Chapter One Introduction - Princeton Universityassets.press.princeton.edu/chapters/s17188.pdf ·...

Chapter One

Introduction

11 An Overview of Classical Thermodynamics

Energy is a concept that underlies our understanding of all physicalphenomena and is a measure of the ability of a dynamical system to producechanges (motion) in its own system state as well as changes in the systemstates of its surroundings Thermodynamics is a physical branch of sciencethat deals with laws governing energy flow from one body to another andenergy transformations from one form to another These energy flow lawsare captured by the fundamental principles known as the first and secondlaws of thermodynamics The first law of thermodynamics gives a preciseformulation of the equivalence between heat (ie the transferring of energyvia temperature gradients) and work (ie the transferring of energy intocoherent motion) and states that among all system transformations thenet system energy is conserved Hence energy cannot be created out ofnothing and cannot be destroyed it can merely be transferred from oneform to another

The law of conservation of energy is not a mathematical truth butrather the consequence of an immeasurable culmination of observations overthe chronicle of our civilization and is a fundamental axiom of the scienceof heat The first law does not tell us whether any particular process canactually occur that is it does not restrict the ability to convert work intoheat or heat into work except that energy must be conserved in the processThe second law of thermodynamics asserts that while the system energy isalways conserved it will be degraded to a point where it cannot produceany useful work More specifically for any cyclic process that is shieldedfrom heat exchange with its environment it is impossible to extract workfrom heat without at the same time discarding some heat giving rise to anincreasing quantity known as entropy

While energy describes the state of a dynamical system entropy is ameasure of the quality of that energy reflecting changes in the status quoof the system and is associated with disorder and the amount of wasted

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For general queries contact webmasterpressprincetonedu

2 CHAPTER 1

energy in a dynamical (energy) transformation from one state (form) toanother Since the system entropy increases the entropy of a dynamicalsystem tends to a maximum and thus time as determined by system entropyincrease [299 392 476] flows in one direction only Even though entropy isa physical property of matter that is not directly observable it permeatesthe whole of nature regulating the arrow of time and is responsible forthe enfeeblement and eventual demise of the universe12 While the laws ofthermodynamics form the foundation to basic engineering systems chemicalreaction systems nuclear reactions cosmology and our expanding universemany mathematicians and scientists have expressed concerns about thecompleteness and clarity of the different expositions of thermodynamics overits long and tortuous history see [697996172184342440447455]

Since the specific motion of every molecule of a thermodynamic systemis impossible to predict a macroscopic model of the system is typicallyused with appropriate macroscopic states that include pressure volumetemperature internal energy and entropy among others One of thekey criticisms of the macroscopic viewpoint of thermodynamics known asclassical thermodynamics is the inability of the model to provide enoughdetail of how the system really evolves that is it is lacking a kineticmechanism for describing the behavior of heat and work energy

In developing a kinetic model for heat and dynamical energy athermodynamically consistent energy flow model should ensure that thesystem energy can be modeled by a diffusion equation in the form ofa parabolic partial differential equation or a divergence structure first-order hyperbolic partial differential equation arising in models involvingconservation laws Such systems are infinite-dimensional and hence finite-dimensional approximations are of very high order giving rise to large-scaledynamical systems with macroscopic energy transfer dynamics Since energyis a fundamental concept in the analysis of large-scale dynamical systemsand heat (energy in transition) is a fundamental concept of thermodynamicsinvolving the capacity of hot bodies (more energetic subsystems) to producework thermodynamics is a theory of large-scale dynamical systems

1Many natural philosophers have associated this ravaging irrecoverability in connection to thesecond law of thermodynamics with an eschatological terminus of the universe Namely thecreation of a certain degree of life and order in the universe is inevitably coupled with an evengreater degree of death and disorder A convincing proof of this bold claim has however neverbeen given

2The earliest perception of irreversibility of nature and the universe along with timersquos arrowwas postulated by the ancient Greek philosopher Herakleitos (sim 535ndashsim 475 BC) Herakleitosrsquoprofound statements Everything is in a state of flux and nothing is stationary and Man cannotstep into the same river twice because neither the man nor the river is the same created thefoundation for all other speculation on metaphysics and physics The idea that the universe isin constant change and that there is an underlying order to this changemdashthe Logosmdashpostulatesthe very existence of entropy as a physical property of matter permeating all of nature and theuniverse



INTRODUCTION 3

High-dimensional dynamical systems can arise from both macroscopicand microscopic points of view Microscopic thermodynamic models canhave the form of a distributed-parameter model or a large-scale system modelcomprised of a large number of interconnected Hamiltonian subsystems Forexample in a crystalline solid every molecule in a lattice can be viewed asan undamped vibrational mode comprising a distributed-parameter modelin the form of a second-order hyperbolic partial differential equation Incontrast to macroscopic models involving the evolution of global quantities(eg energy temperature entropy) microscopic models are based uponthe modeling of local quantities that describe the atoms and molecules thatmake up the system and their speeds energies masses angular momentabehavior during collisions etc The mathematical formulations based onthese quantities form the basis of statistical mechanics

Thermodynamics based on statistical mechanics is known as statisticalthermodynamics and involves the mechanics of an ensemble of many particles(atoms or molecules) wherein the detailed description of the system stateloses importance and only average properties of large numbers of particlesare considered Since microscopic details are obscured on the macroscopiclevel it is appropriate to view a macroscopic model as an inherent modelof uncertainty However for a thermodynamic system the macroscopic andmicroscopic quantities are related since they are simply different ways ofdescribing the same phenomena Thus if the global macroscopic quantitiescan be expressed in terms of the local microscopic quantities then the laws ofthermodynamics could be described in the language of statistical mechanics

This interweaving of the microscopic and macroscopic points of viewleads to diffusion being a natural consequence of dimensionality and henceuncertainty on the microscopic level despite the fact that there is nouncertainty about the diffusion process per se Thus even though as alimiting case a hyperbolic partial differential equation purports to model aninfinite number of modes in reality much of the modal information (egposition velocity energies) is only poorly known and hence such modelsare largely idealizations With increased dimensionality comes an increasein uncertainty leading to a greater reliance on macroscopic quantities so thatthe system model becomes more diffusive in character

Thermodynamics was spawned from the desire to design and buildefficient heat engines and it quickly spread to speculations about theuniverse upon the discovery of entropy as a fundamental physical propertyof matter The theory of classical thermodynamics was predominantlydeveloped by Carnot Clausius Kelvin Planck Gibbs and Caratheodory3

3The theory of classical thermodynamics has also been developed over the last one and a halfcenturies by many other researchers Notable contributions include the work of Maxwell Rankine



4 CHAPTER 1

and its laws have become one of the most firmly established scientificachievements ever accomplished The pioneering work of Carnot [80] was thefirst to establish the impossibility of a perpetuum mobile of the second kind4

by constructing a cyclical process (now known as the Carnot cycle) involvingfour thermodynamically reversible processes operating between two heatreservoirs at different temperatures and showing that it is impossible toextract work from heat without at the same time discarding some heat

Carnotrsquos main assumption (now known as Carnotrsquos principle) was thatit is impossible to perform an arbitrarily often repeatable cycle whose onlyeffect is to produce an unlimited amount of positive work In particularCarnot showed that the efficiency of a reversible cycle5mdashthat is the ratio ofthe total work produced during the cycle and the amount of heat transferredfrom a boiler (furnace) to a cooler (refrigerator)mdashis bounded by a universalmaximum and this maximum is a function only of the temperatures of theboiler and the cooler and not of the nature of the working substance

Both heat reservoirs (ie furnace and refrigerator) are assumed tohave an infinite source of heat so that their state is unchanged by theirheat exchange with the engine (ie the device that performs the cycle)and hence the engine is capable of repeating the cycle arbitrarily oftenCarnotrsquos result (now known as Carnotrsquos theorem) was remarkably arrivedat using the erroneous concept that heat is an indestructible substancethat is the caloric theory of heat6 This theory of heat was proposed byLavoisier and influenced by experiments due to Black involving thermalproperties of materials The theory was based on the incorrect assertionthat the temperature of a body was determined by the amount of caloricthat it contained an imponderable indestructible and highly elastic fluidthat surrounded all matter and whose self-repulsive nature was responsiblefor thermal expansion

Different notions of the conservation of energy can be traced back tothe ancient Greek philosophers Thales (sim 624ndashsim 546 BC) Herakleitos (sim535ndashsim 475 BC) and Empedocles (sim 490ndashsim 430 BC) Herakleitos postulates

Reech Clapeyron Bridgman Kestin Meixner and Giles4A perpetuum mobile of the second kind is a cyclic device that would continuously extract

heat from the environment and completely convert it into mechanical work Since such a machinewould not create energy it would not violate the first law of thermodynamics In contrast amachine that creates its own energy and thus violates the first law is called a perpetuum mobileof the first kind

5Carnot never used the terms reversible and irreversible cycles but rather cycles that areperformed in an inverse direction and order [319 p 11] The term reversible was first introducedby Kelvin [437] wherein the cycle can be run backwards

6After Carnotrsquos death several articles were discovered wherein he had expressed doubt aboutthe caloric theory of heat (ie the conservation of heat) However these articles were not publisheduntil the late 1870s and as such did not influence Clausius in rejecting the caloric theory of heatand deriving Carnotrsquos results using the energy equivalence principle of Mayer and Joule



INTRODUCTION 5

that nothing in nature can be created out of nothing and nothing thatdisappears ceases to exist7 whereas Empedocles asserts that nothing comesto be or perishes in nature8 The mechanical equivalence principle of heatand work energy in its modern form however was developed by manyscientists in the nineteenth century Notable contributions include the workof Mayer Joule Thomson (Lord Kelvin) Thompson (Count Rumford)Helmholtz Clausius Maxwell and Planck

Even though many scientists are credited with the law of conservationof energy it was first discovered independently by Mayer and Joule Mayermdasha surgeonmdashwas the first to state the mechanical equivalence of heat andwork energy in its modern form after noticing that his patientsrsquo bloodin the tropics was a deeper red leading him to deduce that they wereconsuming less oxygen and hence less energy in order to maintain theirbody temperature in a hotter climate This observation in slower humanmetabolism along with the link between the bodyrsquos heat release and thechemical energy released by the combustion of oxygen led Mayer to thediscovery that heat and mechanical work are interchangeable

Joule was the first to provide a series of decisive quantitative studiesin the 1840s showing the equivalence between heat and mechanical workSpecifically he showed that if a thermally isolated system is driven from aninitial state to a final state then the work done is only a function of theinitial and final equilibrium states and is not dependent on the intermediatestates or the mechanism doing the work This path independence propertyalong with the irrelevancy of the method by which the work was done ledto the definition of the internal energy function as a new thermodynamiccoordinate characterizing the quantity of energy or state of a thermodynamicsystem In other words heat or work do not contribute separately to theinternal energy function only the sum of the two matters

Using a macroscopic approach and building on the work of CarnotClausius [87ndash90] was the first to introduce the notion of entropy as a physicalproperty of matter and establish the two main laws of thermodynamicsinvolving conservation of energy and nonconservation of entropy9 Specifi-cally using conservation of energy principles Clausius showed that Carnotrsquosprinciple is valid Furthermore Clausius postulated that it is impossible to

7Μὲν οὗν φησιν εἷναι τὸ πᾶν διαιρετὸν ἀδιαίρετον γενητὸν ἀγένητον θνητὸν ἀθάνατον λὸγον

αίῶνα πατέρα υίὸν ἐστίν ἕν πάντα εἷναι

8Φύσις ουδενός εστίν εόντων αλλά μόνον μίξις τε διάλλαξίς τε μιγέντων εστί φύσις δrsquo επί τοις

ονομάζεται ανθρώποισινmdashThere is no genesis with regard to any of the things in nature but rathera blending and alteration of the mixed elements man however uses the word nature to namethese events

9Clausius succinctly expressed the first and second laws of thermodynamics as ldquoDie energieder Welt ist konstant und die entropie der Welt strebt einem maximum zurdquo Namely the energyof the Universe is constant and the entropy of the Universe tends to a maximum



6 CHAPTER 1

perform a cyclic system transformation whose only effect is to transfer heatfrom a body at a given temperature to a body at a higher temperatureFrom this postulate Clausius established the second law of thermodynamicsas a statement about entropy increase for adiabatically isolated systems (iesystems with no heat exchange with the environment)

From this statement Clausius goes on to state what have becomeknown as the most controversial words in the history of thermodynamicsand perhaps all of science namely the entropy of the universe is tending toa maximum and the total state of the universe will inevitably approach alimiting state Clausiusrsquo second law decrees that the usable energy in theuniverse is locked toward a path of degeneration sliding toward a state ofquietus The fact that the entropy of the universe is a thermodynamicallyundefined concept led to serious criticism of Clausiusrsquo grand universalgeneralizations by many of his contemporaries as well as numerous scientistsnatural philosophers and theologians who followed

Clausiusrsquo concept of the universe approaching a limiting state wasinadvertently based on an analogy between a universe and a finite adiabat-ically isolated system possessing a finite energy content His eschatologicalconclusions are far from obvious for complex dynamical systems withdynamical states far from equilibrium and involving processes beyond asimple exchange of heat and mechanical work It is not clear where the heatabsorbed by the system if that system is the universe needed to define thechange in entropy between two system states comes from Nor is it clearwhether an infinite and endlessly expanding universe governed by the theoryof general relativity has a final equilibrium state

An additional caveat is the delineation of energy conservation whenchanges in the curvature of spacetime need to be accounted for Inthis case the energy density tensor in Einsteinrsquos field equations is onlycovariantly conserved (ie locally conserved in free-falling coordinates)since it does not account for gravitational energymdashan unsolved problemin the general theory of relativity In particular conservation of energyand momentum laws wherein a global time coordinate does not exist hasled to one of the fundamental problems in general relativity Specificallyin general relativity involving a curved spacetime (ie a semi-Riemannianspacetime) the action of the gravitational field is invariant with respect toarbitrary coordinate transformations in semi-Riemannian spacetime with anonvanishing Jacobian containing a large number of Lie groups

In this case it follows from Noetherrsquos theorem [341]10 which derives

10Many conservation laws are a special case of Noetherrsquos theorem which states that forevery one-parameter group of diffeomorphisms defined on an abstract geometrical space (eg



INTRODUCTION 7

conserved quantities from symmetries and states that every differentiablesymmetry of a dynamical action has a corresponding conservation lawthat a large number of conservation laws exist some of which are notphysical In contrast the classical conservation laws of physics which followfrom time translation invariance are determined by an invariant propertyunder a particular Lie group with the conserved quantities correspondingto the parameters of the group And in special relativity conservation ofenergy and momentum is a consequence of invariance through the actionof infinitesimal translation of the inertial coordinates wherein the Lorentztransformation relates inertial systems in different inertial coordinates

In general relativity the momentum-energy equivalence principle holdsonly in a local region of spacetimemdasha flat or Minkowski spacetime Inother words the energy-momentum conservation laws in gravitation theoryinvolve gauge conservation laws with local time transformations wherein thecovariant transformation generators are canonical horizontal prolongationsof vector fields on a world manifold11 and hence in a curved spacetime theredoes not exist a global energy-momentum conservation law Neverthelessthe law of conservation of energy is as close to an absolute truth as ourincomprehensible universe will allow us to deduce In his later work [89]Clausius remitted his famous claim that the entropy of the universe istending to a maximum

In parallel research Kelvin [240 438] developed similar and in somecases identical results as Clausius with the main difference being theabsence of the concept of entropy Kelvinrsquos main view of thermodynamicswas that of a universal irreversibility of physical phenomena occurring innature Kelvin further postulated that it is impossible to perform a cyclicsystem transformation whose only effect is to transform into work heatfrom a source that is at the same temperature throughout12 Withoutany supporting mathematical arguments Kelvin goes on to state that theuniverse is heading toward a state of eternal rest wherein all life on Earth in

configuration manifolds Minkowski space Riemannian space) of a Hamiltonian dynamical systemthat preserves a Hamiltonian function there exist first integrals of motion In other words thealgebra of the group is the set of all Hamiltonian systems whose Hamiltonian functions are thefirst integrals of motion of the original Hamiltonian system

11A world manifold is a four-dimensional orientable noncompact parallelizable manifold thatadmits a semi-Riemannian metric and a spin structure Gravitation theories are formulated ontensor bundles that admit canonical horizontal prolongations on a vector field defined on a worldmanifold These prolongations are generators of covariant transformations whose vector fieldcomponents play the role of gauge parameters Hence in general relativity the energy-momentumflow collapses to a superpotential of a world vector field defined on a world manifold admittinggauge parameters

12In the case of thermodynamic systems with positive absolute temperatures Kelvinrsquos postulatecan be shown to be equivalent to Clausiusrsquo postulate However many textbooks erroneously showthis equivalence without the assumption of positive absolute temperatures Physical systemspossessing a small number of energy levels (ie an inverted Boltzmann energy distribution) withnegative absolute temperatures are discussed in [121260264307373]



8 CHAPTER 1

the distant future shall perish This claim by Kelvin involving a universaltendency toward dissipation has come to be known as the heat death of theuniverse

The universal tendency toward dissipation and the heat death of theuniverse were expressed long before Kelvin by the ancient Greek philosophersHerakleitos and Leukippos (sim480ndashsim420 BC) In particular Herakleitosstates that this universe which is the same everywhere and which noone god or man has made existed exists and will continue to exist asan eternal source of energy set on fire by its own natural laws and willdissipate under its own laws13 Herakleitosrsquo profound statement createdthe foundation for all metaphysics and physics and marks the beginning ofscience postulating the big bang theory as the origin of the universe as wellas the heat death of the universe A century after Herakleitos Leukipposdeclared that from its genesis the cosmos has spawned multitudinous worldsthat evolve in accordance to a supreme law that is responsible for theirexpansion enfeeblement and eventual demise14

Building on the work of Clausius and Kelvin Planck [358362] refinedthe formulation of classical thermodynamics From 1897 to 1964 Planckrsquostreatise [358] underwent eleven editions and is considered the definitiveexposition on classical thermodynamics Nevertheless these editions haveseveral inconsistencies regarding key notions and definitions of reversible andirreversible processes15 Planckrsquos main theme of thermodynamics is thatentropy increase is a necessary and sufficient condition for irreversibilityWithout any proof (mathematical or otherwise) he goes on to concludethat every dynamical system in nature evolves in such a way that the totalentropy of all of its parts increases In the case of reversible processes heconcludes that the total entropy remains constant

Unlike Clausiusrsquo entropy increase conclusion Planckrsquos increase entropyprinciple is not restricted to adiabatically isolated dynamical systemsRather it applies to all system transformations wherein the initial statesof any exogenous system belonging to the environment and coupled tothe transformed dynamical system return to their initial condition Itis important to note that Planckrsquos entire formulation is restricted tohomogeneous systems for which the thermodynamical state is characterizedby two thermodynamic state variables that is a fluid His formulation ofentropy and the second law is not defined for more complex systems that

13Κόσμον (τόνδε) τὸν αὐτὸν ἁπάντων οὔτε τις θεῶν οὔτε ἀνθρώπων ἐποίησεν ἀλλ΄ ᾖν ἀεὶ καὶ

ἔστιν καὶ ἔσται πῦρ ἀείζωον ἁπτόμενον μέτρα καὶ ἀποσβεννύμενον μέτρα

14Είναι τε ώσπερ γενέσεις κόσμου ούτω καί αυξήσεις καί φθίσεις καί φθοράς κατά τινά ανάγκην

15Truesdell [445 p 328] characterizes the work as a ldquogloomy murkrdquo whereas Khinchin [245 p142] declares it an ldquoaggregate of logical and mathematical errors superimposed on a generalconfusion in the definition of the basic quantitiesrdquo



INTRODUCTION 9

are not in equilibrium and in an environment that is more complex than onecomprising a system of ideal gases

Unlike the work of Clausius Kelvin and Planck involving cyclicalsystem transformations the work of Gibbs [163] involves system equilibriumstates Specifically Gibbs assumes a thermodynamic state of a systeminvolving pressure volume temperature energy and entropy among othersand proposes that an isolated system16 (ie a system with no energyexchange with the environment) is in equilibrium if and only if in all possiblevariations of the state of the system that do not alter its energy the variationof the system entropy is negative semidefinite Thus the system entropy ismaximized at the system equilibrium

Gibbs also proposed a complementary formulation of his maximumentropy principle involving a principle of minimal energy Namely for anequilibrium of any isolated system it is necessary and sufficient that in allpossible variations of the state of the system that do not alter its entropythe variation of its energy shall either vanish or be positive Hence thesystem energy is minimized at the system equilibrium

Gibbsrsquo principles give necessary and sufficient conditions for a ther-modynamically stable equilibrium and should be viewed as variationalprinciples defining admissible (ie stable) equilibrium states Thus theydo not provide any information about the dynamical state of the system asa function of time nor any conclusions regarding entropy increase or energydecrease in a dynamical system transformation

Caratheodory [76 77] was the first to give a rigorous axiomaticmathematical framework for thermodynamics In particular using anequilibrium thermodynamic theory Caratheodory assumes a state spaceendowed with a Euclidean topology and defines the equilibrium state ofthe system using thermal and deformation coordinates Next he definesan adiabatic accessibility relation wherein a reachability condition of anadiabatic process17 is used such that an empirical statement of the secondlaw characterizes a mathematical structure for an abstract state spaceEven though the topology in Caratheodoryrsquos thermodynamic frameworkis induced on Rn (the space of n-tuples of reals) by taking the metricto be the Euclidean distance function and constructing the correspondingneighborhoods the metrical properties of the state space do not play a role inhis theory as there is no preference for a particular set of system coordinates

16Gibbsrsquo principle is weaker than Clausiusrsquo principle leading to the second law involving entropyincrease since it holds for the more restrictive case of isolated systems

17Caratheodoryrsquos definition of an adiabatic process is nonstandard and involves transformationsthat take place while the system remains in an adiabatic container this allowed him to avoidintroducing heat as a primitive variable (ie axiomatic element) For details see [7677]



10 CHAPTER 1

Caratheodoryrsquos postulate for the second law states that in every openneighborhood of any equilibrium state of a system there exist equilibriumstates such that for some second open neighborhood contained in thefirst neighborhood all the equilibrium states in the second neighborhoodcannot be reached by adiabatic processes from equilibrium states in the firstneighborhood From this postulate Caratheodory goes on to show that for aspecial class of systems which he called simple systems there exists a locallydefined entropy and an absolute temperature on the state space for everysimple system equilibrium state In other words Caratheodoryrsquos postulateestablishes the existence of an integrating factor for the heat transfer in aninfinitesimal reversible process for a thermodynamic system of an arbitrarynumber of degrees of freedom that makes entropy an exact (ie total)differential

Unlike the work of Clausius Kelvin Planck and Gibbs Caratheodoryprovides a topological formalism for the theory of thermodynamics whichelevates the subject to the level of other theories of modern physicsSpecifically the empirical statement of the second law is replaced by anabstract state space formalism wherein the second law is converted into alocal topological property endowed with a Euclidean metric This parallelsthe development of relativity theory wherein Einsteinrsquos original specialtheory started from empirical principlesmdasheg the velocity of light in freespace is invariant in all inertial framesmdashand then was replaced by an abstractgeometrical structure the Minkowski spacetime wherein the empiricalprinciples are converted into local topological properties of the Minkowskimetric However one of the key limitations of Caratheodoryrsquos work is thathis principle is too weak in establishing the existence of a global entropyfunction

Adopting a microscopic viewpoint Boltzmann [58] was the first to givea probabilistic interpretation of entropy involving different configurationsof molecular motion of the microscopic dynamics Specifically Boltzmannreinterpreted thermodynamics in terms of molecules and atoms by relatingthe mechanical behavior of individual atoms with their thermodynamicbehavior by suitably averaging properties of the individual atoms In par-ticular even though individually each molecule and atom obeys Newtonianmechanics he used the science of statistical mechanics to bridge between themicroscopic details and the macroscopic behavior to try to find a mechanicalunderpinning of the second law

Even though Boltzmann was the first to give a probabilistic interpre-tation of entropy as a measure of the disorder of a physical system involvingthe evolution toward the largest number of possible configurations of thesystemrsquos states relative to its ordered initial state Maxwell was the first



INTRODUCTION 11

to use statistical methods to understand the behavior of the kinetic theoryof gases In particular he postulated that it is not necessary to track thepositions and velocities of each individual atom and molecule but rather itsuffices to know their position and velocity distributions concluding that thesecond law is merely statistical His distribution law for the kinetic theoryof gases describes an exponential function giving the statistical distributionof the velocities and energies of the gas molecules at thermal equilibriumand provides an agreement with classical (ie nonquantum) mechanics

Although the Maxwell speed distribution law agrees remarkably wellwith observations for an assembly of weakly interacting particles that aredistinguishable it fails for indistinguishable (ie identical) particles at highdensities In these regions speed distributions predicated on the principles ofquantum physics must be used namely the Fermi-Dirac and Bose-Einsteindistributions In this case the Maxwell statistics closely agree with theBose-Einstein statistics for bosons (photons α-particles and all nuclei withan even mass number) and the Fermi-Dirac statistics for fermions (electronsprotons and neutrons)

Boltzmann however further showed that even though individualatoms are assumed to obey the laws of Newtonian mechanics by suitablyaveraging over the velocity distributions of these atoms the microscopic(mechanical) behavior of atoms and molecules produced effects visible ona macroscopic (thermodynamic) scale He goes on to argue that Clausiusrsquothermodynamic entropy (a macroscopic quantity) is proportional to thelogarithm of the probability that a system will exist in the state it isin relative to all possible states it could be in Thus the entropy of athermodynamic system state (macrostate) corresponds to the degree ofuncertainty about the actual system mechanical state (microstate) whenonly the thermodynamic system state (macrostate) is known Hence theessence of Boltzmann thermodynamics is that thermodynamic systems witha constant energy level will evolve from a less probable state to a moreprobable state with the equilibrium system state corresponding to a stateof maximum entropy (ie highest probability)

Interestingly Boltzmannrsquos original thinking on the subject of entropyincrease involved nondecreasing of entropy as an absolute certainty and notjust as a statistical certainty In the 1870s and 1880s his thoughts on thismatter underwent significant refinements and shifted to a probabilistic view-point after interactions with Maxwell Kelvin Loschmidt Gibbs PoincareBurbury and Zermelo all of whom criticized his original formulation

In statistical thermodynamics the Boltzmann entropy formula relatesthe entropy S of an ideal gas to the number of distinct microstates



12 CHAPTER 1

W corresponding to a given macrostate as S = k logeW where kis the Boltzmann constant18 Thus the Boltzmann entropy gives thenumber of different microscopic configurations of a systemrsquos states thatleave its macroscopic appearance unchanged and hence it connects theClausius entropy a macroscopic thermodynamic quantity to probability amicroscopic statistical quantity

Even though Boltzmann was the first to link the thermodynamicentropy of a macrostate for some probability distribution of all possiblemicrostates generated by different positions and momenta of various gasmolecules [57] it was Planck who first stated (without proof) this entropyformula in his work on blackbody radiation [359] In addition Planck wasalso the first to introduce the precise value of the Boltzmann constant tothe formula Boltzmann merely introduced the proportional logarithmicconnection between the entropy S of an observed macroscopic state ordegree of disorder of a system to the thermodynamic probability of itsoccurrence W never introducing the constant k to the formula

To further complicate matters in his original paper [359] Planck statedthe formula without derivation or clear justification a fact that deeplytroubled Albert Einstein [130] Despite the fact that numerous physicistsconsider S = k logeW as the second most important formula of physicsmdashsecond to Einsteinrsquos E = mc2mdashfor its unquestionable success in computingthe thermodynamic entropy of isolated systems its theoretical justificationremains ambiguous and vague in most statistical thermodynamics text-books In this regard Khinchin [245 p 142] writes ldquoAll existing attemptsto give a general proof of [Boltzmannrsquos entropy formula] must be consideredas an aggregate of logical and mathematical errors superimposed on a generalconfusion in the definition of basic quantitiesrdquo

In the first half of the twentieth century the macroscopic (classical)and microscopic (statistical) interpretations of thermodynamics underwenta long and fierce debate To exacerbate matters since classical thermody-namics was formulated as a physical theory and not a mathematical theorymany scientists and mathematical physicists expressed concerns about thecompleteness and clarity of the mathematical foundation of thermodynamics

18The number of distinct microstatesW can also be regarded as the number of solutions of theSchrodinger equation for the system giving a particular energy distribution The Schrodinger waveequation describes how a quantum state of a system evolves over time The solution of the equationcharacterizes a quantum wave function whose wavelength is related to the system momentum andfrequency is related to the system energy Unlike Planckrsquos discrete quantum transition theory ofenergy when light interacts with matter Schrodingerrsquos quantum theory stipulates that quantumtransition involves vibrational changes from one form to another and these vibrational changesare continuous in space and time Furthermore if the quantum wave function is known at anygiven point in time then Schrodingerrsquos equation uniquely specifies the quantum wave function atany other moment in time making this constituent part of quantum physics fully deterministic



INTRODUCTION 13

[11 69 447] In fact many fundamental conclusions arrived at by classicalthermodynamics can be viewed as paradoxical

For example in classical thermodynamics the notion of entropy (andtemperature) is only defined for equilibrium states However the theoryconcludes that nonequilibrium states transition toward equilibrium statesas a consequence of the law of entropy increase Furthermore classicalthermodynamics is restricted to systems in equilibrium The second lawinfers that for any transformation occurring in an isolated system theentropy of the final state can never be less than the entropy of the initialstate In this context the initial and final states of the system areequilibrium states However by definition an equilibrium state is a systemstate that has the property that whenever the state of the system starts atthe equilibrium state it will remain at the equilibrium state for all futuretime unless an exogenous input acts on the system Hence the entropy ofthe system can only increase if the system is not isolated

Many aspects of classical thermodynamics are riddled with suchinconsistencies and hence it is not surprising that many formulations ofthermodynamics especially most textbook expositions poorly amalgamatephysics with rigorous mathematics Perhaps this is best eulogized in [447p 6] wherein Truesdell describes the present state of the theory ofthermodynamics as a ldquodismal swamp of obscurityrdquo In a desperate attempt totry to make sense of the writings of de Groot Mazur Casimir and Prigoginehe goes on to state that there is ldquosomething rotten in the [thermodynamic]state of the Low Countriesrdquo [447 p 134]

Brush [69 p 581] remarks that ldquoanyone who has taken a course inthermodynamics is well aware the mathematics used in proving Clausiusrsquotheorem [has] only the most tenuous relation to that known to mathe-maticiansrdquo And Born [61 p 119] admits that ldquoI tried hard to understandthe classical foundations of the two theorems as given by Clausius andKelvin but I could not find the logical and mathematical root of thesemarvelous resultsrdquo More recently Arnold [11 p 163] writes that ldquoeverymathematician knows it is impossible to understand an elementary coursein thermodynamicsrdquo

As we have outlined it is clear that there have been many differentpresentations of classical thermodynamics with varying hypotheses and con-clusions To exacerbate matters there are also many vaguely defined termsand functions that are central to thermodynamics such as entropy enthalpyfree energy quasi-static nearly in equilibrium extensive variables intensivevariables reversible irreversible etc Furthermore these functionsrsquo domainand codomain are often unspecified and their local and global existence



14 CHAPTER 1

uniqueness and regularity properties are unproven

Moreover there are no general dynamic equations of motion noordinary or partial differential equations and no general theorems providingmathematical structure and characterizing classes of solutions Rather weare asked to believe that a certain differential can be greater than somethingthat is not a differential defying the syllogism of differential calculus lineintegrals approximating adiabatic and isothermal paths result in alternativesolutions annulling the fundamental theorem of integral calculus and weare expected to settle for descriptive and unmathematical wordplay inexplaining key principles that have far-reaching consequences in engineeringscience and cosmology

Furthermore the careless and considerable differences in the defini-tions of two of the key notions of thermodynamicsmdashnamely the notionsof reversibility and irreversibilitymdashhave contributed to the widespreadconfusion and lack of clarity of the exposition of classical thermodynamicsover the past one and a half centuries For example the concept of reversibleprocesses as defined by Carnot Clausius Kelvin Planck and Caratheodoryhave very different meanings In particular Carnot never uses the termreversible but rather cycles that can be run backwards Later he added thatthese cycles should proceed slowly so that the system remains in equilibriumover the entire cycle Such system transformations are commonly referred toas quasi-static transformations in the thermodynamic literature Clausiusdefines reversible (umkehrbar) cyclic and noncyclic processes as slowlyvarying processes wherein successive states of these processes differ byinfinitesimals from the equilibrium system states Alternatively Kelvinrsquosnotions of reversibility involve the ability of a system to completely recoverits initial state from the final system state He does not limit his definition ofreversibility to cyclic processes and hence a cyclic process can be reversiblein the sense of Kelvin but irreversible in the sense of Carnot

Planck introduced several notions of reversibility His main notionof reversibility is one of complete reversibility and involves recoverabilityof the original state of the dynamical system while at the same timerestoring the environment to its original condition Unlike Clausiusrsquonotion of reversibility Kelvinrsquos and Planckrsquos notions of reversibility donot require the system to exactly retrace its original trajectory in reverseorder Caratheodoryrsquos notion of reversibility involves recoverability of thesystem state in an adiabatic process resulting in yet another definition ofthermodynamic reversibility These subtle distinctions of (ir)reversibility areoften unrecognized in the thermodynamic literature Notable exceptions tothis fact include [65448] with [448] providing an excellent exposition of therelation between irreversibility the second law of thermodynamics and the



INTRODUCTION 15

arrow of time

12 Thermodynamics and the Arrow of Time

The arrow of time19 and the second law of thermodynamics is one of the mostfamous and controversial problems in physics The controversy betweenontological time (ie a timeless universe) and the arrow of time (ie aconstantly changing universe) can be traced back to the famous dialoguesbetween the ancient Greek philosophers Parmenides20 and Herakleitos onbeing and becoming Parmenides like Einstein insisted that time is anillusion that there is nothing new and that everything is (being) and willforever be This statement is of course paradoxical since the status quochanged after Parmenides wrote his famous poem On Nature

Parmenides maintained that we all exist within spacetime and timeis a one-dimensional continuum in which all events regardless of when theyhappen from any given perspective simply are All events exist endlesslyand universally and occupy ordered points in spacetime and hence realityenvelops past present and future equally More specifically our picture ofthe universe at a given moment is identical and contains exactly the sameevents we simply have different conceptions of what exists at that momentand hence different conceptions of reality Conversely the Heraclitean fluxdoctrine maintains that nothing ever is and everything is becoming Inthis regard time gives a different ontological status of past present andfuture resulting in an ontological transition creation and actualization ofevents More specifically the unfolding of events in the flow of time havecounterparts in reality

Herakleitosrsquo aphorism is predicated on change (becoming) namelythe universe is in a constant state of flux and nothing is stationarymdashTα πάντα ρεί καί oύδέν μένει Furthermore Herakleitos goes on to statethat the universe evolves in accordance with its own laws which are theonly unchangeable things in the universe (eg universal conservation andnonconservation laws) His statements that everything is in a state of fluxmdashTα πάντα ρείmdashand that man cannot step into the same river twice because

19The phrase arrow of time was coined by Eddington in his book The Nature of the PhysicalWorld [123] and connotes the one-way direction of entropy increase educed from the second lawof thermodynamics Other phrases include the thermodynamic arrow and the entropic arrowof time Long before Eddington however philosophers and scientists addressed deep questionsabout time and its direction

20Parmenides (sim515ndashsim450 BC) maintained that there is neither time nor motion His pupilZeno of Elea (sim490ndashsim430 BC) constructed four paradoxesmdashthe dichotomy the Achilles theflying arrow and the stadiummdashto prove that motion is impossible His logic was ldquoimmeasurablysubtle and profoundrdquo and even though infinitesimal calculus provides a tool that explains Zenorsquosparadoxes the paradoxes stand at the intersection of reality and our perception of it and theyremain at the cutting edge of our understanding of space time and spacetime [316]



16 CHAPTER 1

neither the man nor the river is the samemdashΠoταμείς τoίς αυτoίς εμβαίνoμεντε καί oυκ εμβαίνoμεν είμεν τε καί oυκ είμενmdashgive the earliest perception ofirreversibility of nature and the universe along with timersquos arrow The ideathat the universe is in constant change and there is an underlying order tothis changemdashthe Logos (Λόγoς)mdashpostulates the existence of entropy as aphysical property of matter permeating the whole of nature and the universe

Herakleitosrsquo statements are completely consistent with the laws ofthermodynamics which are intimately connected to the irreversibility ofdynamical processes in nature In addition his aphorisms go beyondthe worldview of classical thermodynamics and have deep relativisticramifications to the spacetime fabric of the cosmos Specifically Herakleitosrsquoprofound statementmdashAll matter is exchanged for energy and energy for allmatter (Πυρός τε ἀνταμoιβὴ τὰ πάντα καὶ πῦρ ἁπάντων)mdashis a statement ofthe law of conservation of mass-energy and is a precursor to the principleof relativity In describing the nature of the universe Herakleitos postulatesthat nothing can be created out of nothing and nothing that disappearsceases to exist This totality of forms or mass-energy equivalence iseternal21 and unchangeable in a constantly changing universe

The arrow of time22 remains one of physicsrsquo most perplexing enigmas[122 178 220 254 297 362 377 463] Even though time is one of the mostfamiliar concepts mankind has ever encountered it is the least understoodPuzzling questions of timersquos mysteries have remained unanswered through-out the centuries23 Questions such as Where does time come from Whatwould our universe look like without time Can there be more than onedimension to time Is time truly a fundamental appurtenance woven intothe fabric of the universe or is it just a useful edifice for organizing ourperception of events Why is the concept of time hardly ever found in themost fundamental physical laws of nature and the universe Can we go backin time And if so can we change past events

Human experience perceives time flow as unidirectional the present

21It is interesting to note that despite his steadfast belief in change Herakleitos embracedthe concept of eternity as opposed to Parmenidesrsquo endless duration concept in which all eventsmaking up the universe are static and unchanging eternally occupying fixed points in a frozenimmutable future of spacetime

22Perhaps a better expression here is the geodesic arrow of time since as Einsteinrsquos theory ofrelativity shows time and space are intricately coupled and hence one cannot curve space withoutinvolving time as well Thus time has a shape that goes along with its directionality

23Plato (sim428ndashsim348 BC) writes that time was created as an image of the eternal While timeis everlasting time is the outcome of change (motion) in the universe And as night and day andmonth and the like are all part of time without the physical universe time ceases to exist Thusthe creation of the universe has spawned the arrow of timemdashΧρόνον τε γενέσθαι εἰκόνα τοῦ ἀιδίου

Κἀκεῖνον μὲν ἀεί μένειν τὴν δὲ τοῦ οὐρανοῦ φορὰν χρόνον εἶναι καὶ γὰρ νύκτα καὶ ἡμέραν καὶ μῆνα

καὶ τὰ τοιαῦτα πάντα χρόνου μέρη εἶναι Διόπερ ἄνευ τῆς τοῦ κόσμου φύσεως οὐκ εἶναι χρόνον ἅμα

γὰρ ὑπάρχειν αὐτῶ καὶ χρόνον εἶναι



INTRODUCTION 17

is forever flowing toward the future and away from a forever fixed pastMany scientists have attributed this emergence of the direction of time flowto the second law of thermodynamics due to its intimate connection to theirreversibility of dynamical processes24 In this regard thermodynamics isdisjoint from Newtonian and Hamiltonian mechanics (including Einsteinrsquosrelativistic and Schrodingerrsquos quantum extensions) since these theories areinvariant under time reversal that is they make no distinction betweenone direction of time and the other Such theories possess a time-reversalsymmetry wherein from any given moment of time the governing laws treatpast and future in exactly the same way [258]25 It is important to stress herethat time-reversal symmetry applies to dynamical processes whose reversalis allowed by the physical laws of nature not a reversal of time itself It isirrelevant whether or not the reversed dynamical process actually occurs innature it suffices that the theory allows for the reversed process to occur

The simplest notion of time-reversal symmetry is the statementwherein the physical theory in question is time-reversal symmetric in thesense that given any solution x(t) to a set of dynamic equations describingthe physical laws then x(minust) is also a solution to the dynamic equationsFor example in Newtonian mechanics this implies that there exists atransformation R(q p) such that R(q p) x(t) = x(minust) R(q p) where denotes the composition operator and x(minust) = [q(minust) minusp(minust)]T representsthe particles that pass through the same position as q(t) but in reverseorder and with reverse velocity minusp(minust) It is important to note that if thephysical laws describe the dynamics of particles in the presence of a field(eg an electromagnetic field) then the reversal of the particle velocitiesis insufficient for the equations to yield time-reversal symmetry In thiscase it is also necessary to reverse the field which can be accomplished bymodifying the transformation R accordingly

As an example of time-reversal symmetry a film run backwards of aharmonic oscillator over a full period or a planet orbiting the Sun wouldrepresent possible events In contrast a film run backwards of waterin a glass coalescing into a solid ice cube or ashes self-assembling intoa log of wood would immediately be identified as an impossible event

24In statistical thermodynamics the arrow of time is viewed as a consequence of high systemdimensionality and randomness However since in statistical thermodynamics it is not absolutelycertain that entropy increases in every dynamical process the direction of time as determined byentropy increase has only statistical certainty and not an absolute certainty Hence it cannot beconcluded from statistical thermodynamics that time has a unique direction of flow

25There is an exception to this statement involving the laws of physics describing weak nuclearforce interactions in Yang-Mills quantum fields [471] In particular in certain experimentalsituations involving high-energy atomic and subatomic collisions meson particles (K-mesons andB-mesons) exhibit time-reversal asymmetry [85] However under a combined transformationinvolving charge conjugation C which replaces the particles with their antiparticles parity Pwhich inverts the particlesrsquo positions through the origin and a time-reversal involution R whichreplaces t with minust the particlesrsquo behavior is CPR-invariant For details see [85]



18 CHAPTER 1

Over the centuries many philosophers and scientists shared the views ofa Parmenidean frozen river time theory However since the advent of thescience of thermodynamics in the nineteenth century philosophy and sciencetook a different point of view with the writings of Hegel Bergson HeideggerClausius Kelvin and Boltzmann one involving time as our existentialdimension The idea that the second law of thermodynamics provides aphysical foundation for the arrow of time has been postulated by manyauthors [123 369 377]26 However a convincing mathematical argument ofthis claim has never been given [178254448]

The complexities inherent with the afore statement are subtle andare intimately coupled with the universality of thermodynamics entropygravity and cosmology (see Section 174 and Chapter 18) A commonmisconception of the principle of the entropy increase is surmising that ifentropy increases in forward time then it necessarily decreases in backwardtime However entropy and the second law do not alter the (known) laws ofphysics in any waymdashthe laws have no temporal orientation In the absence ofa unified dynamical systems theory of thermodynamics with Newtonian andEinsteinian mechanics the second law is derivative to the physical laws ofmotion Thus since the (known) laws of nature are autonomous to temporalorientation the second law implies with identical certainty that entropyincreases both forward and backward in time from any given moment intime

This statement however is not true in general it is true only if theprimordial state of the universe did not begin in a highly ordered lowentropy state However quantum fluctuations in Higgs boson particles27

stretched out by inflation and inflationary cosmology followed by the bigbang [183] tells us that the early universe began its trajectory in a highlyordered low entropy state which allows us to educe that the entropic arrowof time is not a double-headed arrow and that the future is indeed in thedirection of increasing entropy This further establishes that the concept oftime flow directionality which almost never enters in any physical theoryis a defining marvel of thermodynamics Heat (ie energy in transition)like gravity permeates every substance in the universe and its radiationspreads to every part of spacetime However unlike gravity the directional

26Conversely one can also find many authors who maintain that the second law ofthermodynamics has nothing to do with irreversibility or the arrow of time [124 231 261]these authors largely maintain that thermodynamic irreversibility and the absence of a temporalorientation of the rest of the laws of physics are disjoint notions This is due to the fact thatclassical thermodynamics is riddled with many logical and mathematical inconsistencies withcarelessly defined notation and terms And more importantly with the notable exception of [195]a dynamical systems foundation of thermodynamics is nonexistent in the literature

27The Higgs boson is an elementary particle (ie a particle with an unknown substructure)containing matter (particle mass) and radiation (emission or transmission of energy) and is thefinest quantum constituent of the Higgs field See Chapter 17 for further details



INTRODUCTION 19

continuity of entropy and time (ie the entropic arrow of time) elevatesthermodynamics to a sui generis physical theory of nature

13 Modern Thermodynamics Information Theory and

Statistical Energy Analysis

In an attempt to generalize classical thermodynamics to nonequilibriumthermodynamics Onsager [347 348] developed reciprocity theorems forirreversible processes based on the concept of a local equilibrium thatcan be described in terms of state variables that are predicated on linearapproximations of thermodynamic equilibrium variables Onsagerrsquos theorempertains to the thermodynamics of linear systems wherein a symmetricreciprocal relation applies between forces and fluxes In particular a flowor flux of matter in thermodiffusion is caused by the force exerted by thethermal gradient Conversely a concentration gradient causes a heat flowan effect that has been experimentally verified for linear transport processesinvolving thermodiffusion thermoelectric and thermomagnetic effects

Classical irreversible thermodynamics [114 272 477] as originallydeveloped by Onsager characterizes the rate of entropy production ofirreversible processes as a sum of the product of fluxes with their associatedforces postulating a linear relationship between the fluxes and forces Thethermodynamic fluxes in the Onsager formulation include the effects ofheat conduction flow of matter (ie diffusion) mechanical dissipation(ie viscosity) and chemical reactions Well-known laws of physicsconfirm Onsagerrsquos reciprocity relationships for near equilibrium systemsFor example Fourierrsquos law of heat conduction asserts that heat flowis proportional to a temperature gradient and Fickrsquos law describes aproportional relationship between diffusion and a chemical concentrationgradient Onsagerrsquos thermodynamic theory however is only correct fornear equilibrium processes wherein a local and linear instantaneous relationbetween the fluxes and forces holds

Casimir [82] extended Onsagerrsquos principle of macroscopic reversibilityto explain the relations between irreversible processes and network theoryinvolving the coupling effects of electrical currents and resistance onentropy production The Onsager-Casimir reciprocal relations treat onlythe irreversible aspects of system processes and thus the theory is analgebraic theory that is primarily restricted to describing (time-independent)system steady states In addition the Onsager-Casimir formalism isrestricted to linear systems wherein a linearity restriction is placed onthe admissible constitutive relations between the thermodynamic forces andfluxes Another limitation of the Onsager-Casimir framework is the difficultyin providing a macroscopic description for large-scale complex dynamical



20 CHAPTER 1

systems In addition the Onsager-Casimir reciprical relations are not validon the microscopic thermodynamic level

Building on Onsagerrsquos classical irreversible thermodynamic theoryPrigogine [166 367 368] developed a thermodynamic theory of dissipativenonequilibrium structures This theory involves kinetics describing thebehavior of systems that are away from equilibrium states HoweverPrigoginersquos thermodynamics lacks functions of the system state and hencehis concept of entropy for a system away from equilibrium does not havea total differential Furthermore Prigoginersquos characterization of dissipativestructures is predicated on a linear expansion of the entropy function abouta particular equilibrium and hence is limited to the neighborhood of theequilibrium This is a severe restriction on the applicability of this theoryIn addition his entropy cannot be calculated nor determined [165 282]Moreover the theory requires that locally applied exogenous heat fluxespropagate at infinite velocities across a thermodynamic body violating bothexperimental evidence and the principle of causality To paraphrase PenrosePrigoginersquos thermodynamic theory at best should be regarded as a trial ordead end

In an attempt to extend Onsagerrsquos classical irreversible thermody-namic theory beyond a local equilibrium hypothesis extended irreversiblethermodynamics was developed in the literature [81 236] wherein inaddition to the classical thermodynamic variables dissipating fluxes areintroduced as new independent variables providing a link between classicalthermodynamics and flux dynamics These complementary thermodynamicvariables involve nonequilibrium quantities and take the form of dissipativefluxes and include heat viscous pressure matter and electric current fluxesamong others These fluxes are associated with microscopic operators ofnonequilibrium statistical mechanics and the kinetic theory of gases andeffectively describe systems with long relaxation times (eg low-temperaturesolids superfluids and viscoelastic fluids)

Even though extended irreversible thermodynamics generalizes clas-sical thermodynamics to nonequilibrium systems the complementary vari-ables are treated on the same level as the classical thermodynamic variablesand hence lack any evolution equations To compensate for this additionalrate equations are introduced for the dissipative fluxes Specifically thefluxes are selected as state variables wherein the constitutive equations ofFourier Fick Newton and Ohm are replaced by first-order time evolutionequations that include memory and nonlocal effects

However unlike the classical thermodynamic variables which satisfyconservation of mass and energy and are compatible with the second law of



INTRODUCTION 21

thermodynamics no specific criteria are specified for the evolution equationsof the dissipative fluxes Furthermore since every dissipative flux isformulated as a thermodynamic variable characterized by a single evolutionequation with the system entropy being a function of the fluxes extendedirreversible thermodynamic theories tend to be incompatible with classicalthermodynamics Specifically the theory yields different definitions fortemperature and entropy when specialized to equilibrium thermodynamicsystems

In the last half of the twentieth century thermodynamics wasreformulated as a global nonlinear field theory with the ultimate objectiveto determine the independent field variables of this theory [92 333 393446] This aspect of thermodynamics which became known as rationalthermodynamics was predicated on an entirely new axiomatic approachAs a result of this approach modern continuum thermodynamics wasdeveloped using theories from elastic materials viscous materials andmaterials with memory [91 106 107 182] The main difference betweenclassical thermodynamics and rational thermodynamics can be traced backto the fact that in rational thermodynamics the second law is not interpretedas a restriction on the transformations a system can undergo but rather asa restriction on the systemrsquos constitutive equations

Rational thermodynamics is formulated based on nonphysical interpre-tations of absolute temperature and entropy notions that are not limited tonear equilibrium states Moreover the thermodynamic system has memoryand hence the dynamic behavior of the system is determined not only by thepresent value of the thermodynamic state variables but also by the historyof their past values In addition the second law of thermodynamics isexpressed using the Clausius-Duhem inequality

Rational thermodynamics is not a thermodynamic theory in theclassical sense but rather a theory of thermomechanics of continuous mediaThis theory which is also known as modern continuum thermodynamicsabandons the concept of a local equilibrium and involves general con-servation laws (mass momentum energy) for defining a thermodynamicstate of a body using a set of postulates and constitutive functionalsThese postulates which include the principles of admissibility (ie entropyprinciple) objectivity or covariance (ie reference frame invariance) localaction (ie influence of a neighborhood) memory (ie a dynamic)and symmetry are applied to the constitutive equations describing thethermodynamic process

Modern continuum thermodynamics has been extended to account fornonlinear irreversible processes such as the existence of thresholds plasticity



22 CHAPTER 1

and hysteresis [118 241 309 310] These extensions use convex analysissemigroup theory and nonlinear programming theory but can lack a clearcharacterization of the space over which the thermodynamical state variablesevolve The principal weakness of rational thermodynamics is that its rangeof applicability is limited to closed systems (see Chapter 2) with a singleabsolute temperature Thus it is not applicable to condensed matter physics(eg diffusing mixtures or plasma) Furthermore it does not provide aunique entropy characterization that satisfies the Clausius inequality

More recently a contribution to equilibrium thermodynamics is givenin [280] This work builds on the work of Caratheodory [76 77] and Giles[164] by developing a thermodynamic system representation involving a statespace on which an adiabatic accessibility relation is defined The existenceand uniqueness of an entropy function is established as a consequenceof adiabatic accessibility among equilibrium states As in Caratheodoryrsquoswork the authors in [280] also restrict their attention to simple (possiblyinterconnected) systems in order to arrive at an entropy increase principleHowever it should be noted that the notion of a simple system in [280] isnot equivalent to that of Caratheodoryrsquos notion of a simple system

Connections between thermodynamics and systems theory as well asinformation theory have also been explored in the literature [35376668192353463464472478] Information theory has deep connections to physics ingeneral and thermodynamics in particular Many scientists have postulatedthat information is physical and have suggested that the bit is the irreduciblekernel in the universe and it is more fundamental than matter itself withinformation forming the very core of existence [167259] To produce change(motion) requires energy whereas to direct this change requires informationIn other words energy takes different forms but these forms are determinedby information Arguments about the nature of reality is deeply rooted inquantum information which gives rise to every particle every force fieldand spacetime itself

In quantum mechanics information can be inaccessible but not anni-hilated In other words information can never be destroyed despite thefact that imperfect system state distinguishability abounds in quantumphysics wherein the Heisenberg uncertainty principle brought the demise ofdeterminism in the microcosm of science The afore statement concerningthe nonannihilation of information is not without its controversy in physicsand is at the heart of the black hole information paradox which resultedfrom the incomplete unification of quantum mechanics and general relativity

Specifically Hawking and Bekenstein [28 208] argued that generalrelativity and quantum field theory were inconsistent with the principle



INTRODUCTION 23

that information cannot be lost In particular as a consequence of quantumfluctuations near a black holersquos event horizon28 they showed that black holesradiate particles and hence slowly evaporate And since matter falling intoa black hole carries information in its structure organization and quantumstates black hole evaporation via radiation obliterates information

However using Richard Feynmanrsquos sum over histories path integralformulation of quantum theory to the topology of spacetime [146] Hawkinglater showed that quantum gravity is unitary (ie the sum of probabilitiesfor all possible outcomes of a given event is unity) and that black holesare never unambiguously black That is black holes slowly dissipate beforethey ever truly form allowing radiation to contain information and henceinformation is not lost obviating the information paradox

In quantum mechanics the Heisenberg uncertainty principle is aconsequence of the fact that the outcome of an experiment is affectedor even determined when observed The Heisenberg uncertainty principlestates that it is impossible to measure both the position and momentum ofa particle with absolute precision at a microscopic level and the product ofthe uncertainties in these measured values is in the order of the magnitudeof the Planck constant The determination of energy and time is alsosubject to the same uncertainty principle The principle is not a statementabout our inability to develop accurate measuring instruments but rathera statement about an intrinsic property of nature namely nature has aninherent indeterminacy And this is a consequence of the fact that anyattempt at observing nature will disturb the system under observationresulting in a lack of precision

Quantum mechanics provides a probabilistic theory of nature whereinthe equations describe the average behavior of a large collection of identicalparticles and not the behavior of individual particles Einstein maintainedthat the theory was incomplete albeit a good approximation in describingnature He further asserted that when quantum mechanics had beencompleted it would deal with certainties In a letter to Max Born he stateshis famous ldquoGod does not play dicerdquo dictum writing ldquoThe theory producesa great deal but hardly brings us closer to the secret of the Old One I amat all events convinced that He does not play dicerdquo [60 p 90] A profoundramification of the Heisenberg uncertainty principle is that the macroscopicprinciple of causality does not apply at the atomic level

Information theory addresses the quantification storage and commu-nication of information The study of the effectiveness of communication

28In relativistic physics an event horizon is a boundary delineating the set of points inspacetime beyond which events cannot affect an outside observer In the present context itrefers to the boundary beyond which events cannot escape the black holersquos gravitational field



24 CHAPTER 1

channels in transmitting information was pioneered by Shannon [406]Information is encoded stored (by codes) transmitted through channelsof limited capacity and then decoded The effectiveness of this process ismeasured by the Shannon capacity of the channel and involves the entropyof a set of events that measure the uncertainty of this set These channelsfunction as input-output devices that take letters from an input alphabetand transmit letters to an output alphabet with various error probabilitiesthat depend on noise Hence entropy in an information-theoretic context isa measure of information uncertainty Simply putmdashinformation is not freeand is linked to the cost of computing the behavior of matter and energyin our universe [39] For an excellent exposition of these different facets ofthermodynamics see [175]

Thermodynamic principles have also been repeatedly used in coupledmechanical systems to arrive at energy flow models Specifically in anattempt to approximate high-dimensional dynamics of large-scale struc-tural (oscillatory) systems with a low-dimensional diffusive (nonoscillatory)dynamical model structural dynamicists have developed thermodynamicenergy flow models using stochastic energy flow techniques In particularstatistical energy analysis (SEA) predicated on averaging system statesover the statistics of the uncertain system parameters has been extensivelydeveloped for mechanical and acoustic vibration problems [78238270297414468] The aim of SEA is to establish that many concepts of energy flowmodeling in high-dimensional mechanical systems have clear connectionswith statistical mechanics of many particle systems and hence the secondlaw of thermodynamics applies to large-scale coupled mechanical systemswith modal energies playing the role of temperatures

Thermodynamic models are derived from large-scale dynamical sys-tems of discrete subsystems involving stored energy flow among subsystemsbased on the assumption of weak subsystem coupling or identical subsys-tems However the ability of SEA to predict the dynamic behavior ofa complex large-scale dynamical system in terms of pairwise subsysteminteractions is severely limited by the coupling strength of the remainingsubsystems on the subsystem pair Hence it is not surprising that SEAenergy flow predictions for large-scale systems with strong coupling canbe erroneous From the rigorous perspective of dynamical systems theorythe theoretical foundations of SEA remain inadequate since well-definedmathematical assumptions of the theory are not adequately delineated

Alternatively a deterministic thermodynamically motivated energyflow modeling for structural systems is addressed in [247ndash249] Thisapproach exploits energy flow models in terms of thermodynamic energy(ie the ability to dissipate heat) as opposed to stored energy and is not



INTRODUCTION 25

limited to weak subsystem coupling A stochastic energy flow compartmentalmodel (ie a model characterized by energy conservation laws) predicatedon averaging system states over the statistics of stochastic system exogenousdisturbances is developed in [37] The basic result demonstrates how linearcompartmental models arise from second-moment analysis of state spacesystems under the assumption of weak coupling Even though these resultscan be potentially applicable to linear large-scale dynamical systems withweak coupling such connections are not explored in [37] With the notableexception of [78] and more recently [273] none of the aforementioned SEA-related works addresses the second law of thermodynamics involving entropynotions in the energy flow between subsystems

Motivated by the manifestation of emergent behavior of macroscopicenergy transfer in crystalline solids modeled as a lattice of identicalmolecules involving undamped vibrations the authors in [44] analyzeenergy equipartition in linear Hamiltonian systems using average-preservingsymmetries Specifically the authors consider a Lie group of phase spacesymmetries of a linear Hamiltonian system and characterize the subgroup ofsymmetries whose elements are also symmetries of every Hamiltonian systemand preserve the time averages of quadratic Hamiltonian functions alongsystem trajectories In the very specific case of distinct natural frequenciesand a two-degree-of-freedom system consisting of an interconnected pair ofidentical undamped oscillators the authors show that the time-averagedoscillator energies reach an equipartitioned state For this limited casethis result shows that time averaging leads to the emergence of damping inlossless Hamiltonian dynamical systems

14 Dynamical Systems

Dynamical systems theory provides a universal mathematical formalismpredicated on modern analysis and has become the prevailing language ofmodern science as it provides the foundation for unlocking many of themysteries in nature and the universe that involve spatial and temporalevolution Given that irreversible thermodynamic systems involve a definitedirection of evolution it is natural to merge the two universalisms ofthermodynamics and dynamical systems under a single compendium withthe latter providing an ideal language for the former

A system is a combination of components or parts that is perceived asa single entity The parts making up the system may be clearly or vaguelydefined These parts are related to each other through a particular set ofvariables called the states of the system that together with the knowledgeof any system inputs completely determine the behavior of the system atany given time A dynamical system is a system whose state changes with



26 CHAPTER 1

time Dynamical systems theory was fathered by Henri Poincare [363ndash365]sturdily developed by Birkhoff [50 51] and has evolved to become oneof the most universal mathematical formalisms used to explain systemmanifestations of nature that involve time

A dynamical system can be regarded as a mathematical modelstructure involving an input state and output that can capture thedynamical description of a given class of physical systems Specificallya closed dynamical system consists of three elementsmdashnamely a settingcalled the state space which is assumed to be Hausdorff29 and in which thedynamical behavior takes place such as a torus topological space manifoldor locally compact metric space a mathematical rule or dynamic whichspecifies the evolution of the system over time and an initial condition orstate from which the system starts at some initial time

An open dynamical system interacts with the environment throughsystem inputs and system outputs and can be viewed as a precise math-ematical object that maps exogenous inputs (causes disturbances) intooutputs (effects responses) via a set of internal variables the state whichcharacterizes the influence of past inputs For dynamical systems describedby ordinary differential equations the independent variable is time whereasspatially distributed systems described by partial differential equationsinvolve multiple independent variables reflecting for example time andspace

The state of a dynamical system can be regarded as an informationstorage or memory of past system events The set of (internal) states ofa dynamical system must be sufficiently rich to completely determine thebehavior of the system for any future time Hence the state of a dynamicalsystem at a given time is uniquely determined by the state of the systemat the initial time and the present input to the system In other wordsthe state of a dynamical system in general depends on both the presentinput to the system and the past history of the system Even though it isoften assumed that the state of a dynamical system is the least set of statevariables needed to completely predict the effect of the past upon the futureof the system this is often a convenient simplifying assumption

Ever since its inception the basic questions concerning dynamicalsystems theory have involved qualitative solutions for the properties of adynamical system questions such as For a particular initial system statedoes the dynamical system have at least one solution What are theasymptotic properties of the system solutions How are the system solutions

29A Hausdorff space is a topological space in which there exists a pair of disjoint openneighborhoods for every pair of distinct points in the space



INTRODUCTION 27

dependent on the system initial conditions How are the system solutionsdependent on the form of the mathematical description of the dynamic ofthe system How do system solutions depend on system parameters Andhow do system solutions depend on the properties of the state space onwhich the system is defined

Determining the mathematical rule or dynamic that defines the stateof physical systems at a given future time from a given present state is oneof the central problems of science Once the flow or dynamic of a dynamicalsystem describing the motion of the system starting from a given initial stateis given dynamical systems theory can be used to describe the behavior ofthe system states over time for different initial conditions Throughout thecenturiesmdashfrom the great cosmic theorists of ancient Greece30 to the present-day quest for a unified field theorymdashthe most important dynamical systemis our vicissitudinous universe By using abstract mathematical models andattaching them to the physical world astronomers mathematicians andphysicists have used abstract thought to deduce something that is true aboutthe natural system of the cosmos

The quest by scientists such as Brahe Kepler Galileo NewtonHuygens Euler Lagrange Laplace and Maxwell to understand theregularities inherent in the distances of the planets from the Sun and theirperiods and velocities of revolution around the Sun led to the science ofdynamical systems as a branch of mathematical physics Isaac Newtonhowever was the first to model the motion of physical systems withdifferential equations Newtonrsquos greatest achievement was the rediscoverythat the motion of the planets and moons of the solar system resulted froma single fundamental sourcemdashthe gravitational attraction of the heavenly

30The Hellenistic period (323ndash31 BC) spawned the scientific revolution leading to todayrsquosscientific method and scientific technology including much of modern science and mathematics inits present formulation Hellenistic scientists which included Archimedes Euclid EratosthenesEudoxus Ktesibios Philo Apollonios and many others were the first to use abstractmathematical models and attach them to the physical world More importantly using abstractthought and rigorous mathematics (Euclidean geometry real numbers limits definite integrals)these ldquomodern minds in ancient bodiesrdquo were able to deduce complex solutions to practicalproblems and provide a deep understanding of nature In his Forgotten Revolution [389] Russoconvincingly argues that Hellenistic scientists were not just forerunners or anticipators of modernscience and mathematics but rather the true fathers of these disciplines He goes on to show howscience was born in the Hellenistic world and why it had to be reborn

As in the case of the origins of much of modern science and mathematics modern engineeringcan also be traced back to ancient Greece Technological marvels included Ktesibiosrsquo pneumaticsHeronrsquos automata and arguably the greatest fundamental mechanical invention of all timemdashtheAntikythera mechanism The Antikythera mechanism most likely inspired by Archimedes wasbuilt around 76 BC and was a device for calculating the motions of the stars and planets aswell as for keeping time and calendar This first analog computer involving a complex arrayof meshing gears was a quintessential hybrid dynamical system that unequivocally shows thesingular sophistication capabilities and imagination of the ancient Greeks and dispenses withthe Western myth that the ancient Greeks developed mathematics but were incapable of creatingscientific theories and scientific technology



28 CHAPTER 1

bodies This discovery dates back to Aristarkhosrsquo (310ndash230 BC) heliocentrictheory of planetary motion and Hipparkhosrsquo (190ndash120 BC) dynamical theoryof planetary motion predicated on planetary attractions toward the Sun bya force that is inversely proportional to the square of the distance betweenthe planets and the Sun [389 p 304]

Many of the concepts of Newtonian mechanics including relative mo-tion centrifugal and centripetal force inertia projectile motion resistancegravity and the inverse square law were known to the Hellenistic scientists[389] For example Hipparkhosrsquo work On bodies thrusting down because ofgravity (Περι τ ων δια βαρvτητα κατω ϕερomicroενων) clearly and correctlydescribes the effects of gravity on projectile motion And in Ploutarkhosrsquo(46ndash120 AD) work De facie quae in orbe lunae apparet (On the light glowingon the Moon) he clearly describes the notion of gravitational interactionbetween heavenly bodies stating that ldquojust as the sun attracts to itself theparts of which it consists so does the earth rdquo31 [389 p 304]

Newton himself wrote in his Classical Scholia [389 p 376] ldquoPythago-ras applied to the heavens the proportions found through these experi-ments [on the pitch of sounds made by weighted strings] and learned fromthat the harmonies of the spheres And so by comparing those weights withthe weights of the planets and the intervals in sound with the intervals of thespheres and the lengths of string with the distances of the planets [measured]from the center he understood through the heavenly harmonies that theweights of the planets toward the sun are inversely proportional to thesquares of their distancesrdquo And this admittance of the prior knowledgeof the inverse square law predates Hookersquos thoughts of explaining Keplerrsquoslaws out of the inverse square law communicated in a letter to Newton onJanuary 6 1680 by over two millennia

It is important to stress here that what are erroneously called Newtonrsquoslaws of motion in the literature were first discovered by Kepler Galileo andDescartes with the latter first stating the law of inertia in its modern formNamely when viewed in an inertial reference frame a body remains in thesame state unless acted upon by a net force and unconstrained motionfollows a rectilinear path Newton and Leibnitz independently advancedthe basic dynamical tool invented two millennia earlier by Archimedesmdashthe calculus32 with Euler being the first one to explicitly write down thesecond law of motion as an equation involving an applied force acting on

31ὡς γὰρ ὁ ἥλιος εἰς ἐαυτὸν ἐπιστρέφει τὰ μέρη ἐξ ὧν συνέστηκε καὶ ἡ γῇ (in Ploutarkhos De

facie quae in orbe lunae apparet 924E)32In his treatise on The Method of Mechanical Theorems Archimedes (287ndash212 BC) established

the foundations of integral calculus using infinitesimals as well as the foundations of mathematicalmechanics In addition in one of his problems he constructed the tangent at any given point fora spiral establishing the origins of differential calculus [29 p 32]



INTRODUCTION 29

a body being equal to the time rate of change of its momentum Newtonhowever deduced a physical hypothesismdashthe law of universal gravitationinvolving an inverse-square law forcemdashin precise mathematical form deriving(at the time) a cosmic dynamic using Euclidian geometry and not differentialcalculus (ie differential equations)

In his magnum opus Philosophiae Naturalis Principia Mathemat-ica [337] Newton investigated whether a small perturbation would makea particle moving in a plane around a center of attraction continue to movenear the circle or diverge from it Newton used his analysis to analyzethe motion of the moon orbiting the Earth Numerous astronomers andmathematicians who followed made significant contributions to dynamicalsystems theory in an effort to show that the observed deviations of planetsand satellites from fixed elliptical orbits were in agreement with Newtonrsquosprinciple of universal gravitation Notable contributions include the workof Torricelli [443] Euler [137] Lagrange [256] Laplace [271] Dirichlet [116]Liouville [286] Maxwell [311] Routh [386] and Lyapunov [294ndash296]

Newtonian mechanics developed into the first field of modern sciencemdashdynamical systems as a branch of mathematical physicsmdashwherein thecircular elliptical and parabolic orbits of the heavenly bodies of our solarsystem were no longer fundamental determinants of motion but ratherapproximations of the universal laws of the cosmos specified by governingdifferential equations of motion And in the past century dynamical systemstheory has become one of the most fundamental fields of modern science asit provides the foundation for unlocking many of the mysteries in nature andthe universe that involve the evolution of time Dynamical systems theoryis used to study ecological systems geological systems biological systemseconomic systems neural systems and physical systems (eg mechanicsfluids magnetic fields galaxies) to cite but a few examples

15 Dynamical Thermodynamics A Postmodern Approach

In contrast to mechanics which is based on a dynamical systems theoryclassical thermodynamics (ie thermostatics) is a physical theory and doesnot possess equations of motion Moreover very little work has been donein obtaining extensions of thermodynamics for systems out of equilibriumThese extensions are commonly known as thermodynamics of irreversibleprocesses or modern irreversible thermodynamics in the literature [113367]Such systems are driven by the continuous flow of matter and energy are farfrom equilibrium and often develop into a multitude of states Connectionsbetween local thermodynamic subsystem interactions of these systems andthe globally complex thermodynamical system behavior are often elusiveThis statement is true for nature in general and was most eloquently stated



30 CHAPTER 1

first by Herakleitos in his 123rd fragmentmdashΦύσις κρύπτεσθαι φιλεί (Natureloves to hide)

These complex thermodynamic systems involve spatio-temporallyevolving structures and can exhibit a hierarchy of emergent system prop-erties These systems are known as dissipative systems [195] and consumeenergy and matter while maintaining their stable structure by dissipatingentropy to the environment All living systems are dissipative systemsthe converse however is not necessarily true Dissipative living systemsinvolve pattern interactions by which life emerges This nonlinear interactionbetween the subsystems making up a living system is characterized byautopoiesis (self-creation) In the physical universe billions of starsand galaxies interact to form self-organizing dissipative nonequilibriumstructures [252 369] The fundamental common phenomenon amongnonequilibrium (ie dynamical) systems is that they evolve in accordancewith the laws of (nonequilibrium) thermodynamics

Building on the work of nonequilibrium thermodynamic structures[166 367] Sekimoto [400ndash404] introduced a stochastic thermodynamicframework predicated on Langevin dynamics in which fluctuation forcesare described by Brownian motion In this framework the classicalthermodynamic notions of heat work and entropy production are extendedto the level of individual system trajectories of nonequilibrium ensemblesSpecifically system state trajectories are sample continuous and are char-acterized by a Langevin equation for each individual sample path and aFokker-Planck equation for the entire ensemble of trajectories

For such systems energy conservation holds along fluctuating trajecto-ries of the stochastic Markov process and the second law of thermodynamicsis obtained as an ensemble property of the process In particular variousfluctuation theorems [5556102103139153223229230255274401] arederived that constrain the probability distributions for the exchanged heatmechanical work and entropy production depending on the nature of thestochastic Langevin system dynamics

Even though stochastic thermodynamics is applicable to a singlerealization of the Markov process under consideration with the first andsecond laws of thermodynamics holding for nonequilibrium systems theframework only applies to multiple time-scale systems with a few observableslow degrees of freedom The unobservable degrees of freedom are assumedto be fast and hence always constrained to the equilibrium manifoldimposed by the instantaneous values of the observed slow degrees of freedom

Furthermore if some of the slow variables are not accessible then



INTRODUCTION 31

the system dynamics are no longer Markovian In this case defining asystem entropy is virtually impossible In addition it is unclear whetherfluctuation theorems expressing symmetries of the probability distributionfunctions for thermodynamic quantities can be grouped into universalclasses characterized by asymptotics of these distributions Moreoverit is also unclear whether there exist system variables that satisfy thetransitive equilibration property of the zeroth law of thermodynamics fornonequilibrium stochastic thermodynamic systems

In an attempt to create a generalized theory of evolution mechanicsby unifying classical mechanics with thermodynamics the authors in[25 26 423 424] developed a framework of system thermodynamics basedon the concept of tribo-fatigue entropy This framework known asdamage mechanics [25 26] or mechanothermodynamics [423 424] involvesan irreversible entropy function along with its generation rate that capturesand quantifies system aging Specifically the second law is formulatedanalytically for organic and inorganic bodies and the system entropyis determined by a damageability process predicated on mechanical andthermodynamic effects resulting in system state changes

In [195] the authors develop a postmodern framework for thermody-namics that involves open interconnected dynamical systems that exchangematter and energy with their environment in accordance with the first law(conservation of energy) and the second law (nonconservation of entropy)of thermodynamics Symmetry can spontaneously occur in such systems byinvoking the two fundamental axioms of the science of heat

Namely i) if the energies in the connected subsystems of an intercon-nected system are equal then energy exchange between these subsystemsis not possible and ii) energy flows from more energetic subsystems toless energetic subsystems These axioms establish the existence of a globalsystem entropy function as well as equipartition of energy [195] in systemthermodynamics an emergent behavior in thermodynamic systems Hencein complex interconnected thermodynamic systems higher symmetry (iesystem decomplexification) is not a property of the systemrsquos parts but ratheremerges as a result of the nonlinear subsystem interactions

The goal of the present monograph is directed toward buildingon the results of [195] to place thermodynamics on a system-theoreticfoundation by combining the two universalisms of thermodynamics anddynamical systems theory under a single umbrella so as to harmonize itwith classical mechanics In particular we develop a novel formulationof thermodynamics that can be viewed as a moderate-sized dynamicalsystems theory as compared to statistical thermodynamics This middle-



32 CHAPTER 1

ground theory involves large-scale dynamical system models characterizedby ordinary deterministic and stochastic differential equations as well asinfinite-dimensional models characterized by partial differential equationsand functional delay differential equations that bridge the gap betweenclassical and statistical thermodynamics

Specifically since thermodynamic models are concerned with energyflow among subsystems we use a state space formulation to develop anonlinear compartmental dynamical system model that is characterizedby energy conservation laws capturing the exchange of energy and matterbetween coupled macroscopic subsystems Furthermore using graph-theoretic notions we state two thermodynamic axioms consistent with thezeroth and second laws of thermodynamics which ensure that our large-scale dynamical system model gives rise to a thermodynamically consistentenergy flow model Specifically using a large-scale dynamical systemstheory perspective for thermodynamics we show that our compartmentaldynamical system model leads to a precise formulation of the equivalencebetween work energy and heat in a large-scale dynamical system

Since our dynamical thermodynamic formulation is based on a large-scale dynamical systems theory involving the exchange of energy withconservation laws describing transfer accumulation and dissipation betweensubsystems and the environment our framework goes beyond classicalthermodynamics characterized by a purely empirical theory wherein aphysical system is viewed as an input-output black box system Furthermoreunlike classical thermodynamics which is limited to the description ofsystems in equilibrium states our approach addresses nonequilibriumthermodynamic systems This allows us to connect and unify the behaviorof heat as described by the equations of thermal transfer and as described byclassical thermodynamics This exposition further demonstrates that thesedisciplines of classical physics are derivable from the same principles and arepart of the same scientific and mathematical framework

Our nonequilibrium thermodynamic framework goes beyond classicalirreversible thermodynamics developed by Onsager [347 348] and furtherextended by Casimir [82] and Prigogine [166 367 368] which as discussedin Section 13 fall short of a complete dynamical theory Specifically theirtheories postulate that the local instantaneous thermodynamic variables ofthe system are the same as that of the system in equilibrium This impliesthat the system entropy in a neighborhood of an equilibrium is dependenton the same variables as those at equilibrium violating Gibbsrsquo maximumentropy principle In contrast the proposed system thermodynamicformalism brings classical thermodynamics within the framework of modernnonlinear dynamical systems theory thus providing information about the



INTRODUCTION 33

dynamical behavior of the thermodynamic state variables between the initialand final equilibrium system states

Next we give a deterministic definition of entropy for a large-scaledynamical system that is consistent with the classical thermodynamicdefinition of entropy and we show that it satisfies a Clausius-type inequalityleading to the law of entropy nonconservation However unlike classicalthermodynamics wherein entropy is not defined for arbitrary states out ofequilibrium our definition of entropy holds for nonequilibrium dynamicalsystems

Furthermore we introduce a new and dual notion to entropymdashnamelyectropy33mdashas a measure of the tendency of a large-scale dynamical systemto do useful work and grow more organized and we show that conservationof energy in an adiabatically isolated thermodynamically consistent systemnecessarily leads to nonconservation of ectropy and entropy Hence for everydynamical transformation in an adiabatically isolated thermodynamicallyconsistent system the entropy of the final system state is greater than orequal to the entropy of the initial system state

Then using the system ectropy as a Lyapunov function candidatewe show that in the absence of energy exchange with the environmentour thermodynamically consistent large-scale nonlinear dynamical systemmodel possesses a continuum of equilibria and is semistable that is ithas convergent subsystem energies to Lyapunov stable energy equilibriadetermined by the large-scale system initial subsystem energies In additionwe show that the steady-state distribution of the large-scale system energiesis uniform leading to system energy equipartitioning corresponding to aminimum ectropy and a maximum entropy equilibrium state

For our thermodynamically consistent dynamical system model wefurther establish the existence of a unique continuously differentiableglobal entropy and ectropy function for all equilibrium and nonequilibriumstates Using these global entropy and ectropy functions we go on toestablish a clear connection between thermodynamics and the arrow of timeSpecifically we rigorously show a state irrecoverability and hence a stateirreversibility34 nature of thermodynamics In particular we show that for

33Ectropy comes from the Greek word εκτρoπη (εκ and τρoπη) for outward transformationconnoting evolution or complexification and is the literal antonym of entropy (εντρoπηmdashεν andτρoπη) signifying an inward transformation connoting devolution or decomplexification Theword entropy was proposed by Clausius for its phonetic similarity to energy with the additionalconnotation reflecting change (τρoπη)

34In the terminology of [448] state irreversibility is referred to as time-reversal non-invarianceHowever since the term time reversal is not meant literally (that is we consider dynamical systemswhose trajectory reversal is or is not allowed and not a reversal of time itself) state reversibilityis a more appropriate expression And in that regard a more appropriate expression for the arrow



34 CHAPTER 1

every nonequilibrium system state and corresponding system trajectory ofour thermodynamically consistent large-scale nonlinear dynamical systemthere does not exist a state such that the corresponding system trajectorycompletely recovers the initial system state of the dynamical system and atthe same time restores the energy supplied by the environment back to itsoriginal condition

This along with the existence of a global strictly increasing entropyfunction on every nontrivial system trajectory gives a clear time-reversalasymmetry characterization of thermodynamics establishing an emergenceof the direction of time flow In the case where the subsystem energiesare proportional to subsystem temperatures we show that our dynamicalsystem model leads to temperature equipartition wherein all the systemenergy is transferred into heat at a uniform temperature Furthermorewe show that our system-theoretic definition of entropy and the newlyproposed notion of ectropy are consistent with Boltzmannrsquos kinetic theoryof gases involving an n-body theory of ideal gases divided by diathermalwalls Finally these results are generalized to continuum thermodynam-ics stochastic thermodynamics and relativistic thermodynamics involvinginfinite-dimensional Markovian and functional energy flow conservationmodels

16 A Brief Outline of the Monograph

The objective of this monograph is to develop a system-theoretic foundationfor thermodynamics using dynamical systems and control notions Themain contents of the monograph are as follows In Chapter 2 we establishnotation and definitions and we develop several key results on nonnegativeand compartmental dynamical systems needed to establish thermodynami-cally consistent energy flow models Furthermore we introduce the notionsof (ir)reversible and (ir)recoverable dynamical systems volume-preservingflows and recurrent dynamical systems as well as output reversibility indynamical systems

In Chapter 3 we use a large-scale dynamical systems perspective toprovide a system-theoretic foundation for thermodynamics Specifically us-ing a system state space formulation we develop a nonlinear compartmentaldynamical system model characterized by energy conservation laws that isconsistent with basic thermodynamic principles In particular using thetotal subsystem energies as a candidate system energy storage function weshow that our thermodynamic system is lossless and hence can deliverto its surroundings all of its stored subsystem energies and can store allof the work done to all of its subsystems This leads to the first law of

of time is system degeneration over time signifying irrecoverable system changes



INTRODUCTION 35

thermodynamics involving conservation of energy and places no limitationon the possibility of transforming heat into work or work into heat

Next we show that the classical Clausius equality and inequality forreversible and irreversible thermodynamics are satisfied over cyclic motionsfor our thermodynamically consistent energy flow model and guaranteethe existence of a continuous system entropy function In addition weestablish the existence of a unique continuously differentiable global entropyfunction for our large-scale dynamical system which is used to define inversesubsystem temperatures as the derivative of the subsystem entropies withrespect to the subsystem energies

Then we turn our attention to stability and convergence Specificallyusing the system ectropy as a Lyapunov function candidate we show thatin the absence of energy exchange with the environment the proposedthermodynamic model is semistable with a uniform energy distributioncorresponding to a state of minimum ectropy and a state of maximumentropy Furthermore using the system entropy and ectropy functionswe develop a clear connection between irreversibility the second law ofthermodynamics and the entropic arrow of time

In Chapter 4 we generalize the results of Chapter 3 to the casewhere the subsystem energies in the large-scale dynamical system modelare proportional to subsystem temperatures and we arrive at temperatureequipartition for the proposed thermodynamic model Furthermore weprovide a kinetic theory interpretation of the steady-state expressions forentropy and ectropy Moreover we establish connections between dynamicalthermodynamics and classical thermodynamics

In Chapter 5 we augment our nonlinear compartmental dynamicalsystem model with an additional (deformation) state representing com-partmental volumes to arrive at a general statement of the first law ofthermodynamics giving a precise formulation of the equivalence betweenheat and mechanical work Furthermore we define the Gibbs free energyHelmholtz free energy and enthalpy functions for our large-scale systemthermodynamic model In addition we use the proposed augmentednonlinear compartmental dynamical system model in conjunction with aCarnot-like cycle analysis to show the equivalence between the classicalKelvin and Clausius postulates of the second law of thermodynamics

In Chapter 6 we address the problems of nonnegativity realizabilityreducibility and semistability of chemical reaction networks Specificallywe show that mass-action kinetics have nonnegative solutions for initiallynonnegative concentrations we provide a general procedure for reducing the



36 CHAPTER 1

dimensionality of the kinetic equations and we present stability results basedupon Lyapunov methods Furthermore we present a state space dynamicalsystem model for chemical thermodynamics In particular we use the lawof mass action to obtain the dynamics of chemical reaction networks

In addition using the notion of the chemical potential we unify ourstate space mass-action kinetics model with our dynamical thermodynamicsystem model involving system energy exchange Moreover we showthat entropy production during chemical reactions is nonnegative and thedynamical system states of our chemical thermodynamic state space modelconverge to a state of temperature equipartition and zero affinity (ie thedifference between the chemical potential of the reactants and the chemicalpotential of the products in a chemical reaction)

In Chapter 7 we merge the theories of semistability and finite-timestability to develop a rigorous framework for finite-time thermodynamicsSpecifically using a geometric description of homogeneity theory wedevelop intercompartmental energy flow laws that guarantee finite-timesemistability and energy equipartition for the thermodynamically consistentmodel developed in Chapter 3

Next in Chapter 8 we address the problem of thermodynamiccritical phenomena and continuous phase transitions In particular toaddress discontinuities in the derivatives of the thermodynamic statequantities we consider dynamical systems with Lebesgue measurableand locally essentially bounded vector fields characterized by differentialinclusions involving Filippov set-valued maps specifying a set of directionsfor the system generalized velocities and admitting Filippov solutionswith absolutely continuous curves Moreover we present Lyapunov-basedtests for semistability finite-time semistability and energy equipartitionfor a discontinuous power balance thermodynamic model characterized bydifferential inclusions

In Chapter 9 we develop thermodynamic models for discrete-timelarge-scale dynamical systems Specifically using a framework analogousto Chapter 3 we develop energy flow models possessing discrete energyconservation energy equipartition temperature equipartition and entropynonconservation principles for discrete-time large-scale dynamical systems

To address thermodynamic critical phenomena and discontinuousphase transitions in Chapter 10 we combine the frameworks of Chapters 3and 9 to develop hybrid thermodynamic models Specifically to capturejump discontinuities in the fundamental thermodynamic state quantities wedevelop a hybrid large-scale dynamical system using impulsive compartmen-



INTRODUCTION 37

tal and thermodynamic dynamical system models involving an interactingmixture of continuous and discrete dynamics exhibiting discontinuous flowson appropriate manifolds

In Chapter 11 we extend the results of Chapter 3 to continuumthermodynamic systems wherein the subsystems are uniformly distributedover an n-dimensional (not necessarily Euclidean) space Specificallywe develop a nonlinear distributed-parameter model wherein the systemenergy is modeled by a conservation equation in the form of a nonlinearpartial differential equation Energy equipartition and semistability areshown using Sobolev embedding theorems and the notion of generalized(or weak) solutions This exposition shows that the behavior of heat asdescribed by the equations of thermal transport and as described by classicalthermodynamics is derivable from the same principles and is part of thesame scientific discipline and thus provides a unification between Fourierrsquostheory of heat conduction and classical thermodynamics

In Chapter 12 we extend the results of Chapter 3 to large-scaledynamical systems driven by Markov diffusion processes to present aunified framework for statistical thermodynamics predicated on a stochasticdynamical systems formalism Specifically using a stochastic state spaceformulation we develop a nonlinear stochastic compartmental dynamicalsystem model characterized by energy conservation laws that is consistentwith statistical thermodynamic principles In particular we show that theaverage stored system energy for our stochastic thermodynamic model is amartingale with respect to the system filtration and is equal to the meanenergy that can be extracted from the system and the mean energy that canbe delivered to the system in order to transfer it from a zero energy level toan arbitrary nonempty subset in the state space over a finite stopping time

Next to effectively address the universality of thermodynamics and thearrow of time to cosmology we extend our dynamical systems framework ofthermodynamics to include relativistic effects To this end in Chapter 13we give a brief exposition of the special and general theories of relativityand review some basic concepts on relativistic kinematics and relativisticdynamics

Then in Chapter 14 we extend our results to thermodynamic systemsthat are moving relative to a local observer moving with the system and afixed observer with respect to which the system is in motion Furthermorethermodynamic effects in the presence of a strong gravitational field are alsodiscussed In addition using the topological isomorphism between entropyand time established in Chapter 3 and Einsteinrsquos time dilation assertion thatincreasing an objectrsquos speed through space results in decreasing the objectrsquos



38 CHAPTER 1

speed through time we present an entropy dilation principle which showsthat the change in entropy of a thermodynamic system decreases as thesystemrsquos speed increases through space

To account for finite subluminal speed of heat propagation inChapter 15 we generalize the results of Chapter 3 to general thermodynamiccompartmental systems that account for energy and matter in transit be-tween compartments Specifically we develop thermodynamic models thatguarantee conservation of energy semistability and state equipartitioningwith directed and undirected thermal flow as well as flow delays betweencompartments

Finally we draw conclusions in Chapter 16 and in Chapter 17 wepresent a high-level scientific discussion of several peripheral albeit key areasof how our dynamical systems framework of thermodynamics can be usedto foster the development of new frameworks in explaining the fundamentalthermodynamic processes occurring in nature explore new hypotheses thatchallenge the use of classical thermodynamics and develop new assertionsthat can provide deeper insights into the constitutive mechanisms thatdescribe the acute microcosms and macrocosms of science



2 CHAPTER 1

energy in a dynamical (energy) transformation from one state (form) toanother Since the system entropy increases the entropy of a dynamicalsystem tends to a maximum and thus time as determined by system entropyincrease [299 392 476] flows in one direction only Even though entropy isa physical property of matter that is not directly observable it permeatesthe whole of nature regulating the arrow of time and is responsible forthe enfeeblement and eventual demise of the universe12 While the laws ofthermodynamics form the foundation to basic engineering systems chemicalreaction systems nuclear reactions cosmology and our expanding universemany mathematicians and scientists have expressed concerns about thecompleteness and clarity of the different expositions of thermodynamics overits long and tortuous history see [697996172184342440447455]

Since the specific motion of every molecule of a thermodynamic systemis impossible to predict a macroscopic model of the system is typicallyused with appropriate macroscopic states that include pressure volumetemperature internal energy and entropy among others One of thekey criticisms of the macroscopic viewpoint of thermodynamics known asclassical thermodynamics is the inability of the model to provide enoughdetail of how the system really evolves that is it is lacking a kineticmechanism for describing the behavior of heat and work energy

In developing a kinetic model for heat and dynamical energy athermodynamically consistent energy flow model should ensure that thesystem energy can be modeled by a diffusion equation in the form ofa parabolic partial differential equation or a divergence structure first-order hyperbolic partial differential equation arising in models involvingconservation laws Such systems are infinite-dimensional and hence finite-dimensional approximations are of very high order giving rise to large-scaledynamical systems with macroscopic energy transfer dynamics Since energyis a fundamental concept in the analysis of large-scale dynamical systemsand heat (energy in transition) is a fundamental concept of thermodynamicsinvolving the capacity of hot bodies (more energetic subsystems) to producework thermodynamics is a theory of large-scale dynamical systems

1Many natural philosophers have associated this ravaging irrecoverability in connection to thesecond law of thermodynamics with an eschatological terminus of the universe Namely thecreation of a certain degree of life and order in the universe is inevitably coupled with an evengreater degree of death and disorder A convincing proof of this bold claim has however neverbeen given

2The earliest perception of irreversibility of nature and the universe along with timersquos arrowwas postulated by the ancient Greek philosopher Herakleitos (sim 535ndashsim 475 BC) Herakleitosrsquoprofound statements Everything is in a state of flux and nothing is stationary and Man cannotstep into the same river twice because neither the man nor the river is the same created thefoundation for all other speculation on metaphysics and physics The idea that the universe isin constant change and that there is an underlying order to this changemdashthe Logosmdashpostulatesthe very existence of entropy as a physical property of matter permeating all of nature and theuniverse



INTRODUCTION 3








4 CHAPTER 1












INTRODUCTION 5












6 CHAPTER 1









INTRODUCTION 7









8 CHAPTER 1











INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 3








4 CHAPTER 1












INTRODUCTION 5












6 CHAPTER 1









INTRODUCTION 7









8 CHAPTER 1











INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






4 CHAPTER 1












INTRODUCTION 5












6 CHAPTER 1









INTRODUCTION 7









8 CHAPTER 1











INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 5












6 CHAPTER 1









INTRODUCTION 7









8 CHAPTER 1











INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






6 CHAPTER 1









INTRODUCTION 7









8 CHAPTER 1











INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 7









8 CHAPTER 1











INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






8 CHAPTER 1











INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 9










10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






10 CHAPTER 1







INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 11








12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






12 CHAPTER 1








INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 13








14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






14 CHAPTER 1







INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 15

arrow of time









16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






16 CHAPTER 1













INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 17








18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






18 CHAPTER 1









INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 19









20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






20 CHAPTER 1








INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 21








22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






22 CHAPTER 1








INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 23









24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






24 CHAPTER 1







INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 25








26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






26 CHAPTER 1









INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 27








28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






28 CHAPTER 1










INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 29








30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






30 CHAPTER 1









INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 31








32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






32 CHAPTER 1







INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 33










34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






34 CHAPTER 1









INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 35









36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






36 CHAPTER 1









INTRODUCTION 37








38 CHAPTER 1






INTRODUCTION 37








38 CHAPTER 1






38 CHAPTER 1






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