the derivatives of Sn evaluated at x the

the spaces T �x � � � � � T

px can be taken to be the eigenspaces of the matrix

limn��Snx ��Snx �

��n� if this limit exists however it should be clear

that this is not the only way in which the tangent plane can be split so that�� holds�This is so because the ��operation �i�e� transposition� depends on the sys�

tem of coordinates and on the metric one sees this by noting that Snx mapsthe tangent space Tx into a di�erent space� namely TSnx� For instance ifwe have a decomposition satisfying �� we get a new one by changingT �x into a new plane forming a di�erent �positive� angle with respect to


the plane of the other vectors T �x � � � � � T p

x � and one can likewise changeT �x � T

�x � � � ��

But the sequence of subspaces T px � T p

x�T p��x � T p

x�� T �x � Tx � T p

x�� T �x � is uniquely determined and metric independent �assuming the existence

of a dynamical base towards the future��

Dynamical bases for the evolution towards the past are de�ned in the sameway if S is invertible �we do not describe here the obvious adjustmentsneeded in the de�nition� e�g� n� �� The same comments about lack ofuniqueness can be made for such bases�

On the other hand in the case of invertible maps dynamical bases whichare such both in the future and in the past may exist and� if they do exist�they are uniquely determined� we shall call them simply dynamical bases orbilateral dynamical bases�

It is remarkable that such bilateral bases exist under rather general condi�tions� For instance

�a� if � is a probability distribution which is S�invariant � then� apart froma set of points of zero ��probability� every point admits a dynamical base�and

�b� if the distribution � is ergodic in the sense that there are no nontrivialmeasurable functions that are constants of motion then the dimensions ofthe planes T j

x � as well as the Lyapunov exponents �j � are x�independentwith ��probability ��

Remarks

�� Properties in �a� and �b� are the content of Oseledec�s theorem�

� � Dynamical bases for the motion towards the future and those for themotion towards the past will be di�erent� in general� However if S is invert�ible and � is an invariant distribution the forward and backward dynamicalbases can be chosen to coincide �apart for a set with zero ��probability� andtheir exponents are opposite� see �Ru��a� p� #��

�� A caveat is that in � � it is essential that � be S�invariant� Thereforethe above is not saying that all points but a set of zero volume will admit adynamical base� because in general the volume measure �� is not invariant�But if �� admits a statistics � in the sense of �� then all points buta set of ��measure � admit a dynamical base� In other words it is impor�tant to note that in general an invariant probability distribution � is notexpressible by means of a density function in phase space �one says that itis not �absolutely continuous� or �nonsmooth�� and therefore one cannotsay that the latter statement �b� holds �apart from a set of points with zerovolume�� even when the volume distribution �� admits a statistics ��

� i�e� the probability ��E� of E and that of S��E are equal for all sets E s�


Considering a dynamical system the question of the existence of the stableand unstable manifolds or just of the local stable and unstable manifolds isof course rather delicate and one would like to have simple critieria guaran�teeing their exixtence and a few basic properties� The real problem is withthe local manifolds� W s��

x �W u��x � in a sphere of radius � around a point x� In

fact once the local manifolds are de�ned one can de�ne the global expandingand contracting manifolds even when S is not invertible� simply as

W ux �f��n�S

nW u��xn if Snxn � xg ��

W sx � fset of points y such that Sny � W s��

Snx for n large enough g

where the �rst union is meant� when S is not invertible� also as a union overthe various possibilities of choosing the xn s�The existence of the local hyperbolic structure may seem a property dif�

�cult to check� This is ideed so� but in many cases the proof is greatlysimpli�ed because of the following su%cient conditions�The existence of the local manifolds W s��

x �W u��x can be deduced from the

existence of a continuous family of cones 'ux�'sx lying in the tangent plane

to every point x with the property that a displacement � that points intothe cone 'ux will be transformed by the evolution map S inside a cone whichis strictly less wide than the cone 'uSx and will have a length jS�j which isstrictly larger than that of �� and a corresponding property holds for cone'u� see Fig�� below�Here �strictly� means� see Fig�� that the length will be larger by a factor� � � with respect to the initial length and � will be independent of thepoint x� and� thinking that the cone is determined by its intersection withthe unit sphere in Tx� �strictly less wide� will simply mean that the imageS'x intersects the unit sphere of TSx in a set which has a distance r � � tothe boundary of 'Sx � TSx� with r being x�independent�The �rst of Fig�� illustrates the parts of the pair of cones 'ux�'

sx around x

�shaded sectors� contained inside a small sphere around x� The evolution Smaps them into the shaded sectors of the second �gure� so that the expand�ing cone �marked by u� ends �well inside� the corresponding one aroundSx �unshaded� while the contracting cone widens around the correspondingcone for Sx�

x

'sx

'ux

S�

Sx

s

u

Fig� ��

�a� �b�

S A similar property is required for the cones 'sx� by using S�� instead ofS�

#� IX� Coarse Graining and Nonequilibrium

The existence of families of cones is su%cient for the system to be anAnosov system� but it is a much simpler condition� conceptually� and infact it can even be easily satis�ed in some cases� �Ru�� Li��

x�� Symbolic Dynamics and Chaos

The key property of mixing Anosov systems is that of admitting a�Markov s partition�� this is an important discovery that was realized inthe classical works of Sinai� �Si�#�� Si� �� heralded by the independent dis�covery of a special case� �AW�#�� We devote this section to the illustrationof this important geometrical notion and to the consequent �symbolic dy�namics�� or �coarse grained� representation of motion�The geometric properties of Anosov systems allow us to imagine a partition

of phase space into rectangular cells E�� EN such that� see Fig� ��

�i� Each cell is de�ned by a �center� c and two �axes� �uc ��

sc it consists

of the points z�� which have the form z�� W u�� W s��

�

def� �� for

some � � �uc and � � �s

c� The boundary E of such a cell E will� therefore�

consist of a shrinking part �or �stable part�� sEdef� �u

c � �sc and of an

expanding part �or �unstable part�� uEdef� �u

c � �sc�

The geometrical construction is illustrated in the two�dimensional case inFig� �� The circles are a neighborhood of c � x of size � very smallcompared to the curvature of the manifolds �so that they look $at�� Fig�

��a shows the axes� Fig� ��b shows the � operation and W u�� W s��

�

�the horizontal and vertical segments through � and �� respectively� have size�� Fig� ��c shows the rectangle E with the axes �dotted lines� and withthe four marked points being the boundaries �u and �s� The picturerefers to the two�dimensional case �which is substantially easier to draw andto conceive� see �Bo�� and the stable and unstable manifolds are drawn as$at� i�e� the � s are very small compared to the curvature of the manifolds�Transversality of W u

x �Wsx is pictorially represented by drawing the surfaces

at ��o angles

x

�s

�u�

�

� � �E

�a� �b� �c�

Fig� ��

�ii� Furthermore we require a covariance property of the various cells withrespect to the action of the evolution map S i�e� we demand that the mapS transforms the shrinking parts of the boundary of E inside the union ofthe shrinking parts of the various cells of the partition� and S�� also enjoysthe corresponding property �with the collection of �expanding� sides of the

IX� Coarse Graining and Nonequilibrium #�

cells now containing the S�� images of the �expanding� sides of each cell��The covariance property is illustrated by Fig� ��

s

�

u u

s

Fig� ��

S�

A partition into such cells is called a Markov partition� Such partitionsenjoy remarkable properties of covariance under the time evolution and aresuitable for a description of the motion�Note that here �shrinking� and �expanding� surface elements are repre�

sented in a �literal� sense� However since these are asymptotic notions�which only as such are metric independent� it may well be that under theaction of S a �shrinking side� actually expands in the metric used� or an�expanding side� actually contracts however under repeated applicationsof S such surface elements do eventually behave as the words� and drawings�we use suggest�Before proceeding it seems useful to discuss brie$y an example� It is the

map of the two�dimensional torus T � �� de�ned by �x� y� � �x �

y� x� y� �modulo �� or

�x�

y�

��

��

��xy

��modulo �� This is the

case in which Markov s partitions were �rst discovered� �AW�#��In this case we see easily that the expanding and contracting planes are

simply the lines through a point �x� y� parallel to the eigenvectors of the

matrix M �

��

�� The stable and unstable manifolds are� therefore�

lines through �x� y� parallel to these eigenvectors and regarded as drawn onthe manifold T �Thus they cover it densely because the slope of these lines is irrational

�being �� p�� We see in this example also why it is �in general�

necessary to distinguish between the local stable and unstable manifoldsand the global ones �which are dense while the local manifolds are not��It is easy� by using ruler and compass� to draw a Markov pavement for the

above map an example is given in Fig� ��

# IX� Coarse Graining and Nonequilibrium

�a� �b�

Fig� ��

This is a partition for the torus map just introduced Fig� ��a representsmany copies of the torus pavement in Fig� ��b �unraveled as a pavementof the plane�� Each copy is a square with side �� The two lines are parallelto the eigenvectors of the above matrix M generating the map� The linewith positive slope is parallel to the eigenvector with eigenvalue larger than�� and the two lines are orthogonal because the matrix is symmetric�In this simple case the expanding and contracting directions are trivially

parallel to the two drawn lines� If we draw the lines of Fig� ��a asthey really appear when wrapped back into the torus we get Fig� ��bwhere the endpoints of the lines are marked as big dots� The lines de�neseveral rectangles �both in the geometrical sense and in the above introduceddynamical sense� which explains the attribute �rectangular� used� with afew exceptions �three in the picture�� They are due to the termination ofthe initial lines �in the middle of nowhere�� The dashed lines continue theoriginal lines �on both ends� until they meet a line� thus completing themissing rectangles� The result depends on the order of the continuationoperations an irrelevant ambiguity �as in any event Markov s partitionsare by no means unique�� when they exist��Fig� ��b could have been obtained directly without any elongation of the

lines had the initial lines been drawn of appropriate size� The constructionshows how to �nd the appropriate size �using only �ruler and the compass��as required by every noble drawing� or by a Postscript program using only�integers and quadratic irrationals��The union of the rectangle boundaries parallel to the line with negative

�positive� slope is transformed into itself by the action of the map �inversemap�� This is so because it is a connected piece of the stable manifold of thetrivial �xed point that is the origin� it shows that the property of Fig� �� is indeed satis�ed� Hence Fig� ��b is a simple example of a �Markovianpavement� also called a �Markov partition� �and its discovery was at thebeginning of the developments discussed here� �AW�#�� Si�#��In fact given a Markov partition we can generate a much �ner partition

simply by transforming it with the various iterates of the map and thenintersecting the collection of pavements thus obtained� see footnote #� Thenew partition of phase space is obviously still Markovian but it can be made

� For instance if P � fEig is a Markov partition then P � � fEi � SEjg� P �� fS��Ei �

Ejg and $P � fS��Ei � Ej � SEmg �and so on� are Markov partitions�


as �ne as we want�

The above de�nitions and related properties can be extended to systemswhich have an attracting set consisting of a smooth surface and the timeevolution restricted to the surface becomes a mixing Anosov system� Theextension can even be pushed to systems with attracting sets that are notsmooth but that are still hyperbolic ��axiom A attractors�� Ru�� Weshall not deal here with this more general notion �for simplicity��

Anosov systems that are mixing are topologically mixing in the sense thatgiven any pair of open sets U� V there is a nV such that the iterates SnUhave a nonempty intersection with V for all n � nV � Hence imposingmixing is the simple way of imposing the condition that the system phasespace cannot be �disconnected� into disjoint parts under the action of S orof any of its iterates �alternative to assuming existence of a dense orbit��

A general theorem ��Smale�s spectral theorem�� deals with cases in whichone does not assume mixing under the assumptions �a� and �b� in thede�nition of x�� and adding the extra assumption

�c� Periodic motions are dense on phase space�

the phase space M can be decomposed into a union of M� � M� � � � � �Mn in each of which there is a dense orbit� If n � � the system is saidto be transitive� Hence each system satisfying assumptions �a�� b�� c�can be regarded as a collection of �nitely many transitive Anosov systems�Furthermore if a Anosov system is transitive then its phase space M can bedecomposed as a union M � M �

� �M �� M �

m on each of which Sm actsas a mixing Anosov system�

In other words given an Anosov system with dense periodic points eithertopological mixing holds for the map Sn for some n� or phase space splitsinto a �nite number of disjoint closed components �called �spectral ele�ments� of the system� in each of which� for some large enough n� the mapSn is topologically mixing� The mixing assumption is not as strong as itmay at �rst appear if it does not hold it is because in some sense wehave chosen the phase space inappropriately not noticing that motion wasactually taking place on a smaller space�

Markov partitions set up a nice �coarse graining� permitting us to thinkof the dynamical system as a copy of something very familiar in statisticalmechanics namely the one�dimensional Ising model� or of one of its exten�sions considered in x�� The correspondence is via the symbolic dynamicsassociated with Markov partitions�

One de�nes for each point x a sequence ��x� of digits each of which cantake N values if N is the number of elements of a Markov partition E withelements so small that the image of every rectangle intersects all the otherrectangles at most in a connected part �i�e� the size of the rectangles isso small that even when stretched by the one time step evolution map� itremains small compared to the curvature of the sides��


One de�nes aN�N transition matrix T�� by setting T�� if the interiorof the rectangle E� evolves under S into a set intersecting the interior ofthe rectangle E� � we set T�� otherwise�

The de�nitions readily imply a few properties� A transition matrix is saidto be transitive if for any pair �� there is a power n such that �Tn�� this means that there is a sequence which is compatible and which containsthe symbols � and� to the right of it� �� A transition matrix is said tobe mixing if for all n large enough �Tn�� A Markov partition for atransitive Anosov system has a transitive matrix which is also mixing if thesystem is mixing�

A sequence f�kg is �allowed� or �compatible�� see x�� if it is a sequence ofsymbols such that T�k�k� � � for all integers k � �� A meditationon Fig� �� i�e� on the covariance property� will convince the reader thatif � � f�ig is allowed there must be a point x such that Skx � E�k for allintegers k � �� and this point must be unique by the hyperbolicity ofthe transformation �if there were two such points they would travel visitingalways the same boxes of E which is impossible because if two pointsvisit the same boxes during the time ��T� T � then� from the de�nitions ofhyperbolicity� their distance must be not bigger than O�Ce��T �� cf� ��

Therefore it follows that we can establish a correspondence between pointsand compatible sequences� There may be exceptionally more than one se�quence representing the same point� but this can happen only if the pointis either on a boundary of a rectangle of E or on that of one of its imagesunder iterates of S� Hence the set of points that are represented by morethan one sequence has zero volume and� therefore� it can be ignored forthe purposes of our discussion �which in any event disregards sets of zerovolume��

This means that we can map� or �code�� phase space into a space of se�quences any function on phase space becomes a function of the sequences�The coding of points into sequences of digits is very similar to the familiarcoding of its coordinates into decimal sequences �which is also well de�nedapart from a zero volume �dense� set of exceptional points� namely thepoints whose coordinates are numbers ending with an in�nite string of � sor of � s�� And it is harder but much better in spite of the fact that the dec�imal representation is the �usual� representation of points in phase space�both in theoretical applications and in numerical experiments�

It is better because it is adapted to the dynamics and turns the mostchaotic dynamics into a �standard� one �still chaotic�� namely the shifton a space of sequences of symbols subject to a nearest neighbor constraint�that T�i��i� � �� also called a hard core� see x�� The latter dynamicalsystems are often called �subshifts of �nite type�� or �one�dimensional spin�or particle� chains� for obvious reasons�

A function F �x� which is mildly regular on phase space� e�g� H�older con�tinuous with exponent a� becomes a function F �x�� of � which has a weakdependence on the digits �k of � with large jkj i�e� if � and �� agree onthe digits between �k and k then the distance between x�� and x�� is


� O�Ce��k� so that F �� F �� is bounded by a constant times e��ak�In particular the expansion rate ,u�x� of the phase space volume over one

time step of the map S �i�e� the determinant of the Jacobian S of the mapS� also called the contraction rate of phase space because ,u�x� is a �real�contraction or a �real� expansion depending on whether it is � � or � ��will satisfy for some C� a

j,u�x�� ,u�x��j � Ce�k a ��

if the digits of � and �� with labels between �k and k agree� This is becausehyperbolicity implies that the stable and unstable tangent planes at x arevery smooth and vary in a H�older continuous fashion with the point x �asmentioned in x�� this is Anosov s theorem�� so that also the functions,u�x��,s�x� are H�older continuous�

One can interpret �� as saying that the function ,u�x��def� e��

has short range as a function of the symbolic sequence � and the followingremarks are worth the e�ort necessary for understanding their formulation�admittedly hard at �rst sight �yielding to a reassuring sense of trivialityafter some thought�

�� The set of points which are symbolically represented by sequences thatagree between � �

�T and ��T is just the set of points in the very small rect�

angle of E� ��T�

��T

consisting of the intersections � ��T

� ��TS�jE�j �with sides

bounded proportionally to e��T �� see footnote # above� Hence a sum over

the elements E � E� ��T�

��T

can be written as a sum over the sequences

��k� � � � � �k� �which are compatible�� The reader will be greatly helped by

attempting to draw a representation of a set in � ��T

� ��TS�jE�j in the manner

of Fig� �� above��

� � A point x � E � � ��T

� ��TS�jE�j is determined by a compatible bi�in�nite

sequence � which continues �� T� � � � � � �

�Ton either side to an in�nite �com�

patible� sequence � � � � �� T��

�T� � � � � � �

�T� � �

�T��

�� Since there are only �nitely many values for each symbol we can de�nefor each symbol � compatible sequences �R��

�� and �L��

�� in�nite to the right and to the left� respectively� and suchthat T��

�� and T��

�� i�e� respectively right and left compatible

with � this can be done in many ways� We shall call a pair of functions� � ��R�� L�� a boundary condition�If we are given a boundary condition ��R�� L�� we can associate with

each element E � � ��T

� ��TS�jE�j a point c�E� whose �bi�in�nite� symbolic

sequence is �L�� T

�� T� � � � � � �

�T� �R��

�T�� We call c�E� the �center�

of E with respect to the boundary condition � � ��R�� L��The bi�in�nite continuation� which we can naturally call �Markovian�� of the

#� IX� Coarse Graining and Nonequilibrium

�nite string �� T� � � � � � �

�T to an in�nite compatible sequence is possible

because of the topological transitivity property supposed for the system�consequence of the mixing��

�� We can interpret a continuation of the sequence �� T� � � � � � �

�Tas the

assignment of a boundary condition to a spin con�guration in the one�dimensional box ��

�T��T � in the sense of the discussion in x�� following

��

�� If x is a point in E � �T�

j� T�

S�jE�j the product of expansion factorsQT�

j� T�

,u�S�jx�� can be written

T�Y

j�T�

,u�S�jx�� e

PT�

k�T�

��k��

with �a�� log ,a�x�� a � u� s�

�� One says that the mixing Anosov system �M�S� admits a symmetry ifthere is an isometry of phase space with I� � � which is either commutingor anticommuting with S i�e� IS � SI or IS � S��I � respectively� Inthe second case the symmetry is called a time reversal symmetry of coursethe same system may admit several time reversal symmetries� A Markovpavement for �M�S� is said to be �I�symmetric� if E � P implies thatIE � P � in this case� again by the transitivity of the compatibility matrix�we can de�ne time reversible boundary conditions � � ��R�� L�� in

such a way that the centers c�E� of the sets E � �T�

j� T�

S�jE�j satisfy the

covariance property Ic�E� � c�IE��In the commuting case this is so because we can choose the continuationof �k to the right and that of i�k to the right to be �consistent� i�e� if�k�� k�� continues �k then i�k�� i�k�� continues i�k� In the an�ticommuting case the continuation of � to the right has to be chosen con�sistent in the same sense with that of i� to the left� Such consistent choicesare Markovian in the above sense�

�� another interesting consequence of mixing is that given a sequence��s� � � � � �s and supposing that p is such that T p

�� for all �� which

Given a symbol �� let n be such that �T n� �� Let n� be such that

�Tn �� this means that there is a sequence �� n��

� � which

is compatible� and also a sequence �$�� $�n � which is compatible� therefore

�R��def� $�� $�n � ��

��

�n��

� � � � where the last dots indicate inde�nite repetition

of the string �� n��

is a in�nite string �continuing the symbol � to the right

into a compatible sequence� Likewise one builds a �L�� continuing th symbol � to theleft into a compatible sequence� The name �Markovian is due to the property that thesequences �L�� R�� share� namely they depend solely on ��


is an analytic form of the mixing property� we can form a compatible se�quence ��s� � � � � �s� �s�� sp � ��s therefore we can form an in�nitesequence in which the latter sequence is repeated inde�nitely into an in�niteand compatible sequence� The point x that corresponds to this sequence is�necessarily� a periodic point �with period s� p�� A consequence is the den�sity of periodic points in phase space every mixing Anosov system admitsa dense set of periodic orbits�

Because of all the above properties Anosov systems �and the related AxiomA attractors� are the paradigm of chaotically behaving systems� The basicidea� developed in the early ��s by D� Ruelle� is that such systems are notjust curiosities but rather they are in some sense the �rule� when dealingwith real dynamics�

x�� Statistics of Chaotic Attractors� SRB Distributions

Perhaps the most important property of mixing Anosov systems or systemswith Axiom A attractors is that they admit stationary states� i�e� the limitsin �� exist and the �statistics� � of almost all data x �i�e� all dataoutside a zero volume set� exists� Furthermore the probability distribution�� called the SRB�distribution� describing them can be characterized quiteexplicitly�

Note that the �almost all� is an essential feature of the de�nition� in fact�see x�� Anosov �or Axiom A� systems will have a dense set of periodicpoints covering phase space �or the attracting set� any such point x� atleast� will of course be an exceptional point as far as the value of the limitin �� is concerned�

The SRB distribution can be given an expression in terms of the kinemat�ical properties discussed in the previous section� This is an expression thatcan play a role similar to that played by the Boltzmann�Gibbs expressionfor the equilibrium distributions� It is an expression that clearly cannotbe computed in any nontrivial case� much like the integrals that expressequilibrium properties in terms of integrals with respect to the canonicaldistribution�However� like the integrals with respect to the canonical distribution� it

can be useful to derive relations that must hold between various averages�Therefore formal expressibility of the SRB distribution seems to be a veryimportant property for nonequilibrium theory�The formula can be rather easily justi�ed at an informal level� however even

this requires good will on the part of the reader� to the extent that he willdevelop it only if convinced of its utility� Therefore we relegate to Appendix��A below the �informal� analysis and we con�ne ourselves to giving herethe SRB distribution expression in a form su%cient for the discussion of afew applications�One needs to de�ne the SRB average of a generic observable F �x� on phase

## IX� Coarse Graining and Nonequilibrium

space hF i �RF �x��dx�� For this purpose we divide phase space into cells

at least so small that F �x� is constant in each of them�The division into cells will be conveniently made by using a Markov par�

tition E �it will not matter which one�� of course the elements E � E willnot� in general� be so small that F is constant in each of them� However wecan re�ne E simply by considering the partition E�T�T � �Tj�TS�jE � asremarked in x�� this is a much �ner partition �and the size of the cells isof the order O�e��T � if � is de�ned in ��Each element of E has the form �Tj�TS�jE�j where ��T � � � � � �T is a

compatible sequence of symbols�Given a boundary condition �L� �R in the sense of x�� which one again

does not matter� we can de�ne a compatible bi�in�nite sequence

�Edef� ��L��T �� T � � � � � �T � �R��T � ��

��

for each E � ET �If c�E� is the point of E whose symbolic representation relative to the

partition E is �E we can de�ne the expansion rate ,u��T �E� of the map S�T

regarded as a map between S�T c�E� and ST c�E� as� cf� ��

,u��T �E� � ,u��T �c�E��def�

T��Yk�T

,u�S�kc�E��

T��Yk�T

e�u�S�kc�E��

�� Then the SRB distribution � can be written as

ZF �y��dy� � lim

T��n�T��n�T ��T

PEE�T�T ,��u��n�T ��c�E��F �c�E��P

EE�T�T ,��u��n�T ��c�E��

where n�T � � T is any sequence tending to � as T ��A particularly convenient choice will be n�T � � T so that� with c � c�E��

ZF �y��dy� � lim

T��

PEE�T�T ,��u��T �c�F �c�P

EE�T�T ,��u��T �c��

Remarks

�� The weight in �� can be written� in terms of the function ��de�ned in x�� see �� and of the shift operation on the in�nitesequences� as

,��u�n�T ��E� � e�Pn�T �

j�n�T ��j��

��

where � � �E is the sequence that is obtained by continuing ��T � � � � � �Tto an in�nite sequence as prescribed by the chosen boundary condition�see �� This is a slightly di�erent rewriting of �� It shows that


the SRB distribution can be interpreted as a probability distribution on thespace of bi�in�nite compatible sequences �� As such it is a Gibbs distributionfor a one�dimensional lattice spin model with short�range interaction andnearest neighbor hard core �due to the compatibility restriction that limits

the allowed con�gurations �� In fact �� says thatPn�T �

j�n�T � �� j��

can be interpreted as the energy of a spin con�guration under a potentialthat is exponentially decreasing at �� see x�� #��

In x��# we have seen that one�dimensional short�range systems are quitetrivial from the statistical mechanics viewpoint and the theory of chaoticmotions inherits quite a few results from the theory of such lattice sys�tems� Such results are often quite nontrivial and surprising when seen asproperties of chaotic motions� For instance uniqueness of SRB distributionscorresponds to the absence of phase transitions in one�dimensional latticesystems� The thermodynamic limit corresponds to the limit as T �� Ex�ponential decay of time correlations in the SRB distributions correspondsto the exponential decay of correlations in one�dimensional short�range lat�tice systems� Large deviation theorems correspond to the analyticity of thethermodynamic functions and so on�For the above reasons the theory of Anosov �and Axiom A� systems hasbeen called thermodynamic formalism� �Ru�#b��

� � If n�T ��T ��T�� then by �� and �� we see that the weight

,��u��n�T ��E� given to the cell E of ET does not change appreciably as the

boundary condition is changed�� This is a kind of �mean value theorem��for the SRB distribution�

�� But if n�T � � T the variation of ,��u��T �E� within E is appreciablebecause� clearly� the sum in �� will undergo variations of order O��when the boundary condition is changed�

�� Hence �� is a deeper property than �� with n�T ��T � �� Itis proved easily in the thermodynamic formalism because it reduces to thestatement that one�dimensional lattice gases with short�range interactionsshow no phase transitions� therefore the boundary condition dependenceof the averages of local observables disappears in the thermodynamic limitT �� see x��#�

�� If the attractor is invariant under the action of a time reversal symmetryI � see x�� we can� and shall� suppose that the Markov partition is I�reversible if E � E then IE � E �� Furthermore the centers c in �� can be chosen so that if c is the center of E then ic is that of IE� see x��comment �� This means that we have to choose a reversible boundary

� Note that at �xed E and as the boundary condition is varied the center point c�E�varies �densely� inside E�

�� If not one could use the �ner partition obtained by intersecting E and IE so that thenew partition will be time reversible�


condition to select the centers�

An expression of the SRB distribution based on a reversible partition E andon a reversible boundary condition will be called a reversible representationof a SRB distribution��

In Appendix ��A below we give details of an informal derivation of ��but the above properties will su%ce for deducing many interesting con�sequences of the chaotic hypothesis� in the form of general properties ofAnosov systems�

It is interesting and important to discuss the connection between the Boltz�manian representation of motion as a cyclic permutation of the phase spacecells and the new symbolic representation of motion in Anosov systems� Infact the chaotic hypothesis is supposed to hold also in the equilibrium casesand therefore one has two di�erent representations of motions for the samesystem�

According to views already repeatedly expressed by Boltzmann� �Bo��this �dualism� is not a priori impossible although in most cases in whichhe envisaged this possibility it seems that he did not really believe in it�The above seems to be a very �ne and nontrivial instance in which a dualrepresentation is possible� This is discussed in Appendix ��A� below� seealso �Ga��a��

x�� Entropy Generation� Time Reversibility and Fluctuation

Theorem� Experimental Tests of the Chaotic Hypothesis

The connection between the general kinematical analysis of chaotic motionsand applications can be established if one accepts that the motions of amany�particle system are so �chaotic� that one can regard the system as amixing Anosov system in the sense of x��

One of the key notions in equilibrium statistical mechanics is that of en�tropy� its extension to nonequilibrium is surprisingly di%cult� assuming thatit really can be extended� In fact we expect that� in a system that reachesunder forcing a stationary state� entropy is produced at a constant rate sothat there is no way of de�ning an entropy value for the system� exceptperhaps by saying that its entropy is ��

Although one should keep in mind that there is no universally acceptednotion of entropy in systems out of equilibrium� even when in a stationarystate� we shall take the attitude that in a stationary state only the entropycreation rate is de�ned the system entropy decreases inde�nitely� but at aconstant rate�� Note that we say �decreases� and not �increases� becausein a nonequilibrium situation nonconservative forces work upon the systemand� since the system is supposed to be in a stationary sate� such work mustbe ceded to the exterior in the form of heat at constant temperature� So

�� De�ning �entropy and �entropy production should be considered an open problem�


entropy �of the system� decreases and entropy of the surroundings increases�as expected��The natural de�nition of entropy creation rate is� following Gibbs and

Boltzmann� that of the time derivative of the entropy of a state of thesystem that evolves towards stationarity� �An# �� Ru��c��We look� temporarily� again at the evolution of our systems in continuous

time so that we imagine that it corresponds to a di�erential equation!x � f�x�� A �state� which at time t � � is described by a distribution ��with density ��x� with respect to the volume element dx on phase spacebecomes at time t a distribution �t�dx� � �t�x�dx with density �t�x� �

�o�S�tx��S�tx�x � where Jt�x�def� �S�tx

�x is the Jacobian determinant at x ofthe map x� S�tx� De�ning the �entropy� of �t as

E�t�def� �

Z�t�x� log �t�x� dx �

� �Z��S�tx�

S�txx

log

��S�tx�

S�txx

�dx

��

we deduce that

E�t� � �Z��S�tx�

S�txx

log ��S�tx� dx�

�Z��S�tx�

S�txx

logS�txx

dx

��

and the �rst term on the righ�hand side does not contribute to !E becauseit equals the constant � R ��y� log ��y�dy �just set S�tx � y�� therefore!E equals the t derivative of the second term� which can be transformed bysetting y � S�tx into

�Zdy ��y� log

S�t�Sty�

Sty�

Zdy ��y� log

Sty

y��

having used the identity

S�t�Sty�

Sty

Sty

y� ��

We now make use of the other identity

d

dt

Sty

y� �Sty

y��Sty� ��

where ��x� is the divergence of �f�x� �writing the equations of motion as!x � f�x�� It follows that the rate of entropy creation is� see �An# �� !E

!E ��Z

��y��Sty� dy �

��Z

��S�tz�S�tzz

��z� dz �

��Z

�t�dz��z��t�� Z

��dz��z�def�

def� � �

��

� IX� Coarse Graining and Nonequilibrium

if �t suitably converges to ��

Hence we see that if �t��t�� the asymptotic average rate of entropy

variation is �def� h�i�� Since the macroscopic state of the system does not

change as t � � we must interpret � as the average entropy increase ofthe thermostat that absorbs the heat created in the system by the forcing�Naturally we must expect � � �� It is reassuring that this a theorem�

�Ru��a�� that � � � for systems with an Axiom A attractor� hence for sys�tems satisfying the stronger chaotic hypothesis� This is not very surprisingbecause� by assumption� our system evolves in a bounded region �i�e� phasespace is bounded� so that � � � would mean that the volume expandsinde�nitely� which is impossible�

Because of the above considerations we shall call � the entropy generationrate and we suppose that it has the property ��x� � � if the system isnot subject to forcing� so that at zero forcing the evolution is volume pre�serving �a property usually true because the nonforced system is� as a rule�Hamiltonian��

Coming back to our previous point of view� with time evolution describedby a map S on a phase space of �timing events� the entropy creation ratewill be� in this case� identi�ed with the phase space contraction betweenone timing event and the next

��x� � � log jS�x�

xj� ��

Our analysis concerns idealized systems of the above type that are alsomixing Anosov maps in the sense of x��We now attempt to deduce other consequences of the chaotic hypothesis�

possibly new �and in any event beyond the existence of the stationary dis�tribution � and the nonnegativity of �� and measurable in at least somesimple cases� The simplest cases to study are systems whose dynamics isreversible not only in the nonforced case� but under forcing as well�Examples of thermostatting mechanisms that generate reversible motions

are provided under rather general circumstances by forces acting on oth�erwise Hamiltonian systems and realizing an anholonomic constraint ac�cording to the principle of least constraint of Gauss� also called minimalconstraint principlesee Appendix ��A�� In the following we shall providesome simple concrete examples� but it is important to note that the theoryis far more general than the few examples that we shall discuss�We consider� therefore� a general reversible mechanical system governed by

a smooth equation

!x � f�x� G � ��#� ��

depending on several parameters G � �G�� Gn� measuring the strengthof the forces acting on the system and causing the evolution x � Stx ofthe phase space point x representing the system state in the phase space Fwhich can be� quite generally� a smooth manifold�


We suppose that the system is �thermostated� so that motions take placeon bounded smooth invariant surfaces H�x� G � � E� which are level sur�faces of some �level function� H � Hence we shall identify� to simplify thenotation� the phase space F with this level surface which we shall sometimescall� somewhat inappropriately� the �energy surface��We suppose also that the $ow St generated by ��#� is reversible� i�e�

there is a volume preserving smooth map I � �time reversal�� of phase spacesuch that I� � � and �anticommuting with time�

StI � IS�t ��

i�e� f�Ix� G � � ��xI��x� � f�x� G ��

We shall further restrict our attention to mixing Anosov systems that arereversible� in the above sense� for all values of the forcing parameters G ofinterest and dissipative at G �� This means that the systems we considerare such that� see ��

� � h�i� � � for G ��

Under the above assumptions one can de�ne� for h�i� � �� the �dimension�less average entropy creation rate� p by setting

p ��

h�i��

�

Z ��

��St� G �dt ��

��

Then the probability distribution of the variable p with respect to the SRBdistribution � can be written for large � as �� p�dp � const e�� p�dp� see�Si�� and the function ��p� � lim�� p� satis�es� if ��h�i� � � andjpj � p for a suitable p � �� the property

��p� � ��p� � p�� jpj � p ��

which is called the �uctuation theorem� and is part of a class of theoremsproved in �GC�� see also �Ga��a�� for discrete time systems� and in �Ge�#��for continuous time systems� This theorem can be considerably extended� asdiscussed in �G��b�� Ga�#b� and the extension can be shown to imply� in thelimit G� � �when also � � �� relations that can be identi�ed in variouscases with Green�Kubo s formulae and Onsager s reciprocal relations� seealso �GR�� Ga�#d� and x�� below�Similar theorems can be proved for suitable nonstationary probability dis�

tributions and� in fact� preceded the above� �ES�� or for nondeterministicevolutions� �Ku�� LS�#�� In the closest cases the relations between the

�� For instance if ��x� G � is the rate of change of the volume element of F near x and ifF is a Euclidean space then ��x� � �

P��f��x� G �� If x � �p� q�� and Ix � ��p� q�

and f is an Hamiltonian part plus a pdependent term due to the �thermostat forcesthen ��Ix�G� � ��x�G��


latter theorems and the above is sometimes �only� an interchange of limitsit is precisely in the analysis of this interchange that the chaotic hypothesisplays a major role� see �CG��

The interest of �� is its universal nature� i�e� its �reversible� systemindependence� and the fact that it contains no free parameter�

The connection with applications of the above results is made via the as�sumption that concrete chaotic dynamical systems can be considered� �forthe purpose of studying macroscopic properties�� as mixing Anosov �ows�

The $uctuation theorem proof is quite simple and it is based on the ex�pression �� it will be discussed in the forthcoming sections�

As a concluding comment we note that the probability distribution of pcan be regarded as the probability distribution of the sum of the �localLyapunov exponents�� if one de�nes it as the sum of the eigenvalues �j�x�of the matrix S� �S�

� x� �ph�i � �Pj �j�x��

If a system is an Anosov system then it has been proved that the proba�bility distribution of the sum p�� of the local Lyapunov exponents has adensity of the form e��p�� Si�� One says that the distribution is multi�fractal if ��p� is not linear� This means that the sum of the local Lyapunovexponents has wide $uctuations around its average �given by �� The$uctuation theorem says that the odd part of the multifractal distributionis always linear� in reversible systems� so that multifractality of the vol�ume contraction rate is� in such systems� related to the even part of itsdistribution�

Is the above �� an observable relation� In fact it was observed in anumerical experiment with �� particles modeling a �reversible� gas in a shear$ow� �ECM�� and the attempt at theoretical prediction of the observedresults led to the chaotic hypothesis and to the derivation discussed in x��

It has then be observed in a sequence of experiments with and �� hardcore particles� �BGG�� moving among �xed obstacles in a periodic boxand subject to a constant �eld and thermostatted with a force necessaryto maintain a constant total kinetic energy in spite of the action of the�eld� The force is selected among the several possible force laws as theone satisfying Gauss principle of minimal constraint �so that the resultingequations are reversible� see Appendix ��A��

One can also consider systems in which the forcing has a �thermal nature�like systems enclosed in boxes whose walls are kept at constant temperature�depending however on which side of the walls one considers�� Also suchsystems can be modeled with equations of motion which can be reversible�for instance see �Ga��b�� A very interesting numerical experiment has beenperformed on a chain of oscillators �in number up to �� interacting withinelastic forces and with the oscillators at the extremes forced to have a�given temperature� by acting on them with suitable forces� �LLP��

An experiment of a completely di�erent kind� on a sample of water inconvective chaotic motion �not too strongly chaotic� however� has been per�formed recently� Its interpretation in terms of the $uctuation theorem �or


rather of its extensions discussed in �Ga��c�� Ga��a�� because a $uid isstrongly dissipative and one cannot expect that the attractor is dense inphase space� even in developed turbulence states� is still under analysis��CL�#��

An important prediction of the $uctuation theorem in strongly chaoticparticle systems is that the slope of the graph of ��p� � ��p� is precisely�� This shows that even if the distribution of p was Gaussian� i�e� ��p� ��D �� p�� for some D � �� the theorem would be nontrivial� In fact itwould be ��p�� p� � �

Dp and there would be no a priori reason to have�D � ��

In general one expects the distribution ��p� to be Gaussian near the averagevalue of p �which is � by de�nition�� Si�� but the Gaussian approximationshould be correct only for jp � �j � O�� central limit theorem��Hence it becomes important to test not only the linearity of ��p� � ��p�but also the slope of this linear law and whether the distribution ��p� canbe regarded as Gaussian�

One �nds in the �rst two experiments the correct value of the slope but onecannot really distinguish whether the distribution of p is Gaussian or not�Although one can see a priori that it is not Gaussian� the non�Gaussiannature of the distribution is not observable because it manifests itself ina region so far away from p � � that the corresponding huge $uctuationscannot be observed� being too rare� The attempt at understanding the rela�tion between the central limit theorem and the large deviation experimentalresults on the $uctuation theorem led to the idea that there was a relationbetween the $uctuation theorem and the linear response theory of Onsagerand Green�Kubo� This in fact was found in �Ga��a��Ga��b��Ga�#d��

In the third experiment� �LLP�� one �nds the correct value of the slopein a situation in which ��p� is manifestly not Gaussian� Hence this is a keyexperiment for the theory�

Finally in the fourth experiment� �CL�#�� one gets a linear graph for the oddpart of the large deviation function ��p�� but the slope is not that of �� but considerably smaller� The system in this case� unlike the previous one� iscertainly so dissipative that the attractor is much smaller than phase space�the space of the temperature and velocity �elds of the sample of water�and the slope was certainly not expected to be �� BGG�� BG�� Thismight be due to the fact that the system is not reversible� or that it is notequivalent to a reversible one� or that the chaotic hypothesis is incorrectin this case� But the matter requires further investigation� because in theearlier work �Ga��c�� Ga��a� it was shown that in such cases one couldexpect a slope P � ��

A �nal comment on the observability of the $uctuation theorem in largesystems since the function ��p� is expected to be proportional to the volumeof the system� or at least to the surface of its container �depending on the sizeof the region where dissipation really occurs�� it is impossible to observe the$uctuation relation in macroscopic systems because the $uctuations havetoo small a probability� However in some cases it is possible to derive a


�local �uctuation theorem� which concerns the $uctuations of the entropycreation rate in a microscopic region� In such cases the $uctuations areobservable� One is in a situation similar to that of density $uctuationsin equilibrium� One cannot see density $uctuations of a gas in a largemacroscopic container� but one can quite easily see density $uctuations in asmall microscopic volume� and the functions � that control such deviationsare simply proportional �their ratio being the ratio of the correspondingvolumes�� It would be interesting to formulate local $uctuation theoremsas generally as possible� beyond the few examples known� �Ga�#e��

x�� Fluctuation Patterns

It is natural to inquire whether there are more direct and physical inter�pretations of the theorem �hence of the meaning of the chaotic hypothesis�when the external forcing is really di�erent from the value �� A result inthis direction is the conditional reversibility theorem� discussed below�Consider an observable F which� for simplicity� has a well�de�ned time

reversal parity F �Ix� � �FF �x�� with �F � �� For simplicity supposethat its time average �i�e� its SRB average� vanishes� F � �� and lett � �t� be a smooth function vanishing for jtj large enough� We lookat the probability� relative to the SRB distribution �i�e� in the �naturalstationary state�� that F �Stx� is close to �t� for t � ��

� �� We say that

F �follows the $uctuation pattern� in the time interval t � ��

��

No assumption on the $uctuation size �i�e� on the size of �� nor on thesize of the forces keeping the system out of equilibrium� will be made� Be�sides the chaotic hypothesis we assume� however� that the evolution is timereversible also out of equilibrium and that the phase space contraction rate� is not zero �the results hold no matter how small � is� and they makesense even if � � �� but they become trivial��We denote by ��p� � the large deviation function for observing in the

time interval ��

�� an average contraction of phase space ��

def� �

�R �� Stx�dt � p� and at the same time a $uctuation pattern F �Stx� �

�t��This means that the probability that the dimensionless average entropycreation rate p is in an interval � � �a� b� and� at the same time� F is in aneighborhood�� U�� of � is given by

supp��U��

e�� p�� #��

to leading order as � �� i�e� the logarithm of the mentioned probabilitydivided by � converges as � �� to supp��U�� p� ��

�� By �neighborhood U �� we mean thatR ��

��t�F �Stx�dt is approximated within

given � � � byR ��

��t��t�dt for � in the �nite collection � � �� m� of test

functions� This is� essentially� what is called in mathematics a �weak neighborhood�


Given a reversible� dissipative� mixing Anosov $ow the $uctuation patternt � �t� and the time reversed pattern t � �F��t� are then related bythe following

Conditional reversibility theorem Consider a function t � �t� and an

observable F with de�ned time reversal parity �F � �� Let � be large

and consider the �uctuation pattern f�t�gt��

� �

and its time reversal

fI�t�gt��

� ��f�F��t�gt��

� � � �� they will be followed with equal like�

lihood if the �rst is conditioned to an entropy creation rate p and the second

to the opposite �p� This is an interpretation of the following result

��p� I� � ��p� � � p � for jpj � p ��#� � ��

with � introduced above and a suitable p � ��

In other words� in these systems� while it is very di%cult to see an �anoma�lous� average entropy creation rate during a time � �e�g� p � �� it is alsotrue that �that is the hardest thing to see�� Once we see it all the observableswill behave strangely and the relative probabilities of time reversed patternswill become as likely as those of the corresponding direct patterns under�normal� average entropy creation regime�A waterfall will go up� as likely as we see it going down� in a world in whichfor some reason� or by the deed of a Daemon� the entropy creation rate haschanged sign during a long enough time� We can also say that the motionon an attractor is reversible� even in the presence of dissipation� once thedissipation is �xed�The proof of the above theorem is similar to that of the $uctuation theorem

to which it reduces if F � � � �and in fact it is a repetition of it�� To becomplete we sketch� in the next section� the proof�The $uctuation and the conditional reversibility theorems can also be for�

mulated for systems whose evolution is studied in continuous time �i�e� forAnosov $ows�� The discrete case is simpler to study than the correspondingAnosov $ows because Anosov maps do not have a trivial Lyapunov expo�nent �the vanishing one associated with the phase space $ow direction�� thetechniques to extend the analysis to Anosov $ows are developed in �BR��Ge�#� �and one achieves the goal of proving the analogue of the $uctuationtheorem for such systems��

x�� Conditional Reversibility and �Fluctuation Theorems

In x�� we have seen that in a Anosov system the stable and unstabletangent planes T s

x � Tux form an integrable family of planes �and their integral

surfaces are the stable and unstable manifolds�� If x is a point and if J�x� �S�x� is the Jacobian matrix of S at x� then the covariance of the stable andunstable planes implies that we can regard its action �mapping the tangentplane Tx onto TSx� as �split� linearly into an action on the stable plane andone on the unstable plane i�e� J�x� restricted to the stable plane becomes


a linear map Js�x� mapping T sx to T s

Sx� Likewise one can de�ne the mapJu�x�� Ru��Let ,u�x��,s�x� be the determinants of the Jacobians� i�e� of Ju�x�� Js�x��

Their product di�ers from the determinant ,�x� of S�x� by the ratio ofthe sine of the angle a�x� between the planes T s

x � Tux and the sine of the

angle a�Sx� between T sSx� T

uSx�

Hence

,�x� �sin a�Sx�

sin a�x�,s�x�,u�x� ��

and we also set� see x��

,u�� x� �

��Yj��

,u�Sjx�� ,s�� x� �

��Yj��

,s�Sjx��

,� �x� �

��Yj��

,�Sjx� � ��

Time reversal symmetry� which we assume here� implies that W sIx �

IW ux �W

uIx � IW s

x and

,� �x� � ,� �Ix�� ,s�� Ix� � ,u�� x�� ,u�� Ix� � ,s�� x��

sin a�x� � sin a�Ix� � ��

In x�� we have seen that� given the above geometric�kinematical notions�the SRB distribution � can be represented by assigning suitable weights tosmall phase space cells� �� This is very similar to the representationof the Maxwell�Boltzmann distributions of equilibrium states in terms ofsuitable weights given to phase space cells of equal Liouville volume�The phase space cells can be made� see x�� consistently as small as we

please and� by taking them small enough� one can achieve an arbitraryprecision in the description of the SRB distribution �� in the same way aswe can approximate the Liouville volume by taking the phase space cellssmall�The key to the construction and to our proof is a Markov partition� in�

troduced in x�� this is a partition E � �E�� EN � of the phase spaceC into N cells which are covariant with respect to the time evolution andwith respect to time reversal in the sense that IEj � Ej� for some j�� seex�� for the notion of covariance and for the properties of Markov partitions�

Given a Markov partition E we can �re�ne� it �consistently�� see x�� asmuch as we wish by considering the partition E �T�T � �T�TS�j E whosecells are obtained by �intersecting� the cells of E and of its S iterates� thecells of E �T�T become exponentially small with T � � as a consequenceof the hyperbolicity� In each Ej � E �T�T one can select a center pointxj � c�Ej� �associated with an arbitrary boundary condition in the sense


of remark �� in x�� see also �� so that Ixj is the point selected inIEj � Then we evaluate the expansion rate ,u��T �xj� of S�T as a map of theunstable manifold of S�Txj to that of STxj � see ��Using the elements Ej � E T as cells we can de�ne approximations �as

good as we wish� to the SRB distribution �� as given by �� becausefor all smooth observables F de�ned on C� �Si�#��BR��

Z��dy�F �y� � lim

T��

ZmT �dy�F �y�

def� lim

T��

PEjE T F �xj�,

��u��T �xj�P

EjE T ,��u��T �xj�

�� where mT �dy� is implicitly de�ned here by the ratio in the righ�hand sideof �� is just a change of notation away from ��Let �p denote an interval �p� p � dp� and U� denote a set of of functionst� ��t� de�ned in ��

� �� and with values in a �tube� of width � around

a given �path� t� �t�� t � ��

�� Let �U be the time reversed set of

paths� i�e� the set of paths t� ��t�� t � ��

�� with values within � of

�F��t�� Here time is discrete �but the same ideas and deductions wouldapply to a continuous time case� so that we use a notation that makes sensein both cases�We �rst evaluate the probability� with respect to the distribution m�� in

�� instead of the mT �� of the event that p � p�xj� � �� xj��h�i� ��p and� also� that fF �Skxj�g�k�� U�� divided by the probability �withrespect to the same distribution� of the time reversed event that p�xj� �� xj��h�i � ��p and� also� k � fF �Skxj�g � �U��Thus we compare the probability of a $uctuation pattern in the presence

of average dissipation p and that of the time reversed pattern in the presenceof average dissipation �p� This is essentially

�� p�

�� p��

Pj� p�xj�p�F �Snxj��n�

,��u�� xj�Pj� p�xj��p�F �Snxj��F��n� ,��u�� xj�

��

Since m�� in �� is only an approximation to � an error is involvedin using �� as a formula for the same ratio computed by using the trueSRB distribution � instead of m��It can be shown that this ��rst� approximation �among the two that will be

made� can be estimated to a�ect the result only by a factor bounded aboveand below uniformly in �� p� This is not completely straightforward in asense this is perhaps the main technical problem of the analysis�� Furthermathematical details can be found in �Ga��c��Ru��c��Ge�#��

�� It can be seen if one interprets �� as a probability distribution on the space of thesymbolic sequences � which� via the Markov partition E� can be used to represent thepoints x in phase space� Such probability distribution can also be interpreted as a Gibbsdistribution over the space of the sequences � with potential �� log !u��x�� if� is the symbolic sequence corresponding to xj � see x�� In this way the propertyunder analysis �i�e� the identity of the limits as � of �� and of the same ratiosevaluated by using mT instead of m��

�� appears simply due to the nonexistence


Remark There are other representations of the SRB distributions thatseem more appealing than the above one based on the Markov partitionsnotion and still make the above analysis possible and apparently more intu�itive� e�g� see �MR�� The simplest is perhaps the periodic orbits represen�tation in which the role of the cells is taken by the periodic orbits� HoweverI do not know a way of making the argument that leads to �� whilekeeping under control the approximations and at the same time not relyingon Markov partitions� and in fact I do not know of any expression of theSRB distribution that is not proved by using the very existence of Markovpartitions�

We now try to establish a one�to�one correspondence between the addendsin the numerator of �� and those in the denominator� aiming at showingthat corresponding addends have a constant ratio which will� therefore� bethe value of the ratio in ��This is possible because of the reversibility property� it will be used in the

form of its consequences given by the relations �� The ratio �� cantherefore be written� by virtue of �� simply as

Pj� p�xj�p�F �Snxj��n�

,��u�� xj�Pj� p�xj��p�F �Snxj��F��n� ,��u�� xj�

�P

j� p�xj�p�F �Snxj��n�,��u�� xj�P

j� p�xj�p�F �Snxj��n�,s�� xj�

�� where xj � Ej is the center in Ej � In deducing the second relation we takeinto account

�� time reversal symmetry I �

� � that the centers xj � xj� of Ej and Ej� � IEj are such that xj� � Ixj �and

�� that �� hold�

and transform the sum in the denominator of the left�hand side of ��into a sum over the same set of labels that appear in the numerator sum�It follows then that the ratios between corresponding terms in the ratio

�� is equal to ,��u�� x�,��s�� x�� This di�ers little from the reciprocal ofthe total change of phase space volume over the � time steps �during whichthe system evolves from the point S��x to S��x��The di�erence is only due to not taking into account the ratio of the sines

of the angles a�S��xj� and� see �� a�S��xj� formed by the sta�ble and unstable manifolds at the points S��xj and S��xj � Therefore

of phase transitions in the onedimensional shortrange Ising models� In fact the tworatios become ratios of expectation values of the same quantities evaluated in presenceof di erent boundary conditions� and in absence of phase transitions one should haveboundary conditions independence which in this case would imply that the two ratiosdi er at most by a factor of order O�� so that their logarithms divided by T or �

�

should have the same limit ��p�� p��


,��u�� xj�,��s�� xj� will di�er from the actual phase space contraction under

the action of S� � regarded as a map between S��xj and S��xj � by a fac�

tor that can be bounded between B�� and B with B � maxx�x�� j sin a�x�jj sin a�x��j

��

which is �nite and positive� by the linear independence of the stable andunstable planes�But for all points xj in �� the reciprocal of the total phase space

volume change over a time � equals e�� p �by the constraint� �� p�imposed on the summation labels� up to a �second� approximation thatcannot exceed a factor which is bounded above and below by � �independentpositive and �nite constants B�� due to the above sine ratio� Hence theratio �� will be the exponential e�� p� up to a � �independently boundedfactor and �� follows�It is important to note that there have been two approximations� as just

pointed out� They can be estimated� see �GC��Ga��c�� Ru��c�� and implythat the argument of the exponential is correct up to p� � � �independentcorrections so that the result can be proved even if the approximations areavoided �this also makes clear that the consideration of the limit � �� isnecessary for the theorem to hold��In the special cases in which there is no F and one only looks at the

probability distribution of the entropy production rate the above becomes

B�� e� p h�i �� p�

�� p�� B e� p h�i ��

or�� p�

�� p�� e� p h�iO�� #� � ��

i�e� we get ��

x�� Onsager Reciprocity and Green�Kubo�s Formula�

The $uctuation theorem degenerates in the limit in which � tends tozero� i�e� when the external forces vanish and dissipation disappears �andthe stationary state becomes the equilibrium state��Since the theorem deals with systems that are time reversible at and out�side equilibrium� Onsager s hypotheses are certainly satis�ed and the systemshould obey reciprocal response relations at vanishing forcing� This led tothe idea that there might be a connection between the $uctuation theo�rem and Onsager reciprocity and also to the related �stronger� Green�Kuboformula�This is in fact true if we de�ne the microscopic thermodynamic �ux j�x�

associated with the thermodynamic force E that generates it� i�e� the pa�rameter that measures the strength of the forcing �which makes the systemnonHamiltonian�� via the relation

j�x� ��x�

E��


�not necessarily at E � �� then in �Ga��b� a heuristic proof shows that thelimit as E � � of the $uctuation theorem becomes simply �in the continuoustime case� a property of the average� or �macroscopic�� ux J � hji�E

J

E

��E�

��

Z �

��hj�Stx�j�x�i�E

��E�

dt ��

where h�i�E denotes the average in the stationary state �E �i�e� the SRBdistribution which� at E � �� is simply the microcanonical ensemble��If there are several �elds E�� E�� acting on the system we can de�ne

several thermodynamic $uxes jk�x�def� Ek��x� and their averages hjki� in

the limit in which all forces Ek vanish a �simple� extension of the $uctuationtheorem is shown� �Ga��b�� to reduce to

Lhkdef�

JhEk

��E�

��

Z �

��hjh�Stx�jk�x�iE� dt � Lkh � ��

therefore we see that the $uctuation theorem can be regarded as an exten�sion to nonzero forcing of Onsager reciprocity and� actually� of the Green�Kubo formula�Certainly assuming reversibility in a system out of equilibrium can be dis�

turbing one can� thus� inquire if there is a more general connection betweenthe chaotic hypothesis� Onsager reciprocity and the Green�Kubo formula�This is indeed the case and provides us with a further consequence of the

chaotic hypothesis valid� however� only in zero �eld� It can be shown thatthe relations �� follow from the sole assumption that at E � � thesystem is time reversible and that it satis�es the chaotic hypothesis for Enear � at E �� it can be� as in Onsager s theory� not reversible �GR��It is not di%cult to see� technically� how the $uctuation theorem� in the

limit in which the driving forces tend to �� formally yields the Green�Kuboformula�We consider time evolution in continuous time and simply note that

��#� implies that� for all E �for which the system is chaotic� heIE i �Pp �� p�ep��

Pp �t��p�eO�� eO�� so that

lim��

�

�logheIE i�E � � ��

where IEdef�R��Stx�dt with ��x� being the divergence of the equations

of motion �i�e� the phase space contraction rate� in the case of continuoustime�� This remark� �Bo�� can be used to simplify the analysis in �Ga��b��and �Ga��a�� as follows�We switch to continuous time� to simplify the analysis� Di�erentiating both

sides with respect to E� not worrying about interchanging derivatives and

�� It says that essentially heIE i�E � � or more precisely it is not too far from � so that

�� holds�


limits and the like� one �nds that the second derivative with respect to Eis a sum of six terms� Supposing that for E � � the system is Hamiltonianand �hence� I� � �� the six terms� when evaluated at E � �� are

�

�

hh�EIEi�E jE� � h�EIE��i�E jE��

�

ZEIE�x��E�x�jE� �

�h�EIE��i�E �

Z� E�E

�jE��

�

ZEIE�x��E�x�jE� �

Z� � �E�E jE�

i ��

and we see that the fourth and sixth terms vanish being derivatives ofR�E�dx� � �� and the �rst vanishes �by integration by parts� because IE

is a divergence and �� is the Liouville distribution �by the assumption thatthe system is Hamiltonian at E � � and chaotic�� Hence we are left with

��

�h�EIE��i�E �

�

ZEIE�x�E�E�x�

�E�

� � ��

where the second term is ��E�hEIEi�E �jE� � EJE jE�� be�cause the distribution �E is stationary� and the �rst term tends toR �� hj�Stx�j�x�iE�dt as � � �� Hence we get the Green�Kubo formula

in the case of only one forcing parameter�The argument could be extended to the case in which E is a vector de�

scribing the strength of various driving forces acting on the system� but oneneeds a generalization of �� The latter is a consequence of the $uctu�ation theorem� and the theorem has to be extended in order to derive fromit also the Green�Kubo formula �hence reciprocity� when there were severalindependent forces acting on the system� see �Ga��b� where the extensionis discussed�The above analysis is unsatisfactory because we interchange limits and

derivatives quite freely and we even take derivatives of �E � which seems torequire some imagination as �E is concentrated on a set of zero volume�On the other hand� under the strong hypotheses in which we suppose tobe working �that the system is mixing Anosov�� we should not need extraassumptions� Indeed the above mentioned nonheuristic analysis� �GR�� isbased on the solution of the problem of di�erentiability with respect to aparameter for SRB distributions� �Ru��b��

x�� Reversible Versus Irreversible Dissipation� NonequilibriumEnsembles�

What is missing are arguments similar to those used by Boltzmann tojustify the use of the ensembles independently of the ergodic hypothesis anhypothesis which in the end may appear �and still does appear to many�as having led to the theory of enembles only �by accident�� The missingarguments should justify the $uctuation theorem on the basis of the extreme


likelihood of its predictions in systems that are very large and that may notbe Anosov systems in the mathematical sense� I see no reason why thisshould prove impossible� a priori� now or in the future�In the meantime it seems interesting to take the same philosophical attitude

adopted by Boltzmann not to consider that �by chance� chaotic systemsshare some selected properties� and try to see if such properties help usachieve a better understanding of nonequilibrium� After all it seems thatBoltzmann himself took a rather long time to realize the interplay of theabove two basic mechanisms behind the equilibrium ensembles and to pro�pose a solution harmonizing them� �All it remains to do� is to explore if thehypothesis has implications more interesting or deeper than the $uctuationtheorem�A system driven out of equilibrium can reach a stationary state �and not

steam out of sight� only if enough dissipation is present� This means thatany mechanical model of a system reaching a stationary state out of equi�librium must be a model with nonconservative equations of motion in whichforces representing the action of the thermostats� that keep the system fromheating up� are present�Thus� as we stressed repeatedly in the previous sections� a generic model of

a system stationarily driven out of equilibrium will be obtained by adding toHamilton s equations �corresponding to the nondriven system� other termsrepresenting forces due to the thermostat action�Here one should avoid attributing a fundamental role to special assump�

tions about such forces� One has to realize that there is no privileged ther�mostat many of them can be considered and they simply describe variousways to take energy out of the system�Hence one can even use stochastic thermostats� and there are many types

considered in the literature� or one can consider deterministic thermostatsand� among them� reversible ones or irreversible ones�Each thermostat requires its own theory� However the same system may

behave in the same way under the action of di�erent thermostatting mecha�nisms if the only action we make on a gas tube is to keep the temperaturesof its extremes �xed� by taking in or out heat from them� the di�erence maybe irrelevant� at least in the limit in which the tube becomes long enoughand as far as what happens in the middle of it is concerned�But of course the mathematical representation of the stationary state may

be very di�erent in the various cases� even when we think that the di�erencesare only minor boundary e�ects�For instance� in the case of the gas tube� if our model is of deterministic

dissipation we expect the SRB state to be concentrated on a set of zerophase space volume�� while if the model is stochastic then the stationarystate will be described by a density on phase space� Nothing could seemmore di�erent�

�� Because phase space will on the average contract� when �� so that any stationarystate has to be concentrated on a set of zero volume� which however could still be denseand often it will be�


Nevertheless it might still be true that in the limit of an in�nite tubethe two models give the same result� in the same sense as the canonicaland microcanonical ensembles describe the same state even though the mi�crocanonical ensemble is supported on the energy surface� which has zerovolume if measured by using the canonical ensemble �which is given by adensity over the whole available phase space��

Therefore we see that out of equilibrium we have in fact much more free�dom to de�ne equivalent ensembles� Not only do we have �very likely� thesame freedom that we have in equilibrium �like �xing the total energy ornot� or �xing the number of particles or not� passing from microcanonicalto canonical to grand canonical� etc� but we can also change the equationsof motion and obtain di�erent stationary states� i�e� di�erent SRB distri�butions� which will however become the same in the thermodynamic limit�

Being able to prove mathematical equivalence of two thermostats willamount to proving their physical equivalence� This again will be a di%�cult task� in any concrete case�

What I �nd fascinating is that the above remarks seem to indicate to us thepossibility that a reversible thermostat can be equivalent in the thermody�namic limit to an irreversible one� I conclude by reformulating a conjecture�see for instance �Ga��c�� Ga��a�� Ga�#c�� which clari�es the latter state�ment�

Consider the following two models describing a system of hard balls ina periodic �large� box in which there is a lattice of obstacles that forbidcollisionless paths �by their arrangement and size�� the laws of motion willbe Newton s laws �elastic collisions with the obstacles as well as betweenparticles� plus a constant force E along the horizontal axis �say� plus athermostatting force�

In the �rst model the thermostatting force is simply a constant times themomentum of the particles it acts on the i�th particle as ��pi if � is a�friction� constant� Another model is a force proportional to the momentumbut via a proportionality factor that is not constant and depends on thesystem con�guration at the point x in phase space� it has the form ��x�piwith ��x� � E �Pi pi�

Pi p

�i �

The �rst model is related to the model used by Drude in his theory ofconduction in metals� see �EGM�#�� The second model has been used veryoften in recent years for theoretical studies and has thus acquired a �re�spected� status and a special importance it was among the �rst modelsused in the experiments and theoretical ideas that led to the connectionbetween Ruelle s ideas for turbulent motion in $uids and nonequilibriumstatistical mechanics� �HHP#�� ECM�� ECM�� I think that the im�portance of such work should be stressed and fully appreciated withoutthis work the recent theoretical developments would have been unthinkable�in spite of the fact that a posteriori they seem quite independent and onecould claim �unreasonably in my view� that everything could have beendone much earlier�


Furthermore the second model can be seen as derived from Gauss leastconstraint principle� see Appendix ��A�� It keeps the total �kinetic� energyexactly constant over time �taking energy in and out� as needed� and iscalled a Gaussian thermostat� Unlike the �rst model the second model isreversible� with time reversal being the usual velocity inversion� Thus theabove theory and results based on the chaotic hypothesis apply�

The conjecture was �and is� that

�� Compute the average energy per particle that the system has in theconstant friction case and call it E�� calling also �� the corresponding SRBdistribution�

� � Call "�E the SRB distribution for the Gaussian thermostat system whenthe total �kinetic� energy is �xed to the value E �

�� Then �� "�E�� in the thermodynamic limit �in which the box sizetends to become in�nitely large� with the number of particles and the totalenergy correspondingly growing so that one keeps the density and the energydensity constant� and for local observables� i�e� for observables that dependonly on the particles of the system localized in a �xed �nite region of thecontainer� This means that the equality takes place in the usual sense ofthe theory of ensembles� see Chap�II�IV and �Ru�#��

It has to be remarked that the idea of equivalence between dynamical en�sembles� in contexts perhaps more limited� seems to circulate for quite along time particularly among those who work on numerical experimentsremarkable are the early papers �ES�� Ev�� SJ�� which certainly pro�pose the same kind of ideas� see also �MR��

The above conjecture opens the way to several speculations as it shows thatthe reversibility assumption might be not so strong after all� And resultsfor reversible systems may carry through to irreversible ones�

I have attempted to extend the above ideas also to cases of turbulentmotions but here I can only give references� �Ga��a��Ga��b��

There are a few other results and many speculations about the conse�quences of the chaotic hypothesis among the �few� related results� I wantto quote the �pairing rule�� valid for a somewhat restricted class of systems�It is a further extremely interesting example of a mathematical theorem dis�covered through physical experiments and� although heralded by a similarresult in a simpler case� �Dr##�� it was proved only later �like the $uctuationtheorem�� ECM�� EM�� DM�� WL�#�� It is related to the chaotichypothesis but it does not depend on it� the relation being that it emergedin the same group of experiments that led to the $uctuation theorem�

Among the �speculations� I quote

�� Several applications to $uid mechanics� like the equivalence of Navier�Stokes equations to a similar reversible� equation �in the limit of largeReynolds number�� Ga��c�� Ga��a�� SJ��


� � stability of time reversal symmetry whereby under assumptions� thatI think are quite natural� �BG�� one deduces that when time reversalis spontaneously broken�� it is replaced by another symmetry with thesame property of anticommuting with time evolution� thus showing thatthe $uctuation theorem might hold� with minor modi�cations� even in casesin which the time reversal symmetry of the equations of motion is broken�see x�� of �BGG��

�� The possibility of equivalence between reversible and irreversible equa�tions describing the same system� in the limit of large systems� �Ga�#c��This gives hope that some result like a� suitably reformulated� $uctuationtheorem can hold even in irreversibly driven systems� Furthermore it seemsto indicate that the theory of ensembles in nonequilibrium is much richerthan what we are used to in equilibrium� An ensemble might be character�ized not only by the choice of a few parameters� as in Chap�II� but also bythe choice of the equations of motion�

The above incomplete list is here only to provide the reader with a guide tothe literature� which is constantly increasing but which does not yet seemestablished enough to be treated as an accomplished theory deserving morespace in a short treatise�

Appendix ��A�� M�ecanique statistique hors �equilibre� l�h�eritage

de Boltzmann

The following is a slightly expanded version of a talk at 3Ecole NormaleSuperieure in Paris� january ��#� and gives an informal overview of thebasic ideas of Chap�IX� see x�� and some supplementary analysis�

Boltzmann entreprit� �Bo�� de prouver l existence des atomes en poursui�vant un programme d3ej5a amorc3e par ses pr3ed3ecesseurs� Son approche 3etaitd 3etablir que la conception de la mati5ere en tant qu agglom3eration d atomesob3eissants aux lois de la m3ecanique conduisait 5a la d3eduction des propri3et3esde la mati5ere que connaissaient alors les exp3erimentateurs et les th3eoriciens�

Ainsi Boltzmann produisit des versions de plus en plus ra%n3ees duth�eor eme de la chaleur� �Bo�#�� Bo�� Bo� �� Bo�� Au d3ebut il s agissaitde faire voir qu il est possible de d3e�nir des quantit3es m3ecaniques associ3ees�par exemple� 5a un gaz enferm3e dans un conteneur cubique de volume V �telles que

T � �energie cin�etique moyenneU � �energie totaleV � volumep � impulsion moyenne transf�er�ee aux parois

par collision et par unit�e de surface

�� Because the attracting set A becomes strictly smaller than the phase space and IA �� A�

��# IX� Coarse Graining and Nonequilibrium

o5u les moyennes sont calcul3ees empiriquement en supposant les particulesind3ependantes et 5a distribution uniforme sur une sph5ere dans l espace desimpulsions et dans le volume V des positions�Le th3eor5eme 5a prouver est alors que si on varie U et V de dU et dV et que

l on calcule la quantit3e dU � p dV

T��A�� A��

o5u p et T d3ependent de U et V � on trouve une di��erentielle exacte c est 5adire qu il existe une fonction S�U� V � telle que dS � dU p dV

T � Suite aux tra�vaux de Boltzmann� Helmholtz consid3era les syst5emes m3ecaniques monocy�cliques� c est�5a�dire les syst5emes dont tout mouvement d 3energie donn3ee estp3eriodique et non d3eg3en3er3e �ce qui veut dire que les mouvements d 3energiedonn3ee ne di�5erent entre eux que par un d3ecalage du temps d observation��He��a��He��b��Il �t voir que� en g3en3eral� si on imagine que les mouvements ��etats�� de

tels syst5emes sont param3etr3es par leur 3energie totale U et par un param5etreV dont les potentiels V des forces qui agissent sur le syst5eme d3ependent�alors en d3e�nissant

T � �energie cin�etique moyenneU � �energie totaleV � volumep � h� �

�V io5u hF i maintenant d3enote pr3ecis3ement la moyenne de F par rapportau temps �et donc n est pas d3e�nie empiriquement comme dans le caspr3ecedent�� on trouve en g�en�eral

dU � p dV

T� di�3erentielle exacte ��A�� A��

Celle�ci aurait pu n )etre rien de plus qu une curiosit3e� Mais Boltzmann avaitune conception discr ete de la nature m)eme s il ne l avait pas explicitementdit dans ses 3ecrits populaires� on le verrait dans ses travaux scienti�queso5u l emploi de l analyse� avec ses int3egrales et ses d3eriv3ees� est souvent vucomme un moyen technique pour venir 5a bout du calcul de sommes et dedi�3erences� �Bo��Donc pour Boltzmann le mouvement n est qu une 3evolution discr5ete o5u

l espace des phases est quadrill3e en petites cellules 5a �N dimensions �N3etant le nombre de mol3ecules� dont une contient le point qui repr3esentel 3etat instantan3e du syst5eme� L 3evolution apparait comme les d3eplacementssuccessifs du point repr3esentatif d une cellule 5a une autre� alors que le tempss 3ecoule d une petite quantit3e discr5ete h� Bien s)ur le d3eplacement doit )etreconforme aux lois du mouvement�C est une repr3esentation tr5es famili5ere aujourd hui 5a qui essaye de simuler

sur ordinateur les mouvements d un gaz de particules� Sur l ordinateurles 3etats microscopiques du gaz sont repr3esent3es par des cellules �car lescoordonn3ees des points sont repr3esent3ees par des nombres qui sont d3eter�min3es avec une pr3ecision qui est loin d )etre in�nie et qui d3epend de la


machine ou plut)ot du logiciel que l on emploit� et l 3evolution se d3eroule parpas discrets� le programme qui e�ectue ces pas est 3ecrit avec les lois dumouvement comme guide�De ce point de vue le mouvement est une permutation des cellules qui

repr3esentent l 3etat microscopique� Le syst5eme est alors toujours en 3evolutionp3eriodique car toute permutation d un nombre �ni d objets �les cellulesd 3energie totale U donn3ee� dans le cas pr3esent� engendre une 3evolution cy�clique�On imagine que l on �xe l 3energie totale U et que les forces agissantes sur

le syst5eme sont param3etr3ees par le volume V en fait on imagine que� quoique l on fasse� les forces entre les particules ne varient pas et seules les forcesentre les particules et les parois peuvent changer �5a cause des mouvementsdes parois et des changements de volume qui en d3ecoulent��Alors l hypoth5ese de monocyclicit3e de Helmholtz� de non�d3eg3en3erescence

des mouvements d 3energie donn3ee� correspondrait 5a dire que l 3evolution estune permutation 5a un seul cycle des cellules et donc on serait dans la situa�tion o5u le syst5eme est monocyclique cette hypoth5ese est connue commel hypoth ese ergodique�Sous cette hypoth5ese on devrait avoir la possiblit3e de trouver� en g�en�eral�

une analogie m3ecanique de la thermodynamique et un th3eor5eme g3en3eral de lachaleur� Puisque les moyennes doivent se calculer� alors� par la distributionuniforme sur l espace des cellules d 3energie donn3ee �car les cellules ont destailles 3egales� on se trouve oblig3e de v3eri�er que dans le cas d un gaz �oum)eme d un liquide ou d un solide� vue la g3en3eralit3es des consid3erations enquestion�

dU � p dV

Test exact si p � �h

VV i� ��A�� A��

Cette propri3et3e� cas particulier d une propri3et3e plus g3en3erale que Boltz�mann appella orthodicit�e� doit )etre accompagn3ee par la propri3et3e suppl3e�mentaire que p est aussi l impulsion moyenne transf3er3ee aux parois par lescollisions� par unit3e de temps et de surface� Si cela est bien le cas on auraprouv3e que en g3en3eral un th3eor5eme de la chaleur est valable�C est ce que Boltzmann �t en �##�� Bo#�� en fondant� en m)eme temps� lath�eorie des ensembles statistiques �qui est souvent attribu3ee 5a Gibbs� maispas par Gibbs lui m)eme� �Gi#��Il est tout 5a fait remarquable que le th3eor5eme de la chaleur� ��A�� est va�

lable tant pour les petits syst5emes �m)eme 5a une particule� si la non�lin3earit3edu mouvement est su%sante de fa6con 5a rendre l hypoth5ese ergodique rai�sonnable� que pour les grands �avec �� particules��

L�exactitude dedU � p dV

Tne d�epend pas de la taille du syst eme� �Bo#��

Cette ind3ependance est d ailleurs une propri3et3e absolument fondamen�tale et elle permit 5a Boltzmann de se d3egager des critiques qui lui 3etaientadress3ees�


Les critiques� par Zermelo et m)eme par Poincar3e� 3etaient subtiles etportaient sur le principe selon lequel il serait impossible de d3eduireles lois macroscopiques �irr3eversibles� d une m3ecanique r3eversible qui estn3ecessairement cyclique et donc apparemment pas irr3eversible �au bout d untemps de r�ecurrence le syst5eme revient 5a son 3etat initial� contre toute intui�tion sur le comportement des syst5emes macroscopiques�� Bo�� Bo��

Ces critiques s adressaient surtout 5a l 3equation de Boltzmann et parcons3equent 5a l approche irr3eversible 5a l 3equilibre� Boltzmann� comme il estbien connu� r3epondit qu on ne pouvait pas ne pas tenir compte des 3echellesde temps n3ecessaires 5a r3eveler des contradictions� Pour voir� au niveau ma�croscopique� les e�ets de la r3eversibilit3e microscopique� le temps qu il fallaitattendre 3etait 3enorme qu on le mesure en heures ou en )ages de l Univers��Bo�� Apr5es quoi on observerait une 3evolution anormale pour revenirpresque imm3ediatement au comportement normal et pour une p3eriode dedur3ee encore aussi longue�

Mais cet argument� 5a la d3efense de l 3equation de Boltzmann� d3etruisaitaussi apparemment la signi�cation du th3eor5eme de la chaleur et la possi�bilit3e de d3eduire la thermodynamique de la m3ecanique et de l hypoth5eseergodique� Car pour que le th3eor5eme de la chaleur ait un int3er)et quel�conque il faut que les moyennes dont il parle soient atteintes dans un lapsde temps raisonnablement court mais si le temps de r3ecurrence �c est 5adire le temps n3ecessaire au point repr3esentatif du syst5eme pour revenir 5a lacellule initiale dans l espace des phases� est 3enorme alors les moyennes desobservables risquent d )etre atteintes sur un temps du m)eme ordre� ce quisigni�erait qu elles n ont pas d int3er)et physique�

Boltzmann aper6cut cette di%cult3e et fut conduit 5a dire que dans un syst5ememacroscopique tout se passe comme si les moyennes sur des temps courts3etaient les m)emes que sur les temps �inobservables� de r3ecurrence� Ceciserait d)u au fait que si le nombre de particules est tr5es grand� les grandeursd int3er)et thermodynamique prennent la m)eme valeur sur presque tout l es�pace des phases ce qui leur permet d atteindre leur valeur moyenne sur destemps tr5es courts qui n ont rien 5a voir avec le temps de r3ecurrence �qui estin�ni 5a tout point de vue�� Elle prennent la m)eme valeur parce qu elles sont5a leur tour des moyennes sur les particules et ne d3ependent pas de l 3etat departicules individuelles�

Donc l hypoth5ese ergodique sugg5ere l ensemble microcanonique pour le cal�cul des moyennes c est un fait g3en3eral que ces moyennes v3eri�ent les re�lations thermodynamiques qui� d un autre c)ot3e� sont observables gr)ace 5ala lois des grands nombres qui fait que ces grandeurs ont la m)eme valeurpartout �ou presque� dans l espace des phases�

Il s en suit que l hypoth5ese ergodique n est pas une justi�cation de la ther�modynamique et ne joue qu un r)ole cin3ematique� La thermodynamiqueest une identit�e m3ecanique qui devient observable au niveau macroscopiquegr)ace 5a la loi des grands nombres� �x��

Une fois achev3ee cette admirable construction conceptuelle on se pose la


question de savoir si on peut faire de m)eme dans le cas des syst5emes hors�equilibre�Ce sont des syst5emes de particules sur lesquels agissent une ou plusieurs

forces conservatives dont le travail est dissip3e dans des thermostats� per�mettant ainsi au syst5eme d atteindre un 3etat stationnaire�C est un probl5eme pas vraiment touch3e par Boltzmann qui 3etudia en d3etail

le probl5eme du retour 5a l 3equilibre d un gaz perturb3e de son 3etat d 3equilibre�retour qui se d3eroule selon l 3equation de Boltzmann�� Et il peut para)7tre3etrange qu un probl5eme si naturel et d une telle importance soit rest3e es�sentiellement ouvert jusqu 5a nos jours�On remarque imm3ediatement une profonde di�3erence par rapport au

probl5eme de la th3eorie des 3etats d 3equilibre il n y a pas une v3eritableth3eorie macroscopique �comparable 5a la thermodynamique classique� quipuisse servir de guide et qui fournisse des r3esultats 5a prouver�Une di�3erence technique importante est que l on peut s attendre 5a ce que le

comportement physique du syst5eme d3epende de la m3ethode qu on emploiepour enlever la chaleur produite par le travail des forces qui agissent� Cequi peut donner le souci qu une th3eorie g3en3erale soit impossible 5a cause dela grande vari3et3e de forces thermostatiques qu on peut imaginer pour unm)eme syst5eme�Mais� 5a mon avis� il ne s agit que d une di%cult3e apparente qui disparait

au fur et 5a mesure qu on pr3ecise la th3eorie�Donc on va imaginer un syst5eme de particules sur lesquelles agissent des

forces externes non conservatives et un m3ecanisme quelconque qui emp5echele r3echau�ement� On va mod3eliser ce thermostat par des forces addition�nelles� Par exemple� si le syst5eme est un gaz de sph5eres dures enferm3eesdans un conteneur p3eriodique avec quelques obstacles �xes et soumises 5aun champ de force E� on peut imaginer que les 3equations du mouvementsoient

m�xi � fi� E � � !xi � �i�x� !x� ��A�� A��

o5u les fi

sont les forces entre particules �sph5eres dures 3elastiques� et entreparticules et obstacles �qui sont aussi des sph5eres dures 3elastiques��Ici � !x � �� !x� !x est le mod5ele de thermostat� La vraie di�cult�e est que

l 3evolution engendre une contraction du volume de l espace des phases car

d

dt�dx d !x� � div� � �dx d !x� ��A�� A��

et la dissipativit3e entraine �hdiv�i � �� et donc l 3etat stationnaire devra)etre une distribution de probabilit3e ��dx� d !x� concentr�ee sur un ensemblede volume nul� Elle ne pourra pas )etre d3ecrite par une densit3e de la forme ��x� !x� dx� d !x�Du coup on ne peut m)eme pas 3ecrire les formules qui expriment formel�

lement les moyennes des observables par rapport 5a l 3etat stationnaire entermes d une fonction de densit3e inconnue�N3eanmoins on voudrait avoir de telles expressions pour pouvoir esp3erer

en tirer des cons3equences g3en3erales� du type du th3eor5eme de la chaleur�


qui puissent )etre observ3ees dans les petits syst5emes �parce que directementobservables� et dans les grands aussi �pour des raison di�3erentes��L id3ee clef a pris forme au d3ebut des ann3ees �� au plus tard� et

est due 5a Ruelle mais dans un contexte apparemment assez di�3erent dun)otre �celui de la m3ecanique des $uides et de la turbulence�� On con6coit lesmouvements turbulents d un $uide stationnaire ou d un gaz de particulescomme des mouvements chaotiques�

Cela ne demande pas 5a premi5ere vue beaucoup d imagination mais lepoint est que l hypoth5ese est pos3ee dans un sens technique pr3ecis� �Ru�#a��Ru#�� Dans l interpr3etation d auteurs successifs� �GC�� on dit que leprincipe est que le syst eme est �hyperbolique� ou d� �Anosov�� C estl hypoth ese chaotique�

Cela veut dire que en tout point x de l espace des phases on peut 3etablirun syst5eme covariant de coordonn3ees locales tel que l 3evolution temporellen � Snx observ3ee dans ce syst5eme voit x comme un point �xe �car on lesuit� hyperbolique� C est�5a�dire on voit depuis x les autres points bougerde la m)eme fa6con qu on les voit si on regarde les mouvements 5a partir dupoint �xe instable d un pendule la di�3erence 3etant que cela est vrai pourtout point �et non pas pour un point isol3e comme dans le cas du pendule��On aura cette propri3et3e valable 5a l 3equilibre aussi bien que hors 3equili�

bre les mouvements des mol3ecules sont chaotiques m)eme dans les 3etatsd 3equilibre� Pour comprendre ce qui se passe il convient de revenir au pointde vue discret de Boltzmann�Si un syst5eme est dissipatif il y a des di%cult3es suppl3ementaires car il

est clair qu on a beau rendre petites les cellules de l espace des phases� onn arrivera jamais 5a un syst5eme dynamique discret qui puisse )etre d3ecritcomme une permutation des cellules la contraction de l espace des phasesentraine que certaines cellules ne seront jamais plus visit3ees m)eme si on les avisit3ees au d3epart �par exemple parce que l on a initi3e le mouvement 5a partird elles�� Les mouvements se d3eroulent asymptotiquement sur un attracteur�qui est plus petit que tout l espace des phases� bien que si on consid5erel espace de phases comme continu l attracteur pourrait )etre dense��Mais si on consid5ere seulement les cellules sur lesquelles se d3eroule le mou�

vement on est dans une situation identique a l��equilibre et hors �equilibre�On imagine que le mouvement est une permutation 5a un cycle� et donc il yaura un 3etat stationnaire unique� Le temps pour parcourir le cycle sera� bien3evidemment� toujours du m)eme ordre de grandeur qu 5a l 3equilibre �dans dessituations pas trop extr)emes des param5etres qui d3eterminent les forces agis�santes sur le syst5eme� donc la raison pour laquelle on peut esp3erer observerles moyennes temporelles et les calculer par int3egration par rapport 5a unedistribution de probabilit3e sur l espace des phases reste la m)eme que celled3ej5a discut3ee dans le cas d 3equilibre �et li3ee 5a la loi des grands nombres��Toutefois il y a une di%cult3e c est une di%cult3e qu on aurait pu discuter

d3ej5a dans le cas de l 3equilibre� On a suppos3e� sans critique� que les cellules

� Ce qui montre seulement que la notion de �grandeur d un attracteur est plut%ot d�elicate


de l espace des phases 3etaient toutes 3egales� Mais m)eme dans le cas del 3equilibre les syst5emes sont chaotiques et donc toute cellule est d3eform3ee parl 3evolution temporelle qui la dilate dans certaines directions et la contractedans d autres�

Il appara)7t alors que la repr3esentation du mouvement comme 3evolutiond une cellule vers une autre de forme et de taille identique est loin d )etre tri�viale� Elle est en fait une hypoth5ese forte sur la dynamique� qui� 5a l 3equilibre�s3electionne l ensemble microcanonique comme distribution correcte 5a utili�ser pour calculer les moyennes temporelles �et qui entraine le th3eor5eme dela chaleur�� Il y a bien d autres distributions invariantes sur l espace desphases �contrairement 5a ce qu on entend dire parfois� et l hypoth5ese appa�remment innocente que le mouvement se repr3esente comme une permutationde cellules identiques en s3electionne une particuli5ere�

Hors 3equilibre la di%cult3e devient plus manifeste� Car le volume des cel�lules ne reste m)eme pas invariant contrairement au cas de l 3equilibre �gr)aceau th3eor5eme de Liouville�� De plus hors 3equilibre il faut s attendre 5a ceque la repr3esentation du mouvement comme 3evolution de cellules iden�tiques conduise 5a s3electionner une distribution de probabilit3e particuli5eresur l espace des phases� concentr3ee sur les cellules qui constituent l attrac�teur� �Ga��a��

L int3er)et et l importance des syst5emes chaotiques au sens de l hypoth5esechaotique est que� en e�et� pour tous ces syst5emes il y a une unique dis�tribution stationnaire � sur l�espace des phases qui donne les moyennes desgrandeurs observ�ees sur les mouvements qui commencent dans la grandemajorit�e des cellules identiques en lesquelles on peut imaginer de diviserl�espace des phases� C est un r3esultat fondamental d)u 5a Sinai et 5a Ruelle�Bowen ainsi la distribution � s appelle distribution SRB� �Si�#�� BR��Dans le cas de l 3equilibre� elle co�7ncide avec la distribution microcanonique�

Ce n est pas ici le lieu de poursuivre la critique de la vision discr5ete du mou�vement� bien qu elle soit int3eressante ne fusse que pour une interpr3etationcorrecte des simulations num3eriques qui se font de plus en plus fr3equentes�voir la note �� page suivante�

L hypoth5ese chaotique conduit naturellement 5a une repr3esentation discr5etedi�3erente du mouvement qui non seulement ne sou�re pas des critiquesqu on vient de mentionner� mais qui nous donne une formule explicite pourla valeur des moyennes des observables� valable 5a la fois 5a l 3equilibre �o5u ellese r3eduit 5a l ensemble microcanonique� et hors 3equilibre�

Cette nouvelle repr3esentation est aussi bas3ee sur des cellules mais ellene sont pas vraiment petites dans le sens qu elles sont consid3erablementplus grandes que les cellules que l on a utilis3ees jusqu 5a maintenant et quiavaient la taille minimale concevable� On peut donc les appeller �cellules5a gros grains� ou grosses cellules� r3eservant le nom de cellules de taille �neaux pr3ec3edentes�

Il est en e�et possible de d3ecouper l espace des phases en cellulesE�� E�� fE�g�� qui forment un pavage ou une partition P et quiont la propri3et3e de covariance�


Leur bords sont constitu3es par une r3eunion d axes des syst5emes locauxde coordonn3ees dont on a parl3e plus haut donc les bords consistent endes surfaces qui soit se contractent sous l action de la dynamique� soit sedilatent� On dira que les fronti5eres des cellules de la partition P � E�� E�� consistent en une partie qui se contracte ou �stable� et en une partie qui sedilate ou �instable�� La propri3et3e de covariance dit alors que sous l actionde l 3evolution les cellules se d3eforment mais les parties stables de leur bords�evoluent de fa!con a terminer comme sous�ensembles de leur r�eunion la�gure suivante illustre cette propri3et3e simple�

s

�

u u

s

S�

Fig� ��A��

Si on a une telle partition �qui s appelle partition markovienne� P on peutla ra%ner en d autres qui ont la m)eme propri3et3e de covariance simple�ment en donnant un entier T et consid3erant la partition constitu3ee par lesensembles S�TE��T � � � � STE�T qui� 5a cause de la contraction et de l ex�pansion de l espace lors de l 3evolution� forment une partition PT dont lescellules deviennent aussi petites que l on veut en prenant T assez grand�

Si F est une observable on peut en calculer la valeur moyenne simplementen consid3erant une partition markovienne P �arbitraire� car il n y a pasd unicit3e� en construisant la partition PT avec T assez grand pour que Fsoit constant dans chaque cellule C de PT et puis en posant

hF i �

PC P �C�F �C�P

C P �C��A�� A��

o5u P �C� est un �poids� convenable� Il est construit en choisissant un pointc � C et en consid3erant son 3evolution entre �� et � o5u � est grand maispetit par rapport 5a T �par exemple � � �

�T ��

On consid5ere le point S��c qui est transform3e en S� c en un temps � � Onvoit que l axe� par S��c� des coordonn3ees qui se dilatent sous l action del 3evolution est dilat3e� au cours d un temps � � par un facteur qu on appelle,��i�c� alors le poids P �C� peut "etre choisi �egal a ,��i�c�

��

L 3equation ��A�� est la formule qui remplace la distribution microca�nonique hors 3equilibre on peut prouver que l on s y ram5ene sous l hy�poth5ese chaotique� La question qui se pose est si l on peut tirer quelques


cons3equences g3en3erales de l hypoth5ese chaotique moyennant l usage de larepr3esentation ��A�� ci�dessus��

Dans ce contexte� mentionnons que r3ecemment on a r3eussi 5a d3eduire unecons3equence qui apparemment 5a un certain int3er)et� On va la formuler pourun syst5eme d3ecrit par une 3equation di�3erentielle �et donc en temps continu�!x � f�x� qui engendre un $ot t� Stx dans l espace des phases� On supposeaussi que l��evolution est r�eversible c est�5a�dire qu il y a une transformationisom3etrique I de l espace des phases qui anti�commute avec l 3evolution ISt � S�tI �Imaginons un syst5eme pour lequel l hypoth5ese chaotique soit valable� donc

d3ecrit par une 3equation !x � f�x� et soit ��x� � �div f�x� la contractionde l espace des phases associ3ee� Supposons que l on mesure la quantit3e��Snx� au cours du temps mais avec le syst5eme dans son 3etat stationnaire�Appellons � sa moyenne temporelle que l�on suppose non nulle �alors ellene peut )etre que positive par un th3eor5eme g3en3eral� �Ru��a�� et

p ��

��

Z ��

� ��

��Stx�dt ��A�� A��

� L expression ��A�� pour la distribution SRB permet d �eclaircir la r�epr�esentation del �evolution comme permutation des cellules &a taille �ne� On doit imaginer que chaque�element C ��cellule &a gros grain� de la partition markovienne PT � avec un T tr&es grandde fa'con &a ce que toute observable F �pertinente pour le comportment macroscopique�reste constante sur chaque C � pour une repr�esentation �d&ele du mouvement� on imagine que chaque C est quadrill�e par des cellules tr&es petites �de taille �ne en nombreproportionnel �a P �C�� Par l �evolution les cellules de taille �ne se r�epartissent entre les�el�ements C� de PT qui intersectent SC� On fait �evoluer de la m%eme fa'con les autres cellules �nes des �el�ements de PT � la th�eorie des distributions SRB montre que le nombredes cellules de taille �ne qui viennent se trouver dans chaque C � PT ne change pas� &aune tr&es bonne aproximation pr&es� c est la stationnarit�e de la distribution SRB� �Ga��a��Alors on peut d�e�nir l �evolution des cellules de taille �ne simplement en disant qu unecellule �ne dans C �evolue dans une des cellules �nes qui sont dans la C� qui contientS � il faut seulement faire attention &a ne pas associer une m%eme cellule �ne de C� &adeux cellules �nes appartenant &a di e�rentes C �parmi celles telles que SC�C� �� onpeut s arranger de fa'con telle que la permutation des cellules �nes ainsi d�e�nie soit &a unseul cycle� car les d�etails du mouvement &a l int�erieur des cellules C n ont pas d importance parce que les observables qui nous int�eressent sont constantes dans les C� Mais lam%eme construction peut %etre faite en rempla'cant le poids P �C� par P �C�� avec � �� on obtient ainsi d autres distributions stationnaires di��erentes de la SRB� et on peutm%eme en construire d autres� �Si�� Bo�� On peut repr�esenter de la m%eme fa'con aussices autres distributions � mais on doit imaginer que les cellules de taille �ne que l onutilise pour en repr�esenter une soient di��erentes de celles utilis�ees pour repr�esenter lesautres� En �n de compte toutes les cellules �nes ainsi introduites repr�esentent l�attrac�teur� Si on divise l espace entier en �beaucoup de� cellules �nes� de fa'con &a ce que toutesdistributions stationnaires puissent %etre repr�esent�ees par une permutation des cellules�nes qui se trouvent dans les C � PT � alors on obtient une r�epr�esentation disc&ete tr&es�d&ele du mouvement� Mais toutes les cellules ne feront pas partie d un cycle� car ladynamique est en g�en�eral dissipative et une grande partie d entre elles ne reviennentpas sur elles m%emes mais �tombent sur l attracteur o&u� d&es lors� elles �evoluent dansun cycle� La th�eorie de la distribution SRB montre que si on consid&ere un ensembleouvert dans l espace des phases le comportement asymptotique du mouvement de toutpoint� sauf un ensemble de volume nul� est bien r�epr�esent�e par la distribution SRB� cequi lui fait jouer un r%ole particulier� au contraire des autres distributions que l on peutde�nir � c est&adire que la grande majorit�e �en volume� des cellules �nes tombant surl attracteur vont se trouver parmi celles que l on a associ�ees aux cycles de la distributionSRB�


et soit �� p� � e��p� la distribution de probabilit3e de cette observable�Alors

��p�� p�

�� A��#� �A��

c est le th�eor eme de �uctuation� �GC��Je ne peux pas discuter ici la signi�cation physique du th3eor5eme et de l hy�

poth5ese de r3eversibilit3e� mais il est int3eressant de souligner sa g3en3eralit3e�son ind3ependance du syst5eme consid3er3e et aussi l absence� dans sa formu�lation� de param5etres libres� Ce qui le rend en un certain sens analogue auth3eor5eme de la chaleur� qui lui aussi est g3en3eral et sans param5etres libres�Il su%ra de dire que le th3eor5eme de $uctuation est une propri3et3e qu il faut

quand m)eme v3eri�er exp3erimentalement en e�et une partie de la th3eorieci�dessus est n3ee 5a la suite d une exp3erience de simulation num3erique etpour en interpr3eter th3eoriquement les r3esultats� �ECM�� Il y a eu aussiquelques v3eri�cations ind3ependantes� �BGG�� LLP��La raison pour laquelle des exp3eriences sont n3ecessaires est qu il n y a au�

cun espoir de prouver que des syst5emes r3eels v3eri�ent au sens math3ematiquedu mot l hypoth5ese chaotique� moins encore de prouver que des syst5emesr3eel v3eri�ent l hypoth5ese ergodique� Il n y a m)eme pas d espoir de prouverque des syst5emes int3eressants en simulation num3erique ou dans la r3ealit3ev3eri�ent des propri3et3es qui soient assez proches de celles des syst5emes hy�perboliques pour en d3eduire des cons3equences telles que le th3eor5eme de $uc�tuation� Mais on peut croire que n3eanmoins �les choses se passent commesi l hypoth5ese chaotique 3etait litt3eralement vraie��Il y a donc une n3ecessit3e d un contr)ole exp3erimental car on est dans la

m)eme situation qu 5a l 3equilibre o5u tout en croyant� avec Feynman� que�if we follow our solution �i�e� motion� for a long enough time it tries eve�rything that it can do� so to speak� �see p� �� in �Fe�� vol� I�� il a 3et3en3eanmoins n3ecessaire de faire de bonnes v3eri�cations exp3erimentales pourne plus avoir de r3eserves ou de doutes sur l hypoth5ese ergodique dans lath3eorie de l 3equilibre�Quelques r3ef3erences sont donn3ees ici pour guider le lecteur dans la

litt3erature r3ecente et ancienne� Elles sont loin d )etre exhaustives �HHP#��EM�� ECM�� DPH�� Ga��c�� Ga��a�� Ga��b�� Ge�#�� GR��Ga�#a�� Ga�#b�� Ga��c�� Ga��a�� Ru��a�� Ru��c�� BG�� MR��b��Ku��

Appendix ��A� Heuristic Derivation of the SRB Distribution

The discussion below follows �Ga��a��Ga��d��Ga�#c�� see also �Ga#�� Aball B containing a unit mass uniformly spread in it with density � andcentered around a �xed point�� O for an Anosov map S which is a Anosov

�� It is not restrictive to suppose that there is a �xed point for S� In fact Anosov systemsalways admit periodic points� they are always dense on phase space F � If � is a periodicpoint of period N then it is a �xed point for SN � Clearly the map SN is still an Anosov


mixing map �hence such that the stable and unstable manifolds of O �lldensely the phase space / on which S acts� see x�� will be elongatedalong the unstable manifold of O� in so doing the map S will compress themass so that after T iterations the image STB will coat a large portion ofW u

O with a thin coating of mass�The mass around an in�nitesimal surface element � around a point x �W u

O

which is reached by the spreading coating� i�e� which has the form x � ST yfor some y in the connected part of W u

O � B which contains O� will be theone that at time � was above the image S�T � of � �of extremely small size�area j�j and very close to the origin for T large and x �xed� i�e� it willbe proportional to the area of the image times � and the proportionalityconstant will be essentially the power n of the radius h of the ball if n is thedimension of the stable manifold of O� In formulae the mass d� in questionwill be

d� � �hn,��u�T �S�Tx� j�j � ��A �� A��

For T large we see therefore that the mass initially in B will coat a �nite �butvery large and increasing with T � surface of the �dense� manifold W u

O � Hencewe see that the SRB distribution� which should be the limit to which thedescribed distribution of mass should tend� will be in a sense �concentrated�along the unstable manifold W u

O �Let �� be two �in�nitesimal� surface elements on W u

O centered aroundx� x� respectively with x� x� close and on the same stable manifold W s

x � Andsuppose that the stable manifolds through the points of � intersect �� andvice�versa �i�e� �� are the bases of a �tube� whose generators are the stablemanifolds�� See Fig� ��A��

x

x�

�

�� Fig� ��A��

where the vertical lines represent stable manifolds and the horizontal partsof the unstable manifold� The surface elements �� are in�nitesimal partsof a connected surface which� being very large and winding around on themanifold� �almost �lls� the whole manifold� The surface �i�e� the unstablemanifold of O� is not drawn� it would connect the two surface elements� Wesee that the ratio of the masses originally in B and coating � and �� is

,��u�T �S�Tx�

,��u�T �S�Tx��j�jj��j ��A � � �A��

where j�j� j��j denote the surface areas of the elements �� but the ratioj�j�j��j can be expressed as

j�jj��j �

S�T �ST ��

S�T �ST ��

,��u�T �STx�

,��u�T �STx��jST �jjST ��j ��A �� A��

map� so that the following discussion will apply to the map SN and even to S providedwe take T an integer multiple of N �

��# IX� Coarse Graining and Nonequilibrium

and by the composition rule of derivatives �and Jacobian determinants� wesee that the ��A � � can be written

,��u�T �S�Tx�,��u�T �STx�

,��u�T �S�Tx��,��u�T �STx��jST �jjST ��j ��A �� A��

but the last ratio approaches � as T �� because �� are in�nitesimal andST �� ST �� get closer and closer as T �� The limit as T �� is

limT��

��A � � ��Y

k��

,��u�S�kx�

,��u�S�kx�� A �� A��

It is not di%cult to check that the in�nite product on the right convergesbecause the points Skx� Skx� tend to get close exponentially fast both in thefuture and in the past� being at the same time on the stable and on theunstable manifolds of each other�Thus the coating generated by the splashing �due to the time evolution�

of the mass initially in the ball B of the unstable manifold W uO will have

the formal densityQ�

k�� ,��u��S�kx� which can be written �formally� as

e�H with H �P�

k�� log ,u��S�kx�� Since x can be represented as a

sequence � via its symbolic coding� x � x�� on a Markov partition E wecan de�ne �u�� log ,u��x�� and obtain that the SRB distribution� will be represented as a distribution on the space of the �compatible�sequences associated with the Markov partition and it will have the formalexpression

��d�� const e�P

�

k��u��

k��A �� A��

if is the shift on the bilateral sequences ��Recalling x�� we see that the distribution � can be interpreted as a Gibbs

state for a one�dimensional Ising model with short�range interaction in thesense of the discussion in Chap�V following �� see x��The function �u�� has in fact all the properties needed for the lat�

ter interpretation� as a consequence of the discussion in x�� Theabove heuristic discussion establishes the connection between the statis�tical mechanics of one�dimensional spin systems with short�range interac�tions and the apparently highly nontrivial dynamics of an Anosov system�The above remarks lay the foundations of the thermodynamic formalism��Si�#��Ru��Bo��Ru�#b��In a sense the reduction of the system to an Ising model is the chaotic

dynamics analogue of the integration by quadratures of classical mechanics�

Appendix ��A� Aperiodic Motions Can be Begarded as Periodic

with In�nite Period�

This famous and criticized statement of Boltzmann� �Bo�� is the heart ofthe application of the heat theorem for monocyclic systems to a gas in a


box� Imagine the box containing the gas to be covered by a piston of sectionA and located to the right of the origin at distance L� so that V � AL�The microscopic model for the piston will be a potential �L � �� if x �

�� are the coordinates of a particle� The function �r� will vanish forr � r�� for some r� � L� and diverge to �� at r � �� Thus r� is the widthof the layer near the piston where the force of the wall is felt by the particlesthat happen to roam there�Noting that the potential energy due to the walls is �

Pj �L� �j� and

that V � A��L we must evaluate the time average of

L�x� � �Xj

��L� �j� � ��A�� A��

As time evolves the particles with �j in the layer within r� of the wall willfeel the force exercised by the wall and bounce back� Fixing the attentionon one particle in the layer we see that it will contribute to the average ofL�x� the amount

�

total time

Z t�

t�

��L� �j�dt ��A�� A��

if t� is the �rst instant when the point j enters the layer and t� is theinstant when the ��component of the velocity vanishes �against the wall��Since ��L��j� is the ��component of the force� the integral is mj !�j j �by

Newton s law�� provided !�j � � of course� One assumes that the density islow enough so that no collisions between particles occur while the particlestravel within the range of the potential of the wall i�e� the mean free pathis much greater than the range of the potential de�ning the wall�The number of such contributions to the average per unit time is therefore

given by �wallARv� mv f�v� v dv if �wall is the density �average� of the

gas near the wall and f�v� is the fraction of particles with velocity between vand v�dv� Using the ergodic hypothesis �i�e� the microcanonical ensemble�and the equivalence of the ensembles to evaluate f�v� it follows that

pdef� � hV i � �wall�

�� A�� A��

where �� kBT with T the absolute temperature and kB Boltmann sconstant� Hence we see that ��A�� yields the correct value of the pres�sure� see Chap�I� Chap�II� in fact it is often even taken as the microscopicde�nition of the pressure� �MP� ��On the other hand we have seen in x�� repeating the analysis

in Appendix ��A�� Chap�I�� that if all motions are periodic the quantity pin ��A�� is the right quantity that would make the heat theorem work�Hence regarding all trajectories as periodic �i�e� the system as monocyclic�leads to the heat theorem with p� U� V� T having the right physical inter�pretation� And Boltzmann thought since the beginning of his work that

� � IX� Coarse Graining and Nonequilibrium

trajectories con�ned into a �nite region of phase space could be regardedas periodic possibly with in�nite period� �Bo��

Appendix ��A�� Gauss� Least Constraint Principle

Let � !x� x� � �� !x� x� � f !xj � xjg be a constraint and let R� !x� x� be theconstraint reaction and F � !x� x� the active force�Consider all the possible accelerations a compatible with the constraints

and a given initial state !x� x� Then R is ideal or satis�es the principle ofminimal constraint if the actual accelerations ai � �

mi�F i � Ri� minimize

the e�ort

NXi�

�

mi�F i �miai�

� ��NXi�

�F i �miai� � �ai � � ��A�� A��

for all possible variations �ai compatible with the constraint � Since all

possible accelerations following !x� x are such thatPN

i� �xi� !x� x� � �ai � �we can write

F i �miai � � �xi� !x� x� � � ��A�� A��

with � such thatd

dt� !x� x� � �� A�� A��

i�e�

� �

Pi � !xi � xi � �

miF i � �xi�P

im��i � �x

i��

��A�� A��

which is the analytic expression of the Gauss principle� see �Wi#��Note that if the constraint is even in the !xi then � is odd in the velocities

therefore if the constraint is imposed on a system with Hamiltonian H �K � V � with K quadratic in the velocities and V depending only on thepositions� and if other purely positional forces �conservative or not� act onthe system then the resulting equations of motion are reversible if timereversal is simply de�ned as velocity reversal�

The Gauss principle has been somewhat overlooked in the Physics litera�ture in statistical mechanics its importance has again only recently beenbrought to the attention of researchers� see the review �HHP#�� A notable�though by now ancient� exception is a paper of Gibbs� �Gi#�� which developsvariational formulas which he relates to Gauss principle of least constraint�Conceptually this principle should be regarded as a de�nition of ideal non�holonomic constraint� much as D Alembert s priciple or the least actionprinciple are regarded as the de�nition of ideal holonomic constraint�

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�Ma�� Martin�L(of� A� On the spontaneous magnetization in the Ising model� Communications in Mathematical Physics� ��

�Ma�� Maxwell� J�� On Boltzmann�s theorem on the average distribution of energy ina system of material points� in The scienti�c papers of J�C� Maxwell� ed� W� Niven�Cambridge University Press� �� vol� II� p� ��

�Ma�� Mathieu� M�� On the origin of the notion �Ergodic Theory�� Expositiones Mathematicae� ��

�Ma�� Mastropietro� V�� Renormalization group and correlation functions for the XYZmodel� preprint� Roma� ��

�Me�� Mermin� D�� Absence of ordering in certain classical systems� Journal of Mathematical Physics� ��

�Me�� Mermin� D�� Solvable model of a vaporliquid transition with sigular coexistencecurve diameter� Physical Review Letters� ��

�Mi�� Minlos� R�A�� Lectures in Statistical Mechanics� in Russian Mathematical Surveys��

�Mi�� Miracle�Sol�e� S�� On the microscopic theory of phase coexistence� � Years ofNon�Equilibrium Statistical Mechanics� ed� J�J� Brey� Lecture Notes in Physics� �� pp�� Springer� Berlin� �� And Surface tension� step free energy and facets in theequilibrium crystal shape� Journal Statistical Physics� � ��

�Mo�� Morrey� C�B�� On the derivation of the equations of hydrodynamics from Statis�tical Mechancs� Communications in Pure and Applied Mathematics� ��

�NM�� G�F� Newell G�F�� Montroll E�W�� On the theory of the Ising model of ferromag�netism� Reviews of Modern Physics� ��

�Ne�� Nelson� E�� Dynamical theories of Brownian motion� Princeton University Press��

�On�� Onsager L�� Crystal statistics� I� A two dimensional Ising model with an orderdisorder transition� Physical Review� ��

�Or�� Orbach� R�� Linear antiferromagnetic chain with anisotropic coupling� PhysicalReview� ��

�Or�� Ornstein� D�� Ergodic theory� randomness and dynamical systems� Yale UniversityPress� ��

�PS�� Pirogov� S�A�� Sinai� J�G�� Phase diagrams of classical lattice systems� Theoreticaland Mathematical Physics� �� and �� Both reprinted in�Si��

�PV�� P�ster� C� Velenik� Y�� Mathematical theory of the wetting phenomenon in the�DIsing model� Helvetica Physica Acta� �� P�ster� C�� Velenik� Y��Large deviations and continuum limit in the �D Ising model� Probability Theory andRelated �elds� ��

�PV�� P�ster� C� Velenik� Y�� Interface� surface tension and reentrant pinning transitionin the �D Ising model� preprint �� in print on Communications in MathematicalPhysics�

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�Pe�� Peierls� R��On Ising�s model of ferromagnetism� Proceedings of the CambridgePhilosophical Society� ��

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�Yo�� Young� L�S�� Ergodic theory of di�erentiable dynamical systems� in Real and com�plex dynamical systems� ed� B� Branner� P� Hjorth� Nato ASI series� Kluwer� Dordrecht�� And Ergodic theory of chaotic dynamical systems� Proceedings of the InternationalCongress of Mathematical Physics� Brisbane� �� ed� A� Bracken�

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Names index ��

Names index

A

Anosov �� Avogadro �

B

Bachelier �� Baxter �� Bethe ��Boltzmann�� Borel ��Bowen ��Brown ��Brush �� B(ohm ��

C

Cantoni �� Cesaro ��Clausius ��Cohen ��

D

Dobrushin ��

E

Ehrenfest �� Einstein �� Euler ��Eulero ��

F

Fisher ��

G

Galgani ��Gallavotti ��Gibbs �� Grad ��Green ��Gri�ths ��

H

Helmholtz �� Hilbert ��Hoover ��Hulth�en ��Hurst ��

J

Jeans ��

K

Kac ��Kadano ��Kasteleyn ��Kaufman ��Kintchin �� Knudsen ��Kramers ��Krylov �� Kuhn ��

L

Lagrange ��Lanford �� Langevin ��Lawler ��Lebowitz ��Lee ��Leibnitz ��Lieb �� Liouville ��Lucretius ��

M

Mattis ��Maxwell ��Minlos ��

N

Nelson ��N(ageli ��

O

Onsager �� Ornstein �� Oseledec ��

P

�� Names index

Pauling �� Peierls �� Perrin �� Pirogov ��Planck �� Plato ��Poincar�e �� Poisson ��

R

Raoult ��Ruelle ��

S

Schultz ��Sinai �� Smale ��Smoluchowski �� Spohn ��Sutherland ��Sykes ��

T

Thomson ��

U

Uhlenbeck ��

V

Van der Waals ��Van der Waerden ��Van t Ho ��

W

Wannier ��Ward ��Wiener �� Wilson ��

Y

Yang CN �� Yang CN� Yang CP ��Yang ��

Z

Zeno �Zermelo ��

v

van der Waals ��

Analytic index ��

Analytic index

A

Anosov map or system ��Anosov system �� Anosov system� def ��Arnold s constant ��Avogadro s number �� Axiom A systems ��

B

Bachelier ��Bethe ansatz ��Bethe s ansatz �� Birkho s ergodic theorem ��BogoliubovValatin transformation ��Boltzmann equation �� Boltzmann s Htheorem ��Boltzmann s constant�� Boltzmann s constant� universality ��Boltzmann s equation �� Boltzmann s formula ��Boltzmann s sea ��Borel set ��Bose condensation ��BoyleMariotte law ��

C

Carnot cycle ��Carnot s principle ��Cesaro s limit ��Clausius virial theorem ��Couette s �ow ��Coulomb interaction ��Coulomb s interaction �� CurieWeiss theory ��

D

DLR equation �� Debye s screening ��Dirac s delta ��Doppler regime ��Doppler s resistance �� Dulong Petit law ��

E

Einstein s formula ��

Einstein�Smoluchowski relation ��Euler s equation ��

F

F model ��Fermi s condensation ��Fermi s sphere ��Feynman�Kac formula ��

G

Gauss principle �� Gaussian distribution ��Gaussian process ��Gaussian thermostat �� Gibbs distribution �� Gibbs distribution� def ��Gibbs potential �� Grad s conjecture �� Grad s limit �� GradBoltzmann limit ��GreenKubo formula ��

H

Htheorem ��Hamilton equation ��Hamilton s equation �Hausdor dimension ��Heisenberg s model ��Hilbert s space ��H(older s continuity ��

I

Ising model� �D solution ��Ising s model �� Ising s model� many spin interaction ��

K

KDP model ��KDPmodel ��KH�PO ��Kac s potential ��Kirkwood Salsburg equations ��KirkwoodSalsburg s equation ��Knudsen s model ��

L

Landau s argument �� Lanford s theorem ��

�� Analytic index

Langevin equation ��Lawler s theorem ��LeeYang s theorem �� Lennard�Jones potential ��LennardJones potential �� Levy s regularity law ��Liouville distribution ��Liouville s distribution ��Liouville s theorem �� Liouville s volume ��Lorentz gas �� Lorentz model ��Lyapunov s exponent ��

M

Markov partition �� Markov pavement �� Markov s partition ��Markov s partition� def ��Maxwell s daemon ��Maxwell s equations ��Maxwell s rule �� MaxwellBoltzmann distribution ��MaxwellBoltzmann law ��Maxwellian distribution ��Maxwellian distribution� def ��MerminWagner theorem ��MerminWagner s theorem ��Moon ��

N

Navier�Stokes equation ��NavierStokes equation ��NavierStokes �uid ��Newton s equation ��

O

Onsager s reciprocity ��Ornstein�Uhlenbeck s process ��Oseledec s theorem ��

P

PauliJordan s transformation ��Peierls argument ��PerronFrobenius theorem ��PirogovSinai theory ��Planck s constant �� Planck s distribution ��Poisson s distribution �� Postscript ��

R

Raoult�van t Ho s law ��Raoult�van t Ho s law ��Raoultvan t.Ho s law ��RayleighJeans formula ��Ruelle s theorem ��

S

SRB distribution �� Schr(odinger s operator �� Smoluchowski s theory ��Stirling s formula �� Stokes law ��Stokes regime ��Stokes law �� Sun ��

U

Uhlenbeck�Ornstein theory ��

V

Van Hove s theorem �� Van der Waals equation ��

W

Wien s distribution ��Wiener process �� Wiener theorem ��Wiener s theory ��Wul s construction ��

Z

Zeno s paradox �

a

absolute continuity ��abstract nonsense ��activity �� age of the Universe ��allowed sequence ��almost all points ��analytic error �� angular momentum ��annoying error ��ansatz ��anthropocentrism ��approach to equilibrium ��approximate symmetry ��


artifact ��atomic hypothesis �� atomic size ��atomism �� attracting set �� attracting set� def ��attractive potential ��attractor ��attractor� def ��average kinetic energy ��average over time ��average value �� axiom A attractor ��axiom of choice ��

b

big nonchaotic island ��black body �� boson ��boundary condition � ��boundary condition � ��boundary condition�� boundary condition� periodic �� broken symmetry ��

c

canonical absolute temperature ��canonical distribution �� canonical ensemble �� canonical entropy ��canonical free energy ��canonical internal energy ��canonical pressure �� canonical speci�c volume ��cell in hase space ��cell in phase space�� cell of Markov partition ��cell size �� cell� coarse ��cells permutation ��chain of oscillators ��chaos ��chaotic hypothesis �� chaotic system ��charge ��charged particles ��chemical potential ��classical thermodynamics �� close packing density �� cluster property� def ��coalescence catastrophe ��coarse grained cell ��

coding ) symbolic dynamics ��coexistence line ��coexisting phases �� collapse ��collision crosssection ��collision volume ��collision ��colloid ��combinatorial error �� combinatorial factor ��compatibility matrix ��compatible sequence ��compatible spins ��compatible symbols ��compressibility ��condenser ��conditional probability ��conditional reversibility theorem �� cone property ��conservation law ��constant of Boltzmann �constant of Planck �� constant of gas ��constant of motion ��continuum �contraction rate of phase space ��convexity ��correlation function �� covariance ��critical exponent ��critical parameter ��critical point �� crystalliquid transition ��crystalline solid ��cylindrical boundary condition ��

d

daemon �� de�ection angle ��degeneracy ��degeneration ��degree of freedom� frozen ��degree of freedom� oscillatory ��density matrix ��determinism �deterministic principle �diatomic gas ��di usion coe�cient ��discrete space ��discrete time ��distribution SRB �� distribution of Liouville ��doom ��duality of KramersWannier ��dynamical base towards future ��dynamical base ��dynamical system ��dynamics cycle ��


e

elastic collision �energy indetermination ��enlarged ensembles ��ensemble equivalence �� ensemble �� entropy creation ��entropy decrease� forget ��entropy generation rate ��entropy increase ��entropy �� equation of Boltzmann �� equation of state �� equation of van der Waals ��equilibrium property ��equilibrium state �� equilibrium� approach to ��equipartition �� equivalence of ensembles �� equivalence of statistical ensembles �� ergode �� ergodic decomposition �� ergodic distribution ��ergodic hypothesis �� ergodic hypothesis� critique ��ergodic hypothesis� di�culties ��ergodic problem �� ergodic set ��ergodic theorem �� ergodic theory ��ergodic ��ergodicity� def �� errors negligibility ��essence of equivalence ��etymology ��evaporation catastrophe ��evolution map ��exact solution� Ising �D ��exact solution� six vertex ��expansion rate of phase space ��extremal distribution ��extreme degeneracy ��

f

fallacious argument ��fermion �� fermionic system ��ferroelectricity ��nite volume orthodicity ��xed particle boundary condition ��xed particle boundary conditions �� xed particles boundary condition ��

�xed particles boundary conditions ��xed spin boundary condition ��xed spins boundary condition ��ight time ��uctuation theorem ��uctuation�dissipation theorem ��formalism ��free boundary conditions ��free energy �� free gas ��free microcanonical energy ��frozen degree ��function� measurability ��

g

gamma function �� gap ��gas constant ��gaussian process� def ��ghastly generality ��global iterated logarithm law ��grand canonical ensemble �� gravitation constant ��gravitational force ��ground state ��

h

hand waving ��hard core interaction �� hard core lattice system ��hard core potential ��hard core �� hard core� def ��hard sphere �� harmonic oscillator ��heat equation ��heat theorem �� hidden variables ��holode �� homogeneous phase ��hydrogen bond ��hypothesis� ergodic ��

i

ice model ��ice rule ��imponderable ��indetermination principle �� indetermination ��indistinguishability �� in�nite energy ��in�nite system ��in�nite volume ��infrared bounds ��


infrared catastrophe ��instability and boundary conditions ��integrating factor ��intensive quantity ��interaction potential ��interaction range �interaction strength �interesting case ��intrinsic pulsation ��invariant distribution ��irreversibility �� iterated logarithm ��

j

joint probability ��

k

kinetic energy �� kinetic theory ��

l

large deviation ��large matrix ��large system ��latent heat ��lattice gas ��lattice model ��lattice spin system ��law of conservation of di�culties ��law of large numbers ��law of motion �� least constraint principle �� limit of GradBoltzmann ��linear operator ��liquid gas transition ��local distribution� def ��local observable �� local observable� def ��locally �nite con�gurations� def ��long range order ��longrange order ��longrange order� def ��

m

macroscopic equilibrium state �� macroscopic equilibrium �� macroscopic observable �� macroscopic states ��macroscopically negligible ��maxwelliam distribution ��maxwellian momentum distribution ��

mean �eld theory �� mean �eld �� mean free path �� measurable function� def ��measurable set� def ��measure� def ��metal ��metaphysics ��metastability �� microcanonical absolute temperature ��microcanonical distribution ��microcanonical ensemble �� microcanonical entropy �� microcanonical internal energy ��microcanonical partition function ��microcanonical pressure ��microcanonical speci�c volume ��microscopic con�guration �� microscopic dynamics �� microscopic state �� minimal constraint principle�� minus boundary condition ��mixing� topological ��mixture of phases ��model of thermodinamics ��model of thermodynamics �� mole �molecular chaos �� monoatomic gas ��monocyclic system �� monocyclic �� monocyclicity ��monode �� motion reversibility �multifractal distribution ��multiple collision ��

n

nearest neighbor ��neon ��no regret ��noble picture ��noise ��non equivalence of ensembles �� non ergodicity ��normal conditions �normal mode �� nuclear size ��number of cells ��numerical density ��numerical experiment �numerical simulation ��

o


observable �� occupation number ��open boundary condition �� optimistic viewpoint �order of transition ��order parameter� def ��orthode �� orthodic ensemble ��orthodic �� orthodicity in �nite volume ��orthodicity �� oscillators chain ��oscillatory degree of freedom ��

p

pair correlation ��pair interaction �paradox of Zeno �paradox �� partition function �� partition of phase space ��partition sum ��perfect gas ��perfect wall boundary condition ��periodic boundary condition �� periodic boundary �� periodic orbit �� perpetual motion ��pfa�an ��phase coexistence ��phase separation line ��phase separation ��phase space contraction ��phase space �� phase transition�� phase transition� def �� phase� def ��physical grounds ��plus boundary condition ��point process ��positive de�nite potential ��positive type potential ��pot ��potential energy �� potential of Kac ��pressure ensemble �� pressure ��principle of D Alembert ��principle of Gauss ��principle of Ruelle ��principle of equipartition ��principle of indetermination �principle of least action ��

principle of least constraint ��principle of su�cient reason ��probability distribution� def ��process ��propagator ��proper frequencies ��pure phase �� pure phase� def ��

q

quadrature ��quantum e ects ��quantum grand canonical ensemble ��quantum mechanics �� quantum microcanonical ensemble ��quantum statistical mechanics ��quasiergodic hypothesis ��

r

radiating cavity �� radiation distribution ��radiation ��range of interaction �� rare�ed gas ��reciprocity of Onsager ��rectangular cell ��recurrence time �� recurrence ��renormalization group ��repulsive potential ��reversibility paradox ��reversibility �� reversible representation ��reversible system ��reversible thermostat �� rigid sphere �� roughening transition ��roundo error �ruler and compass ��

s

scale invariance ��scale of time �second law �� second principle ��sense of Fisher ��set �measurable� def ��simplex ��sinister axiom ��six vertex model ��six vertex model� solution ��small system ��soluble model ��speci�c heat ��


spectral theorem ��spin con�guration ��spin system ��spontaneous magnetization ��spontaneous polarization ��spontaneous symmetry breakdown ��stability of a potential �� stability of a potential� def ��stability of matter ��stability �� stable manifold ��stable manifold� global def ��star �� stationary distribution �� statistical ensemble �� statistical ensembles equivalence ��statistics of BoseEinstein ��statistics of motion ��statistics of motion� def ��statistics ��statistics� quantum symmetry ��stochastic process ��stochastic process� def �� stochastic thermostat ��stock market �� stosszahlansatz �� straight line ��strange idea ��subshift of �nite type ��superstability �� superstability� def ��surface tension �� symbolic dynamics �� symmetry and phase transitions ��symmetry of time reversal ��symmetry ��symmetry� def ��

t

tautology ��temperature �� temperedness of a potential ��temperedness of a potential� def ��temperedness �� theorem H of Ruelle ��theorem H ��theorem of Anosov �� theorem of Lanford �� theorem of Liouville �� theorem of MerminWagner ��theorem of heat ��theorem of van Hove ��theory of speculation ��thermodynamic formalism �� thermodynamic limit �� thermodynamics model ��thermostat ��

third law ��time average �� time of �ight ��time reversal �� time scale �� timing event ��topological mixing ��total energy �trace of matrix ��tranformation of BogoliubovValatin ��transfer matrix ��transfer matrix� def ��transition matrix ��transition matrix� mixing ��transition matrix� transitive ��transitive Anosov system ��translation invariance ��translation of an observable ��translational degree of freedom ��triviality theorem ��typical cell �

u

ultraviolet catastrophe ��universal constant ��universal law ��universe ��unstable manifold ��unstable manifold� global def ��upsetting contradiction ��

v

van Hove s theorem ��van t.Ho s law ��variational principle �� vertex model �V ��vertex model �V ��vertex models ��virial series �� virial theorem ��viscosity �� volume preserving map �

w

wall potential ��water ��waterfall ��wave functions ��weight ��white noise �� wind tree model ��wrong prediction ��

z

�� Citations index

zzz ��Citations index

�AA�� ACG�� AGM�� AL�� AM�� AR�� AW�� Ab�� Ab��a� ��Ab��b� ��Ab�� Ai�� An�� An�� As�� BC�� BCS�� BF�� BG�� BG�� BGG�� BGG�� BGG�� BGJS�� BGJS�� BH�� BJS�� BK�� BLPO�� BR�� BS�� Ba�� Ba�� Ba�� Be�� Be�� Be�� Be�� Be�� Bo�� Bo�� Bo�� Bo�� Bo�� Bo��a� ��Bo��b� �� Bo��c� ��Bo�� Bo�� Bo�� Bo�� Bo�� Bo�� Br��

Citations index ��

�Br�� Br�� Br�� CC�� CCO�� CELS�� CF�� CL�� Ca�� Ca�� Ce�� Cl�� Co�� Co�� DKS�� DL�� DLS�� DM�� DM�� DPH�� Do�� Do��a� �� Do��b� ��Do��c� �� Do�� Do�� Dr�� Du�� Dy�� ECM�� ECM�� EE�� EG�� EGM�� EM�� ES�� ES�� Ei�� Ev�� FKG�� FPU�� FR�� FS�� Fe�� Fe�� Fe�� Fi�� Fi�� Fi�� Fi��a� ��Fi��b� ��Fi��c� ��Fi�� Fr�� Fr�� Fr�� G��b� ��GC�� GG�� GH��

�GHM�� GM��a� ��GM��b� �� GMM�� GMR�� GR�� GS�� GV�� Ga�� Ga��a� ��Ga��b� ��Ga�� Ga�� Ga�� Ga��a� �� Ga��b� ��Ga��c� �� Ga��d� ��Ga��a� �� Ga��b� �� Ga��c� �� Ga��a� �� Ga��b� �� Ga��a� ��Ga��b� �� Ga��c� �� Ga��d� �� Ga��e� ��Ge�� Ge�� Gi�� Gi��b� ��Gi�� Gi�� Gr�� Gr�� Gr�� HHP�� HKU�� HS�� He��a� �� He��b� �� Hi�� Hi�� Hu�� IM�� Is�� Ja�� Jo�� KM�� KM�� KO�� KO�� KP�� KSiS�� KW�� Ka�� Ka�� Ka��

�� Citations index

�Ka�� Ka�� Kl�� Kl�� Kl�� Kr�� Ku�� Ku�� Ku�� LL�� LL�� LL�� LLP�� LM�� LMMRS�� LMP�� LP�� LP�� LP�� LR�� LS�� LS�� LS�� LT�� LV�� LW�� LY�� LY�� La�� La�� La�� La�� Le�� Li��a� �� Li��b� �� Li��c� �� Li�� Li�� ML�� MLP�� MM�� MMR�� MMRS�� MOS�� MP�� MR�� MR�� MR�� MR��b� ��MS�� MS�� MTW�� MW�� MW�� MW�� Ma�� Ma�� Ma�� Ma��

�Ma�� Me�� Me�� Mi�� Mi�� Mo�� NM�� Ne�� On�� Or�� Or�� PS�� PV�� PV�� Pe�� Pe�� Pe�� Po�� RM�� RT�� RT�� Ru�� Ru�� Ru�� Ru��a� ��Ru��b� ��Ru��b� �� Ru�� Ru�� Ru��a� �� Ru��b� �� Ru��c� �� Ru�� Ru��a� ��Ru��b� ��Ru�� Ru�� Ru�� Ru��a� �� Ru��a� ��Ru��b� ��Ru��c� �� SJ�� SML�� Sc�� Si�� Si�� Si�� Si�� Si�� Sp�� Sp�� Su�� Su�� TF�� Th�� UF�� UO�� Ul��

Citations index ��

�VB�� VB�� VH�� VK�� VN�� VN�� VN�� VP�� VS�� VW�� VW�� WF�� WL�� WL�� WMTB�� Wa�� We�� Wi�� Wi�� YY�� Ya�� Za��

Statistical Mechanics by Gallavotti

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