Statistical PhysicsETH Zurich Herbstsemester 2013

Manfred Sigrist, HIT K23.8Tel.: 3-2584, Email: sigrist@itp.phys.ethz.chWebpage: http://www.itp.phys.ethz.ch/research/condmat/strong/

Literature: Statistical Mechanics, K. Huang, John Wiley & Sons (1987). Introduction to Statistical Physics, K. Huang, Chapman & Hall Books (2010). Equilibrium Statistical Physics, M. Plischke and B. Bergersen, Prentice-Hall International(1989). Theorie der Warme, R. Becker, Springer (1978). Thermodynamik und Statistik, A. Sommerfeld, Harri Deutsch (1977). Statistische Physik, Vol. V, L.D. Landau and E.M Lifschitz, Akademie Verlag Berlin(1975). Statistische Mechanik, F. Schwabl, Springer (2000). Elementary Statistical Physics, Charles Kittel,John Wiley & Sons (1967). Statistical Mechanics, R.P. Feynman, Advanced Book Classics, Perseus Books (1998). Statistische Mechanik, R. Hentschke, Wiley-VCH (2004). Introduction to Statistical Field Theory, E. Brezin, Cambridge (2010). Statistical Mechanics in a Nutshell, Luca Peliti, Princeton University Press (2011). Principle of condensed matter physics, P.M. Chaikin and T.C. Lubensky, Cambridge University Press (1995). Many other books and texts mentioned throughout the lecture

Lecture Webpage:http://www.itp.phys.ethz.ch/education/hs13/StatPhys

1

Contents1 Classical statistical Physics1.1 Gibbsian concept of ensembles . . . . . . . .1.1.1 Liouville Theorem . . . . . . . . . . .1.1.2 Equilibrium system . . . . . . . . . . .1.2 Microcanonical ensemble . . . . . . . . . . . .1.2.1 Entropy . . . . . . . . . . . . . . . . .1.2.2 Relation to thermodynamics . . . . .1.2.3 Ideal gas - microcanonical treatment .1.3 Canonical ensemble . . . . . . . . . . . . . . .1.3.1 Thermodynamics . . . . . . . . . . . .1.3.2 Equipartition law . . . . . . . . . . . .1.3.3 Ideal gas - canonical treatment . . . .1.4 Grand canonical ensemble . . . . . . . . . . .1.4.1 Relation to thermodynamics . . . . .1.4.2 Ideal gas - grand canonical treatment1.4.3 Chemical potential in an external field1.5 Fluctuations . . . . . . . . . . . . . . . . . . .1.5.1 Energy . . . . . . . . . . . . . . . . . .1.5.2 Particle number . . . . . . . . . . . .1.5.3 Magnetization . . . . . . . . . . . . .

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2 Quantum Statistical Physics2.1 Basis of quantum statistical physics . . . . . . . . . . . . . . . . . . . . . .2.2 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3 Ensembles in quantum statistics . . . . . . . . . . . . . . . . . . . . . . . .2.3.1 Microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . .2.3.2 Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.3 Grand canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . .2.4 Ideal quantum paramagnet - canonical ensemble . . . . . . . . . . . . . . .2.4.1 Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4.2 Spin S - classical limit . . . . . . . . . . . . . . . . . . . . . . . . . .2.5 Ideal quantum gas - grand canonical ensemble . . . . . . . . . . . . . . . . .2.6 Properties of Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.6.1 High-temperature and low-density limit . . . . . . . . . . . . . . . .2.6.2 Low-temperature and high-density limit: degenerate Fermi gas . . .2.6.3 Spin-1/2 Fermions in a magnetic field . . . . . . . . . . . . . . . . .2.7 Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.7.1 Bosonic atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.7.2 High-temperature and low-density limit . . . . . . . . . . . . . . . .2.7.3 Low-temperature and high-density limit: Bose-Einstein condensation2.8 Photons and phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.8.1 Blackbody radiation - photons . . . . . . . . . . . . . . . . . . . . .2

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6678891011131415151617181819192021

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222223242424252525262830313234353535363941

2.9

2.8.2 Phonons in a solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4344

3 Identical Quantum Particles - Formalism of Second Quantization3.1 Many-body wave functions and particle statistics . . . . . . . . . . . .3.2 Independent, indistinguishable particles . . . . . . . . . . . . . . . . .3.3 Second Quantization Formalism . . . . . . . . . . . . . . . . . . . . . .3.3.1 Creation- and annihilation operators . . . . . . . . . . . . . . .3.3.2 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4 Observables in second quantization . . . . . . . . . . . . . . . . . . . .3.5 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7 Selected applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7.1 Spin susceptibility . . . . . . . . . . . . . . . . . . . . . . . . .3.7.2 Bose-Einstein condensate and coherent states . . . . . . . . . .3.7.3 Phonons in an elastic medium . . . . . . . . . . . . . . . . . . .

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484849505052535556565860606165

4 One-dimensional systems of interacting degrees4.1 Classical spin chain . . . . . . . . . . . . . . . . .4.1.1 Thermodynamics . . . . . . . . . . . . . .4.1.2 Correlation function . . . . . . . . . . . .4.1.3 Susceptibility . . . . . . . . . . . . . . . .4.2 Interacting lattice gas . . . . . . . . . . . . . . .4.2.1 Transfer matrix method . . . . . . . . . .4.2.2 Correlation function . . . . . . . . . . . .4.3 Long-range order versus disorder . . . . . . . . .

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707070717374747677

5 Phase transitions5.1 Ehrenfest classification of phase transitions . . . . . . . . . . . . . . . . . .5.2 Phase transition in the Ising model . . . . . . . . . . . . . . . . . . . . . . .5.2.1 Mean field approximation . . . . . . . . . . . . . . . . . . . . . . . .5.2.2 Instability of the paramagnetic phase . . . . . . . . . . . . . . . . .5.2.3 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3 Gaussian transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3.1 Correlation function and susceptibility . . . . . . . . . . . . . . . . .5.4 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.1 Ginzburg-Landau theory for the Ising model . . . . . . . . . . . . .5.4.2 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.3 Range of validity of the mean field theory - Ginzburg criterion . . .5.5 Self-consistent field approximation . . . . . . . . . . . . . . . . . . . . . . .5.5.1 Renormalization of the critical temperature . . . . . . . . . . . . . .5.5.2 Renormalized critical exponents . . . . . . . . . . . . . . . . . . . . .5.6 Long-range order - Peierls argument . . . . . . . . . . . . . . . . . . . . . .5.6.1 Absence of finite-temperature phase transition in the 1D Ising model5.6.2 Long-range order in the 2D Ising model . . . . . . . . . . . . . . . .

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797981818285868991919394959597989899

6 Superfluidity6.1 Quantum liquid Helium . . . . . . . . . .6.1.1 Superfluid phase . . . . . . . . . .6.1.2 Collective excitations - Bogolyubov6.1.3 Gross-Pitaevskii equations . . . . .

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101101102104108

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. . . .. . . .theory. . . .

of freedom. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .

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6.2

Berezinskii-Kosterlitz-Thouless transition . . . . . . . . . . . . . . . . . . . . . . 1116.2.1 Correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2.2 Topological excitations and BKT transition . . . . . . . . . . . . . . . . . 113

7 Linear Response Theory7.1 Linear Response function . . . . . . . . . . . . .7.1.1 Kubo formula - retarded Greens function7.1.2 Information in the response function . . .7.1.3 Analytical properties . . . . . . . . . . . .7.1.4 Fluctuation-Dissipation theorem . . . . .7.2 Example - Heisenberg ferromagnet . . . . . . . .7.2.1 Tyablikov decoupling approximation . . .7.2.2 Instability condition . . . . . . . . . . . .7.2.3 Low-temperature properties . . . . . . . .

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115115116117118119121122123124

8 Renormalization group1258.1 Basic method - Block spin scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.2 One-dimensional Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.3 Two-dimensional Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A 2DA.1A.2A.3

Ising model: Monte Carlo method and Metropolis algorithm133Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Monte Carlo methods in thermodynamic systems . . . . . . . . . . . . . . . . . . 133Example: Metropolis algorithm for the two site Ising model . . . . . . . . . . . . 134

B High-temperature expansion of the 2D Ising model: Finding the phase transition with Pade approximants137B.1 High-temperature expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.2 Finding the singularity with Pade approximants . . . . . . . . . . . . . . . . . . . 139

4

IntroductionThermodynamics is a phenomenological, empirically derived description of the equilibrium properties of macroscopic physical systems. These systems are usually considered large enough suchthat the boundary effects do not play an important role (we call this the thermodynamiclimit). The macroscopic states of such systems can be characterized by several macroscopicallymeasurable quantities whose mutual relation can be cast into equations and represent the theoryof thermodynamics.Much of the body of thermodynamics has been developed already in the 19th century before themicroscopic nature of matter, in particular, the atomic picture has been accepted. Thus, theprinciples of thermodynamics do not rely on the input of such specific microscopic details. Thethree laws of thermodynamics constitute the essence of thermodynamics.Statistical Physics provides a road towards thermodynamics from a microscopic models of matter. We consider a macroscopic systems consisting of an extremely large number of (oftenidentical) microscopic entities which can be found in different microstates, whose dynamics isoften understood through a Hamiltonian. The number of degrees of freedom (variables) is largeenough such that following the evolution of each microscopic entity individually becomes inaccessible. Rather a statistical approach taking averages over the microstates is appropriateto derive the essential and desired information on the macroscopic properties of a macroscopicsystem.There are two basic viewpoints to this: (1) kinetic theory such as the Boltzmann kinetic gastheory and (2) statistical physics based on the Gibbsian concept of ensembles.(1) The kinetic theory is based on statistical time evolution of a macroscopic system using the socalled Master equations. This allows us to discuss systems out of equilibrium which graduallyapproach equilibrium after long time when they are left on their own . The motion towardsequilibrium is most impressively displayed in Boltzmanns H-theorem. Equilibrium is reachedwhen macroscopic quantities do not show any time-dependence. Average properties are obtainedthrough time averages over microstates.(2) The Gibbsian approach considers a large number of identical copies of the system each onebeing in another microstate of the system, corresponding to the same macroscopic parameters(energy, volume, etc). Here time does not play a role and the averages are taken over ensembles, as we will see below. In many aspects this is the more accessible approach to derivethermodynamics and is for equilibrium physics equivalent to the kinetic theory if the ergodicityof the system is guaranteed. This means any microstate in the ensembles is (approximately)connected to any other microstate through temporal evolution following the dynamics given bythe Hamiltonian.

5

Chapter 1

Classical statistical PhysicsStatistical physics deals with the equilibrium properties of matter and provides the microscopicunderstanding and basis for thermodynamics. This chapter develops a new practical approachto equilibrium state of macroscopic systems. Time is not a variable anymore and measurementsmay be considered as averages over a large ensemble of identical systems in different possiblemicroscopic states under the same external parameters.

1.1

Gibbsian concept of ensembles

As a practical example, we consider the state of a gas of N classical particles, given by 3N canonical coordinates q1 , . . . , q3N and by the corresponding 3N conjugate momenta p1 , . . . , p3N . Thesedefine a 6N -dimensional space , where each point in represents a state of the microscopicsystem. Considering the whole system of N particles under certain macroscopic conditions,given external parameter such as temperature, pressure, volume, internal energy, .... , we findthat an infinite number of states in are compatible with the same external condition, andwould not be distinguishable by macroscopic measurements.

[q]

t=0

t>0

[p]

ensemble

time evolution

Fig. 1.1: Time averages are replaced by averages over an ensemble of systems in differentmicroscopic states, but with the same macroscopic conditions.If we would like to calculate a certain macroscopic quantity we could perform temporal averageof the microscopic variables over a very long evolution time. Based on the states in thisis, however, not a practical method as it involves the discussion of the temporal evolution.Gibbs introduced the concept of ensembles to circumvent this problem. Taking a large (infinite)number of systems under identical macroscopic conditions, we can find macroscopic quantitiesby averaging over an ensemble of states. That this scheme is equivalent to a temporal averageis a hypothesis, as it is assumed that the time evolution of the system would lead to all possiblestates also represented in the ensemble. This is the ergodicity hypothesis.1 The set of states1Ergodicity hypothesis: We consider the subspace defined by H(p, q) = E in . Due to energy conservation apoint in this subspace, evolving in time, does not leave this subspace. The ergodicity hypothesis states, that apoint on H = E can reach in its temporal evolution every point of this subspace (Boltzmann, 1887). This is not

6

for given macroscopic parameters is then represented by a distribution of points in the space .This distribution is generally continuous for a gas of particles.For the calculation of averages we introduce the density function (p, q) providing the measureof the density of points in -space ((p, q) stands for the whole state (p1 , . . . , p3N ; q1 , . . . q3N )).Then(p, q)d3N pd3N q(1.1)gives the number of representative points contained in the small volume d3N pd3N q in , veryanalogous to the distribution function in Boltzmann theory. This can now be used to calculateaverages of any desired quantity which can be expressed in the variable (p, q), A(p, q):RdpdqA(p, q)(p, q)RhAi =.(1.2)dpdq(p, q)We will use from now on the short notation dpdq for d3N pd3N q where it is not misleading.

1.1.1

Liouville Theorem

The dynamics of the system of N particles shall be described by a Hamiltonian H(p, q) whichyields the equation of motion in the Hamiltonian formulation of classical mechanics,pi =

Hqi

and qi =

Hpi

(i = 1, . . . , 3N ) .

(1.3)

This equation describes the motion of the points in . If H does not depend on time derivativesof pi and/or qi , then the equations of motion are time reversal invariant. They determine theevolution of any point in uniquely.Now let us consider the points in the space and note that their number does not change intime, as if they form a fluid. Thus, they satisfy the continuity equation ~+ (~v ) = 0(1.4)t~ = (/p1 , . . . , /p3N ; /q1 , . . . , /q3N ). Introwhere ~v = (p1 , . . . , p3N ; q1 , . . . , q3N ) and ducing the generalized substantial derivative we can rewrite this equation asD~ ~v = 0 .+ Dt

(1.5)

The divergence of the velocity ~v is X

3N 3N X qi pi H H~ ~v =+== 0.(1.6)qi piqi pipi qii=1i=1 |{z}=0This means that the points in space evolve like an incompressible fluid. This property isknown as Liouvilles theorem. We may write this also in the form

3N 3N D X X H H 0==+qi+ pi=+.(1.7)Dttqipitpi qiqi pii=1

Using Poisson

brackets2

i=1

this equation reads= {H, } .t

(1.9)

rigorously valid, however. Ehrenfests version of this hypothesis (1911) states: The trajectory of a point comesany other point in the same subspace arbitrarily close in the course of time. Of course, it is possible to find(singular) counter examples, for which only a tiny fraction of the subspace is accessed.2The Poisson bracket is defined as

X u vu v{u, v} == {v, u} .(1.8)qi pipi qii

7

1.1.2

Equilibrium system

A satisfactory representation of a system in equilibrium requires for an ensemble that the densityfunction does not depend on time t, i.e. /t = 0. From Liouvilles theorem we get thecondition that

3N X~ = {H, } .+ pi= ~v (1.10)0=qiqipii=1

A general way to satisfy this is to take a density function which depends only on quantitiesconserved during the motion, such as energy or particle number. Then the system would evolvewithin a subspace where is constant.We may use this feature of now to consider averages of certain quantities, based on the abovementioned equivalence between the temporal and ensemble averages. Defining the temporalaverage of A(p, q) asZ1 ThAi = limA(p(t), q(t))dt(1.11)T T 0for any starting point (p(t = 0), q(t = 0)) in the space and (p(t), q(t)) obeying the equationof motion (1.3). The hypothesis of ergodicity, even in its restricted sense, implies that thisaverage can be taken as an average of an ensemble of an infinite number of different microscopicstates (points in -space). As the evolution of (p, q) conserves the energy, this leads naturallyto consider an ensemble of states of a fixed energy (internal energy). We call such ensemblesmicrocanonical. Although the microcanonical ensemble is not so frequently used in practicethan other ensembles which we will consider later, it is very useful for illustrative purposes. Itdescribes an isolated closed system with no energy exchange with the environment.We postulate that in equilibrium any state of a macroscopic system satisfying the externalconditions appears with equal probability. In our microcanonical description with fixed energy,number of particles N and volume V , we postulate const. E H(p, q) E + E(p, q) =(1.12)0otherwisewhere E is small. The average value of A is then given by (1.2). The validity of this approachis based on the assumption of small mean square fluctuations (standard deviation)h{A hAi}2 i1.hAi2

(1.13)

Such fluctuations should be suppressed by the order N 1 as we will see below.

1.2

Microcanonical ensemble

We consider a macroscopic system of N particles in a volume V which is isolated and closed.The microcanonical ensemble for a given energy E consists of all systems of this kind, whoseenergy lies in the range [E, E + E]. First we define the phase volumeZ(E) = Ndpdq ,(1.14)H(p,q)E

which contains all point in space with energy lower than or equal to E. Moreover, N is arenormalization factor1(1.15)N =N !h3Nwhich compensates for the over-counting of the phase space in the integral by dividing by thenumber of equivalent states reached by permutations of the particles, N !. The factor h3N corrects8

for the dimension integral to produce a dimensionless (E). Thus, h has the units of action([pq] = Js). While this could be Planck constant h, its magnitude is completely unimportant inclassical statistical physics. From this we obtain the volume of the microcanonical ensemble as(E) = (E + E) (E) =

d(E)E .dE

(1.16)

Thus we now can renormalize (p, q) with the condition,ZZNdp dq1 = N dp dq (p, q) =(E) EH(p,q)E+Esuch that(p, q) =

1(E)

0

E H(p, q) E + E

(1.17)

.

(1.18)

otherwise

As postulated (p, q) is constant in the energy range [E, E + E].

1.2.1

Entropy

We use (E) to define the entropyS(E, V, N ) = kB ln (E) .

(1.19)

We can consider (E) or S as a measure of the imprecision of our knowledge of the state of thesystem. The more states are available in microcanonical ensemble, the less we know in whichstate the system is at a given time and the larger is the volume and the entropy.We consider a composite system consisting of two subsystems, 1 and 2,H(p, q) = H1 (p1 , q1 ) + H2 (p2 , q2 ) ,

(1.20)

with (N1 , V1 ) and (N2 , V2 ), resp., for the corresponding particle number and volume. Each ofthe two systems is characterized by 1 (E1 ) and 2 (E2 ), respectively. The volume (E) of themicrocanonical ensemble is the product of the subsystems under the conditionsE = E1 + E2 ,

N = N1 + N2

and V = V1 + V2 .

(1.21)

For simplicity we assume that the volumes and particle numbers of the subsystems may be fixed,while they can exchange energy, such that E1 and E2 can fluctuate. Therefore the volume ofthe microcanonical ensemble of the total system for given E reads,X(E) =1 (E 0 ) 2 (E E 0 ) ,(1.22)0E 0 E

where we assume for the sum a discrete mesh of equally spaced E 0 -values of mesh spacingE ( E). We claim that this sum is well approximated by a single value E00 giving rise toa sharp maximum among the summands (E1 = E00 and E2 = E E00 )3 . The largest term is3

Note that ln i Ni and Ei Ni (i = 1, 2) as both quantities are extensive. We then find quickly thefollowing bounds:E1 (E00 )2 (E E00 ) (E) 1 (E00 )2 (E E00 ) .(1.23)ENote that E/E is the number of summands. Thus, we obtain for the entropykB ln 1 (E00 )2 (E E00 ) S(E) kB ln 1 (E00 )2 (E E00 ) + kB ln

E.E

(1.24)

It is important to see that the last term only scales with system size as ln N (keeping E fixed), while the firstterm scales with N . Therefore in the very large-N limit the last term is irrelevant. Which proves that S(E) isgiven by the maximal term in the sum (1.22).

9

obtained by

1 (E 0 ) 2 (E E 0 ) 0 0 =0E 0E =E

(1.25)

0

such that

0 =

2 (E E 0 )11 1 (E 0 )+1 (E 0 ) E 02 (E E 0 )E 0

E 0 =E00

(1.26)

ln 2 (E2 ) ln 1 (E1 ) .=

E1E2E1 =E 0E2 =EE 00

0

1 = E 0 and E2 = E E 0From this we obtain with E00

S1 (E1 ) S2 (E2 ) =E1 E1 =E1E2 E2 =E2

(1.27)

which can be considered as the equilibrium condition. Note that this is equivalent to the statement that the equilibrium state corresponds to the macrostate with the largest number of microscopic realizations.Identifying E as the internal energy U we define the temperatureS1=UT

11=,T1T2

(1.28)

leading to T as an equilibrium state variable. We have ignored the other variables V, N whichwe will consider later.Let us assume that there is some impediment which forces the two subsystems to specific variables such that each subsystem independently is at equilibrium, but not the total system. Forexample, a big vessel of gas may be separated by a wall into two subvessels. Then we find forthe corresponding of the combined system, E = E1 + E2V = V1 + V2 (E, V, N ) = 1 (E1 , V1 , N1 )2 (E2 , V2 , N2 )with(1.29)N = N1 + N2such that the entropy is given byS(E,V, N ) = S1 (E1 , V1 , N1 ) + S2 (E2 , V2 , N2 ) S(E, V, N ) ,

(1.30)

i.e. the entropy is concave. This means also the equilibrium is obtained by the maximal entropy,which is a consequence of the second law of thermodynamics. In terms of the volume of themicrocanonical volume the equilibrium state assumes among all volumes the maximal one,the most likely one in terms of probability.

1.2.2

Relation to thermodynamics

With (1.19) we have a definition of the entropy which for the variables E(= U ), V and N is athermodynamic potential and allows us to calculate a variety of state variables and relations,

SSS1pdS =dE +dV +dN = dE + dV dN .(1.31)E V,NV E,NN E,VTTTThis allows also to calculate the thermodynamic equation of state,

Sp=TV E,N10

(1.32)

and to determine other thermodynamic potentials. The caloric equation of state is obtainedfrom

S1= ,(1.33)U V,NTby using U = E as the internal energy. The derivative with respect to N yields finally thechemical potential

S = T,(1.34)N E,Vthe energy in order to add a particle to the system.

1.2.3

Ideal gas - microcanonical treatment

We consider a classical gas of N independent mono-atomic particles in the fixed volume V ,which is closed and isolated. The Hamiltonian is simply given byNXp~i2H(p, q) = H(p) =.2m

(1.35)

i=1

Hence we obtain the volumeZ(E) = N

dpdq = N V

N

H(p)E

Zdp .

(1.36)

H(p)E

This p-integral corresponds to the volume of a sphere of radius R in 3N -dimensional space.4The integral is straightforward,(E) = N V N C3N (2mE)3N/2

with

Cn =

n/2

n2 + 1

(1.39)

4Volume of sphere in -dimensional space: V(R) = CR . The volume enclosed in a thin shell of width R atthe radius R is given by

R(1.37)vshell = V(R) V(R R) = CR 1 1 R

with R R, see Fig. 1.2 . In the limit very large, R R, we findvshell V(R) .

(1.38)

Thus, the main part of the volume in the sphere is concentrated at the outermost shell.

000001111111110000000001111100001111000001111100001111000001111100001111R000001111100001111000001111100001111000001111100001111000001111100001111R

Rn

10.80.60.40.20

n=5n = 10n = 40

0

0.2 0.4 0.6 0.8

vshell,1% /CRn1.00.80.60.40.20R20012

400

600

800 n

Fig. 1.2: The volume of the sphere is located close to the surface in high dimensions, see the central diagram.Thus, the fraction of the volume close to the shell and the total volume converges towards one as n (diagram on the right hand side).

11

where C3N is the proper prefactor for the volume of an 3N -dimensional sphere.5 This leads to(E) =

(E)3NE = N C3N V N2m (2mE)3N/21 E .E2

(1.42)

Remarkably, for very large N ( 1023 ) we find that the following definitions for the entropy areidentical up to terms of order ln N and constants:S = kB ln (E, V, N )

and

S = kB ln (E, V, N )

(1.43)

leading to

N

S = kB ln(N V C3N ) + kB

3N3N 1 ln(2mE) + kB ln2mE22

(1.44)3N= kB ln(N V N C3N ) + kBln(2mE) + O(ln N ) = S + O(ln N ) .2Since we can drop terms of order ln N for the extensive entropy, we will continue, for convenience,using S instead of S ,( )3N3N3N2mE 3/2kB ln+kB N kB ln N + N kB (1.45)S(E, V, N ) = N kB ln V2h222where we used Stirlings formula1ln(2n)for n ,2and neglected all terms of order ln N . We then rewrite( )V 4mE 3/25S(E, V, N ) = N kB ln+ N kB .N 3N h22ln n! n ln n n +

(1.46)

(1.47)

This equation may now be solved for E so that we obtain the internal energy as a thermodynamicpotential

53N 5/3 h2SU (S, V, N ) = E =exp.(1.48)3N kB34mV 2/3The thermodynamic quantities are obtained by derivation: the temperature

2U3U=U = N kB T ,(1.49)T =S V,N3N kB2the pressure

p=

UV

=S,N

2UN kB T=3VV

pV = N kB T ,

and the chemical potential(

)UU 5 2 SV 2mkB T 3/2=== kB T ln.N S,VN 3 3 N kBNh2

(1.50)

(1.51)

Through further derivatives it is possible to obtain various response functions. The ideal gas isreadily described by means of the microcanonical ensemble.5

Prefactor Cn : Use the n-dimensional Gaussian integral in Cartesian coordinatesZ +nZ +Z +222I=dx1 dxn e(x1 ++xn ) =dxex= n/2 .

(1.40)

The same integral in spherical coordinates is given byZ Z nn

2nnnI = nCndrrn1 er = Cndt t 2 1 et = Cn = Cn +1222200such that we obtain Cn given in (1.39). Note, (n + 1) = n! for n 0 as an integer.

12

(1.41)

1.3

Canonical ensemble

We change to a macroscopic system for which we control the temperature by connecting it to avery large heat reservoir. The system together with the reservoir forms a closed system of giventotal energy. Therefore we consider two subsystems, system 1 describing our system and system2 being the heat reservoir,H(p, q) = H1 (p1 , q1 ) + H2 (p2 , q2 ) .

(1.52)

The heat reservoir is much larger than system 1, N2 N1 , such that energy transfer betweenthe two subsystems would be too small to change the temperature of the reservoir. Withinthe microcanonical scheme we determine the phase space of the combined system in the energyrangeE E1 + E2 E + E .(1.53)

E2 , S2 , TE1 , S1 , T

Fig. 1.3: The two systems 1 and 2 are coupled thermally. System 2 acts as a huge reservoirfixing the temperature of the system 1 which we want to investigate.

Therefore the volume of the microcanonical ensemble of the total system isX(E) =1 (E1 )2 (E E1 ) .

(1.54)

0E1 E

1 = E 0 (E2 = E E 0 )Analogously to our previous discussion in section 1.2.1, there is one value E002 E1 is valid. Thewhich provides the by far dominant contribution. In addition here E1 ) and 2 (E2 ) and (E) 1 (E1 )2 (E E1 ). Duecorresponding volumes in -space are 1 (Eto this simple product form we can determine the density function 1 (p1 , q1 ) of the system 1, byconsidering the mean value of A(p, q) in system 1,RRR, q1 ) 2 dp2 dq2 (p, q)dp1 dq1 A(p1 , q1 )1 (p1 , q1 )1 dpR1 dq1 A(p1 RhAi1 == 1 R.(1.55)1 dp1 dq1 2 dp2 dq2 (p, q)1 dp1 dq1 1 (p1 , q1 )Taking into account that (p, q) is constant in the range E H1 (p1 , q1 ) + H2 (p2 , q2 ) E + Ewe obtainRdp1 dq1 A(p1 , q1 )2 (E H1 (p1 , q1 ))hAi1 = 1 R.(1.56)1 dp1 dq1 2 (E H1 (p1 , q1 ))2 E E1 we may expand 2 (E H1 (p1 , q1 )) in H1 (p1 , q1 )) ,Using the assumption that E

2 ) S2 (E

kB ln 2 (E H1 (p1 , q1 )) = S2 (E H1 (p1 , q1 )) = S2 (E) H1 (p1 , q1 )+ E2 E2 =E= S2 (E)

H1 (p1 , q1 )+ T

13

(1.57)

from which we derive2 (E H1 (p1 , q1 )) = eS2 (E)/kB eH1 (p1 ,q1 )/kB T .

(1.58)

Here T is the temperature of both systems which are in equilibrium.Within the canoncial ensemble, taking the temperature T as a given parameter, we write generally for the density function, the probability to find the system of N particles in the microstate(p, q),1(p, q) = eH(p,q)/kB T ,(1.59)Zwhere we introduced the partition function ZZ1Z = N dp dqeH(p,q)with =,(1.60)kB Twhich, up to prefactors, corresponds to the volume of the ensemble of system 1, called canonicalensemble, again renormalized by N .6

1.3.1

Thermodynamics

The connection to thermodynamics is given by the relationZ = eF (T,V,N ) ,

(1.61)

where F (T, V, N ) is the Helmholtz free energy, a thermodynamical potential. Note, F is anextensive quantity, because obviously scaling the system by a factor would yield Z . Moreover,F = U TSwith

(1.62)

U = hHi

S=

and

FT

.

(1.63)

V,N

This can be proven using the equation,Z1 = N

dp dqe(F H) ,

(1.64)

which through differentiation with respect to on both sides gives,)(

ZF(F H)0 = N dp dqeF +H V,N

F (T, V, N ) U (T, V, N ) T

FT

(1.65)

=0.V,N

Using this formulation for the free energy we obtain for the pressure

Fp=,V T,Nwhich in the case of a gas leads to the thermodynamic equation of state.The internal energy is easily obtained from the partition function in the following way,ZN1 ZU (T, V, N ) = hHi =dp dq HeH = =ln Z .ZZ

(1.66)

(1.67)

This is the caloric equation of state.6

Note that there is, rigorously speaking, the constraint H1 (p1 , q1 ) < E. However, ignoring this constraint is agood approximation, as the main contribution is from the valid range.

14

1.3.2

Equipartition law

We now consider a set of special average values which will lead us to the so-called equipartitionlaw, the equal distribution of energy on equivalent degrees of freedom. We examine the meanvalue,

ZZNH HN HH=dp dqqe=dp dqqeqqZqZqN=Z

0

Z

0

dp d q q e|{z=0

H

N +Z

Z

dpdqeH = kB T ,

(1.68)

}

whereused integration by parts leading to the boundary terms in the q -coordinate (expressedR 0 we0by d q...), which we assume to vanish. Analogously we find for the momentum

Hp= kB T .(1.69)pIf the Hamiltonian is separable into a p-dependent kinetic energy and a q-dependent potentialenergy part and, moreover, if the following scaling behavior is validwith Ekin (p) = 2 Ekin (p)

H(p, q) = Ekin (p) + V (q)

and V (q) = V (q)

(1.70)

then we can use the above relations and find for mono-atomic particleshEkin i =

3NkB T2

hV i =

and

3NkB T .

(1.71)

The total energy is given by the sum of the two contributions.

1.3.3

Ideal gas - canonical treatment

Consider a gas of N particles without external potential and mutual interactions described bythe HamiltonianNXp~i 2H(p) =.(1.72)2mi=1

The partition function is given byZ = N

N ZY

3

3

~pi 2 /2mkB T

d pi d qi e

Z= N

3

3

~p 2 /2mkB T

N

d pd qe

(1.73)

i=1

= N V N {2mkB T }3N/2 .From this we obtain the free energy and the internal energy using Stirlings formula,( )V 2mkB T 3/2F (T, V, N ) = kB T ln Z = N kB T ln N kB T ,Nh2(1.74)U (T, V, N ) =

3Nln Z =kB T2

(caloric equation of state) .

The entropy is given by

S(T, V, N ) =

FT

(

= N kB lnV,N

15

VN

2mkB Th2

3/2 )+

5NkB2

(1.75)

and the pressure by

p=

FV

N kB TV

=T,N

(1.76)

which corresponds to the thermodynamic equation of state. Finally the chemical potential isobtained as(

)

FV 2mkB T 3/2== kB T ln.(1.77)N T,VNh2An important aspect for the ideal system is the fact that the partition function has a productform because each particle is described independently. In this way it leads to an extensive freeenergy and internal energy.

1.4

Grand canonical ensemble

We consider now a new situation by allowing beside the heat exchange also the exchange ofmatter of our system with a very large reservoir, see Fig. 1.4 . Thus we take the system 1 withN1 particles in a volume V1 coupled to the reservoir 2 with N2 particles in the volume V2 withN1 N2

V1 V2 ,

and

(1.78)

and N = N1 + N2 and V = V1 + V2 fixed.

TN2 , V2T,

N1 , V1T,

Fig. 1.4: The two systems 1 and 2 can exchange matter between each other while thesurrounding heat reservoir fixes the temperature of system 1 and 2. System 2 acts as a hugeparticle reservoir fixing the chemical potential of the system 1 which we want to investigate.The Hamiltonian can be decomposed into two partsH(p, q, N ) = H(p1 , q1 , N1 ) + H(p2 , q2 , N2 )

(1.79)

such that the corresponding partition function for given temperature (everything is coupled toan even larger heat reservoir) is given byZ1ZN (V, T ) = 3Ndp dq eH(p,q,N ) .(1.80)h N!The factor 1/N ! takes into account that all possible commutation of the particles give the samestates (distinguishable classical particles). Now we segregate into the subsystems fixing thevolumes and particle numbers (N2 = N N1 ),ZN

=

NX

1h3N N !

N1 =0

N!N1 !N2 !

Z

Zdp1 dp2

Z

dq2 e{H(p1 ,q1 ,N1 )+H(p2 ,q2 ,N2 )}

dq1V1

V2

(1.81)=

NXN1 =0

1h3N1 N1 !

Z

dp1 dq1 eH(p1 ,q1 ,N1 )

V1

16

1h3N2 N2 !

ZV2

dp2 dq2 eH(p2 ,q2 ,N2 ) .

Note that the distribution of the particles into the two subsystems is not fixed yielding thecombinatorial factor of N !/N1 !N2 ! (number of configurations with fixed N1 and N2 by permutingthe particles in each subsystem). From this we define the probability (p1 , q1 , N1 ) that we canfind N1 particles in the volume V1 at the space coordinates (p1 , q1 ),ZeH(p1 ,q1 ,N1 )(p1 , q1 , N1 ) =dp2 dq2 eH(p2 ,q2 ,N2 )(1.82)ZN N1 !h3N2 N2 ! V2which is renormalized asNXN1 =0

1

Z

h3N1

dp1 dq1 (p1 , q1 , N1 ) = 1 .

(1.83)

ZN2 1 H(p1 ,q1 ,N1 )e,ZN N1 !

(1.84)

V1

We may write(p1 , q1 , N1 ) =

where we now use the relationZN2 (V2 , T )= e{F (T,V V1 ,N N1 )F (T,V,N )}ZN (V, T )

(1.85)

with

F (T, V V1 , N N1 ) F (T, V, N )

FV

T,N

V1

FN

T,V

N1 = N1 + pV1 . (1.86)

Thus we definez = e

(1.87)

which we call fugacity. Thus within the grand canonical ensemble we write for the densityfunctionz N {pV +H(p,q,N )}(p, q, N ) =e.(1.88)N! is the chemical potential as introduced earlier. We now introduce the grand partition functionZ(T, V, z) =

X

z N ZN (V, T ) ,

(1.89)

N =0

which incorporates all important information of a system of fixed volume, temperature andchemical potential.

1.4.1

Relation to thermodynamics

We use now (1.88) and integrate both sidesZXzNdp dq H(p,q,N )1 = epVe= epV Z(T, V, z)N!h3N

(1.90)

N =0

which leads tothe grand potential

(T, V, ) = pV = kB T ln Z(T, V, z) ,

(1.91)

d = SdT pdV N d .

(1.92)

The average value of N is then given by

1 XhN i = = kB Tln Z = zln Z =N z N ZN . T,VzZ

(1.93)

N =0

It is also convenient to derive again the internal energyU =ln Z

17

CV =

UT

.V,

(1.94)

1.4.2

Ideal gas - grand canonical treatment

For the ideal gas, it is easy to calculate the grand partition function (here for simplicity we seth = 1),Z(T, V, z) =

XN =0

noXzN NV (2mkB T )3N/2 = exp zV (2mkB T )3/2 . (1.95)z ZN (T, V ) =N!N

N =0

We can also derive the probability PN of finding the system with N particles. The average valueis given byhN i = z

zV (2mkB T )3/2 = zV (2mkB T )3/2z

Z = ehN i .

(1.96)

From this we conclude that the distribution function for the number of particles is given byPN = ehN i

hN iN12e(N hN i) /2hN ipN!2hN i

(1.97)

which is strongly peaked at N = hN i 1.7 The fluctuations are given byhN 2 i hN i2 = z

hN i= hN iz

T =

vkB T

=

1.p

(1.98)

The grand potential is given by(T, V, ) = kB T e V (2mkB T )3/2 = kB T hN i = pV .

(1.99)

The chemical potential is obtained by solving Eq.(1.96) for = kB T ln

1.4.3

hN i(2mkB T )3/2V

!.

(1.100)

Chemical potential in an external field

In order to get a better understanding of the role of the chemical potential, we now consider anideal gas in the gravitational field, i.e. the particles are subject to the potential (h) = mg h,where h and g denote the altitude and We introduce a reference chemical potential 0 as aconstant. Then we write for the chemical potential,on + kB T ln n(2mkB T )3/2 = 0 + mg h(1.101)where we define n = hN i/V as the local number density of particles. In equilibrium the temperature and the chemical potential shall be constant. We may determine by the condition = 0 the density is n = n0 ,that at hnon 0(T ) = 0 + kB T ln n0 (2mkB T )3/2 mgh = kB T ln.(1.102)nWe can now solve this equation for n = n(h),

= n0 emgh ,n(h)

(1.103)

and with the (local) equation of state we find

= n(h)kB T = p0 emgh .p(h)

(1.104)

This is the famous barometer formula.7Note that for > 0 it pays the energy to add a particle to the system. Therefore as T goes to 0 the averageparticle number hN i increases (diverges). Oppositely, hN i decreases for T 0, if < 0 and energy has to bepaid to add a particle.

18

1.5

Fluctuations

Changing from one type of ensemble to the other we have seen that certain quantities whichhave been strictly fixed in one ensemble are statistical variables of other ensembles. Obviousexamples are the internal energy which is fixed in the microcanonical ensemble but not in theother two, or the particle number which is fixed in the microcanonical and canonical ensemblesbut not in the grand canonical ensemble. The question arises how well the mean values ofthese quantities are determined statistically, which is connected with the equivalence of differentensembles. In this context we will also introduce the fluctuation-dissipation theorem whichconnects the fluctuations (statistical variance) of a quantity with response functions.

1.5.1

Energy

In the canonical ensemble the internal energy is given as the average of the Hamiltonian U = hHi.Therefore the following relation holds:ZN dp dq [U H] e(F H) = 0 .(1.105)Taking the derivative of this equation with respect to we obtain

ZUUF0=+ N dp dq (U H) F T H e(F H) =+ h(U H)2 i .T

(1.106)

This leads to the relation for the fluctuations of the energy around its average value U ,hH2 i hHi2 = h(U H)2 i =

UU= kB T 2= kB T 2 CV .T

(1.107)

Because CV is an extensive quantity and therefore proportional to N , it follows that1hH2 i hHi22hHiN

(1.108)

which is a sign of the equivalence of microcanonical and canonical ensembles. In the thermodynamic limit N , the fluctuations of the energy vanish compared to the energy itself.Therefore the internal energy as U = hHi = E is a well defined quantity in the canonicalensemble as well as it was in the microcanonical ensemble.We now consider the partition functionZZ ZH(p,q)EZ = N dp dq e=dE (E)e=0

dE e

E+ln (E)

Z

=

0

dE e(T S(E)E)

0

(1.109)where the entropy S(E) is defined according to the microcanonical ensemble. The maximum ofthe integrand at E = E0 is defined by the condition, see Fig. 1.5,

S 2 S T=1and .The fermions occupy states within a sphere in momentum space, the Fermi sphere (Fig.2.2).The particle density n isZ(2s + 1)kF3N2s + 12s + 1 4 3n==d3 phnp~ i =pF =(2.78)33Vhh36 2where pF is the Fermi momentum (p~F = (T = 0) = F ), isotropic, and kF = pF /~ is the Fermiwavevector. The groundstate energy isZ32s + 1Vd3 pp~ hnp~ i = (2s + 1)N FU0 =(2.79)h310where F denotes the Fermi energy. The zero-point pressure is obtained through (2.56),p0 =

2 U01N= (2s + 1) F .3V5V

(2.80)

In contrast to the classical ideal gas, a Fermi gas has finite zero-point pressure which is againa consequence of the Pauli principle and is responsible for the stability of metals, neutron starsetc.

Next we turn to finite temperatures for which the occupation number broadens the step at pF .We use now (2.70, 2.78) to obtain the relation

FkB T

3/2

3 3 N 3/2 2 1/2==++ ,4 2s + 1 VkB T8 kB T32

(2.81)

p

z

p

py

F

pxFigure 2.2: Fermi sphere of occupied single particle states. Fermi radius pF .

which at constant density n = N/V can be solved for the chemical potential,!

2 kB T 2(T ) = F 1 + ,12F

(2.82)

and analogously we obtain for the pressure,p(T ) = p0

5 21+12

!

2

kB TF

+

.

(2.83)

Again we derive the internal energy from the relation (2.56)3U = pV = U02

5 21+12

kB TF

!

2+

,

(2.84)

which also leads to the heat capacity for fixed NCN = kB N

2kB T(2s + 1)+ ,4F

(2.85)

see Fig.2.3. This is the famous linear temperature dependence of the heat capacity, which canbe well observed for electrons in simple metals. Obviously now the third law of thermodynamicsT 0is satisfied, CN 0. Also the entropy goes to zero linearly in T .CVU

3N kB23N kB T2

quantumclassic

3N F5

0

0

quantumclassic

kB T

0

0

kB T

Figure 2.3: The internal energy U and the heat capacity CV in the quantum situation comparedwith the classical situation.

33

2.6.3

Spin-1/2 Fermions in a magnetic field

We consider now the magnetic response of spin s = 1/2 fermions in a magnetic field (idealparamagnetic gas). The Hamiltonian has to be extended by a Zeeman term. Taking the fieldalong the z-axis this reads,NgB X zsi H(2.86)HZ = ~i=1

with g = 2 as gyromagnetic ratio and B the Bohr magneton. This can be absorbed into aspin-dependent fugacity,7z = eB H ,(2.90)such that the density of fermions is given byN11 = = 3 f3/2 (z+ ) + f3/2 (z ) = n+ + nVv

(2.91)

MB = B (n+ n ) = 3 f3/2 (z+ ) f3/2 (z ) .V

(2.92)

n=and the magnetizationm=

Let us now calculation the spin susceptibility for zero magnetic field, given by

f3/2 (z) 2B22Bm

==2z=f (z)H H=0 3 kB Tz H=0 3 kB T 1/2

(2.93)

with z = e . We may now again consider limiting cases.High-temperature limit: We replace z 1 in Eq.(2.93) using Eq.(2.71) with n = 1/v and find

2B n3 n=1 5/2 .(2.94)kB T2The first term is the result for particles with spin and has a Curie like behavior and the secondterm is the first quantum correction reducing the susceptibility.Low-temperature limit: Taking only the lowest order for z 1 we obtain,2244 = 3 B (ln z)1/2 = 3 B kB T kB T

FkB T

1/2

= 2B

3n.2F

(2.95)

This is the famous Pauli spin susceptibility for a Fermi gas, which is temperature independent.7

We calculate the grand canonical partition functionnp~Y X + H np~ X H np~ Y Y X BBz ep~Z=ze p~ze p~=p~

np~

np~

(2.87)

p~ =+, np~

where z is defined as in Eq.(2.90). The grand canonical potential is given by = kB T ln Z =

kB T f5/2 (z+ ) + f5/2 (z )3

(2.88)

from which we determine the magnetizationm

1 kB T X kB T X z == 3f5/2 (z ) = 3f5/2 (z)V HHHzz=z(2.89)

B XB X= 3zf5/2 (z)= 3f3/2 (z ) z z=z

corresponding to Eq.(2.92).

34

2.7

Bose gas

There are two situations for Bosons: (1) a system with well-defined particle number, e.g. bosonicatoms, 4 He, ... ; (2) Bosons which results as modes of harmonic oscillators (no fixed particlenumber), e.g. photons, phonons, magnons, etc..

2.7.1

Bosonic atoms

We consider Bosons without spin (S = 0) for which 4 He is a good example. Analogously to thefermions we introduce functions of z to express the equation of state and the particle number,p11 X zl= 3 g5/2 (z) = 3kB Tl5/2l=1

(2.96)

1N11 X zl.== 3 g3/2 (z) = 3vVl3/2l=1

For small z both functions grow linearly from zero and g3/2 (z) has a divergent derivative forz 1. We concentrate on the range 0 < z 1, such that (T ) 0. For z = 1 we obtaing3/2 (1) =

X 1= (3/2) 2.6123/2ll

and g5/2 (1) =

X 1= (5/2) 1.3425/2ll

(2.97)

where (x) is Riemanns -function (see Fig.2.4).

g 3/2g 5/2

Figure 2.4: Functions g3/2 (z) and g5/2 (z).

2.7.2

High-temperature and low-density limit

It is easy to see that (like the fermions) the bosons behave in this limit as a classical ideal gas.An intriguing aspect occurs, however, in the quantum corrections. For the pressure we find

NN 3p(T ) = kB T 1 5/2+ .(2.98)VV2The quantum correction reduces the classical ideal gas pressure and yields the compressibility

1 VV1T = =.(2.99)3NV p T,NN kB T 1 3/22

35

V

In contrast to the fermions where the quantum nature of the particles diminishes the compressibility, here the compressibility is enhanced. Actually, in this approximation the compressibilityeven diverges ifN23/2 = 3 ,(2.100)Vi.e. at low enough temperature or high enough density. We will now see that this indeed indicatesan instability of the Bose gas.

2.7.3

Low-temperature and high-density limit: Bose-Einstein condensation

Let us now consider Eq. (2.96) carefully. The function g3/2 (z) is monotonically increasing withz. If T is lowered, T 1/2 increases, such that z has to increase too in order to satisfy (2.96).Therefore approaches the singular point at 0 (z = 1). The critical point is determined byh2,T=c2kB m[(3/2)V /N ]2/3N 3(2.101)g3/2 (1) = (3/2) = V3Nh Vc =.(3/2) (2mkB T )3/2This defines a critical temperature Tc and critical volume Vc below which a new state of theBose gas occurs. Note that this equation is qualitatively very similar to (2.100) and evenquantitatively not so far ((3/2) 2.612 23/2 2.85) . The question arises what happens forT < Tc or V < Vc . Actually the problem occurring in (2.96) and (2.101) arises in the stepZX1V1N= 3 d3 p ( ).(2.102)()p~p~he1e1p~The integral does not count the occupation of the state p~ = 0, because the momentum distribution function entering the integral,(p) =

p2e(p~ ) 1

(0) = 0 .

(2.103)

This is fine as long as the occupation of the ~p = 0-state (single-particle groundstate) is vanishingly small compared to N . However, for T < Tc (V < Vc ) the occupation becomes macroscopic,hnp~=0 i/N > 0 and we cannot neglect this contribution in the calculation of N (see Fig.2.5). Thus,the correct density isN1= 3 g3/2 (z) + n0 (T ) = nn (T ) + n0 (T )(2.104)Vwith n0 (T ) denoting the density of bosons in the single-particle groundstate (~p = 0). Theseparticles form a condensate, the Bose-Einstein condensate. What happens at Tc is a phasetransition. We encounter here a two-fluid system for which the total particle density splitinto a condensed fraction n0 and a normal fraction nn . From (2.104) we find the temperaturedependence of n0 (see Fig.2.5)," 3/2 #NT1.(2.105)n0 (T ) =VTc

Next we also determine the equation of state,kB T 3 g5/2 (z) , V > Vcp= kB T g5/2 (1) , V < Vc336

.

(2.106)

n0

n1

n0

nn

p

Tc

T

Figure 2.5: Occupation: Left panel: A macroscopic fraction of particle occupy the momentump = 0-state for T < Tc . Right panel: Temperature dependence of the condensate fraction.

We now consider the compressibility for V > Vc . For this purpose we first determineg 0 (z)V3 3/2= N ,zg3/2 (z)2

(2.107)

and considerpkB T 0z= 3 g5/2(z)VV

T =

0 (z)g3/2N 60 (z) ,V kB T g3/2 (z)2 g5/2

(2.108)

where we use the notation gn0 (z) = dgn (z)/dz. As anticipated earlier the compressibility diverges0 (z) for z 1. In theat the transition V Vc (or T Tc ), since the derivative g3/2condensed phase the pressure is independent of V as is obvious from (2.106). Therefore thecondensed phase is infinitely compressible, i.e. it does not resist to compression.Some further thermodynamic quantities can be derived. First we consider the entropy S fromthe grand canonical potential

5vg (z) ln z , T > Tc ,N kB

23 5/2pVS(T, V, ) = ==(2.109) 3/2T V,T V, g(1)5T5/2 N kB,T < Tc ,2 g3/2 (1) Tcwhere we used (2.96).8 For the heat capacity at fixed particle number N we find from the8

Calculation of the temperature derivatives: (1) Fixed chemical potential: V kB T5V kBg5/2 (z) =g5/2 (z) +T 33

V kB T g3/2 (z) z3T}{zz| V= kB 3 g3/2 (z) = N kB ln z

(2.110)

where we used zg5/2 = g3/2 /z.(2) Fixed particle number: we use

g3/2 (z) =which leads to the relation

N 3V

dg3/2g1/2 (z) dz3 N 3==dTzdT2V T

dg5/2g3/2 (z) dz9 g3/2 (z) N 3==.dTzdT4 g1/2 (z) V

This leads to the expression for the heat capacity.

37

(2.111)

(2.112)

internal energy U = 32 pV ,

CV =

UT

=V,N

9 g3/2 (z)15vg (z) , T > Tc ,N kB43 5/24 g1/2 (z) 15 g5/2 (1) T 3/2, N kB4 g3/2 (1) Tc

(2.113)

T < Tc .

CVNk B3/2

Tc

T

Figure 2.6: Heat capacity: CV has a cusp at the transition and vanishes as T 3/2 towardszero-temperature. In the high-temperature limit Cv approaches 3N kB /2 which corresponds tothe equipartition law of a mono-atomic gas.

In accordance with the third law of thermodynamics both the entropy and the heat capacity goto zero in the zero-temperature limit. The entropy for T < Tc can be viewed asS=sN

TTc

3/2=

nn (T )sn

with

5 g5/2 (1)s = kB2 g3/2 (1)

(2.114)

where s is the entropy per normal particle (specific entropy), i.e. a non-vanishing contributionto the entropy is only provided by the normal fraction (two-fluid model). The heat capacity hasa cusp at T = Tc .p

transition line

ptransition line

isothermal

TBECvc(T)

v

T

Figure 2.7: Phase diagrams; left panel: p-v-diagram; the isothermal lines reach the transitionline with zero-slope, i.e. the compressibility diverges. Right panel: p-T -diagram; the condensedphase corresponds to the transition line, there is no accessible space above the transition line.

We consider now the phase diagram of various state variable.

38

(1) p-v-diagram: phase transition linep0 v 5/3 =

g5/2 (1)h2,2m [g3/2 (1)]5/3

(2.115)

(2) p-T -diagram: phase transition linep0 =

kB Tg (1) T 5/23 5/2

(2.116)

which is the vapor pressure (constant for T < Tc ) (see Fig.2.7). We use this line to determinethe latent heat l per particle via the Clausius-Clapeyron relation,dp0l=dTT v

with

l = T s .

(2.117)

The condensate takes no specific volume compared to the normal fraction. Thus, v = vc .Therefore we obtainl = T vc

dp05 kB g5/2 (1)5 g5/2 (1)= T vc= T kB3dT22 g3/2 (1)

(2.118)

where we used the relation 3 = vc g3/2 (1). Note that this is consistent with our result onthe specific entropy s. The condensed phase has no specific entropy such that s = s and,consequently, l = T s using (2.114).Examples of the Bose-Einstein condensates is the quantum fluid 4 He which shows a condensationbelow T 2.18K into a superfluid phase. We will discuss this in more detail in Chapt. 6. Afurther very modern example are ultracold atoms in optical traps, e.g. 87 Rb (37 electrons +87 nucleons = 124 Fermions Boson). For 2000 atoms in the trap the critical temperature toBose-Einstein condensation is as low as 170 nK (for the measured momentum distribution seeFig.2.8).

Figure 2.8: Velocity distribution of Rb-atoms: Left panel: T > Tc ; middle panel: T Tc ;right panel T Tc . A peak at the center develops corresponding to the fraction of particleswith zero-velocity and at the same time the fraction of particles with finite velocity shrinks.(Source: http://www.nist.gov/public affairs/gallery/bosein.htm)

2.8

Photons and phonons

We consider now classes of Bose gases whose particle numbers is not conserved. They are derivedas normal modes of harmonic oscillators. Thus we first consider the statistical physics of the39

harmonic oscillator. The most simple example is the one-dimensional harmonic oscillator whosespectrum is given by

1withn = 0, 1, 2, . . .(2.119)n = ~ n +2and the eigenstates |ni.9 The quantum number n is considered as the occupation number ofthe oscillator mode. We analyze this within the canonical ensemble formulation with a giventemperature T . The partition function readsZ = treH =

X

X

hn|eH |ni =

n=0

en = e~/2

n=0

X

e~n =

n=0

e~/2.1 e~

(2.123)

The internal energy is obtained throughU =

ln Z1~.= ~ + ~2e1

The heat capacity isC=

dU= kBdT

~2kB T

2

(2.124)

1,sinh (~/2)2

(2.125)

with the limiting properties

C=

kB

kB T ~

kB

~kB T

2

(2.126)~

e

kB T ~ .

In the high-temperature limit the heat capacity approaches the equipartition law of a onedimensional classical harmonic oscillator. The mean quantum number is given by1 X n1hni == ~ne.Ze1

(2.127)

n=0

This corresponds to the Bose-Einstein distribution function. Thus we interpret n as a numberof bosons occupying the mode .9

Harmonic oscillator with the HamiltonianH=

P 22 21+Q = ~ a a +222

with a and a as the lowering and raising operators, respectively,r~Q=(a + a )2[a, a ] = 1r~P = i(a a ) 2

(2.120)

P ] = i~[Q,

The stationary states |ni are defined by H|ni = n |ni and obeya|ni = n|n 1ianda |ni = n + 1|n + 1i .

(2.121)

(2.122)

We can interpret a and a as creation and annihilation operator for a particle of energy ~. In the language ofsecond quantization the commutation relation of a and a corresponds to that of bosons.

40

2.8.1

Blackbody radiation - photons

Electromagnetic radiation in a cavity is a good example of a discrete set of independent harmonicoscillators. Consider a cubic cavity of edge length L. The wave equation is expressed in termsof the vector potential

1 22~=0~(2.128) Ac2 t2and the electric and magnetic field are then~~ = 1 A~ =~ A~,EandB(2.129)c t~ A~ = 0 and = 0. This can be solved by a plane wave,where we used the Coulomb gauge = ~k = c|~k| ,noX1i~k~rit i~k~r+it~withA(~r, t) = A~k~e~k e+ A~k~e~k eV ~~e~k ~k = 0 ,k,(2.130)i.e. a linear dispersion relation and a transverse polarization ~e~k (: polarization index). Assuming for simplicity periodic boundary conditions in the cube we obtain the quantization ofthe wavevector,~k = 2 (nx , ny , nz )withni = 0, 1, 2, . . .(2.131)LEach of the parameter set (~k, ) denotes a mode representing an independent harmonic oscillator.These oscillators can be quantized again in the standard way.10 The states of a mode differ byenergy quanta ~~k . The occupation number n~k is interpreted as the number of photons in thismode.The partition function is then derived from that of a harmonic oscillator!2Y e~~k /2Ye~~k /2Z==(2.135)1 e~~k1 e~~k~~k,

k

where the exponent 2 originates from the two polarization directions. The internal energy followsfromZZX ~~ ln Z~kU (T ) = =2=dD()=Vdu(, T )(2.136)e~ 1e~~k 1~k

where we have neglected the zero point motion term (irrelevant constant). The density of modesin (2.136) is denoted asZX22V2D() =( ~k ) =4dkk(ck)=V(2.137)(2)3 2 c3~k,

10

Canonical quantization of the radiation field: Introduce the variables

i~1Q~k = A~k + A~kand P~k = k A~k A~k4c4c

which leads to the following expression for the HamiltonianZX ~ ~2 + B~22

E1X 2k H = d3 r=A~ =P~k + ~k2 Q~2k .82c k2~k,

(2.132)

(2.133)

~k,

This is the Hamiltonian of a harmonic oscillator for each mode which we can quantize and obtain the new form

X

X11H=~~k a~k a~k +=~~k n~k +(2.134)22~k,

~k,

where A~k a~k annihilates and A~k a~k creates a photon in the mode (~k, ).

41

which leads to the spectral energy density2~,(2.138)23~ c e1which is the famous Planck formula (Fig.2.9). There are two limits2~ kB T Rayleigh-Jeans-lawkTB 2 c3(2.139)u(, T ) 3~e~ ~ kB T Wiens law 2 c3whereby the Rayleigh-Jeans law corresponds to the classical limit. The maximum for given Tfollows Wiens displacement law,~0 = 2.82kB T .(2.140)u(, T ) =

The total internal energy density leads to the Stefan-Boltzmann lawZU 2 (kB T )4= du(, T ) = T4 .V15 (~c)3

(2.141)

The energy current density of a blackbody is defined asenergyUc=.(2.142)Varea timeThus the emission power of electromagnetic radiation per unit area for the surface of a blackbodyis defined byZ 0Z 0~k ~nU 1U 1Uc 2 (kB T )4Pem = cd~k= cd~k cos === T 4(2.143)V 4V 44V60 ~3 c2|~k|where for the current density the componentperpendicular to the surface counts (~n: surfaceR0normal vector). Note that the integral d~k only extends over the hemisphere with cos > 0.

RayleighJeansPlanckWien

h /kB TFigure 2.9: Spectral density of black body radiation.

This blackbody radiation plays an important role for the energy budget of the earth. The suncan be considered a blackbody emitting an energy current at the temperature of T 6000K.This delivers an energy supply of 1.37kW/m2 to the earth. The earth, on the other hand, hasto emit radiation back to the universe in order not to heat up arbitrarily. The earth is not ablack body but a gray body as it is strongly reflecting in certain parts of the spectrum. Afurther example of blackbody radiation is the cosmic background radiation at a temperature2.73 K which originates from the big bang.42

2.8.2

Phonons in a solid

We consider Debyes theory of the lattice vibration and their influence on the thermodynamics ofa solid. A solid consists of atoms which form a lattice. They interact with each other through aharmonic potential. Let us assume that the solid consists of NA atoms arranged in a cube of edgelength L, i.e. there are 3NA degrees of freedom of motion. For our purpose it is convenient andsufficient to approximate this solid as a homogeneous isotropic elastic medium whose vibrationare described by the following equations of motion:1 2 ~u ~ ~ ( ~u) = 0 longitudinal sound mode ,c2l t2 2 ~u

1~ 2 ~u = 0c2t t2

(2.144)

transversal sound mode .

There are two independent transversal (~k ~u = 0) and one longitudinal (~k ~u = 0) soundmode. These equation can be solved by plane waves and yield linear dispersion analogous tothe electromagnetic waves:(l)~ = cl |~k|

and

k

(t)~ = ct |~k| .

(2.145)

k

The density of states is obtained analogously using periodic boundary conditions for the waves,

2V 2 1+.(2.146)D() =2 2 c3lc3tA difference occurs due to the finite number of degrees of freedom. In the end we get 3NAmodes. Thus there must be a maximal value of and |~k|. We take the sum3NA =

X|~k|kD

3V43=(2)3

Z0

kD

3V kDdk k =2 22

kD =

6 2 NAV

1/3(2.147)

and define in this way the Debye wave vector kD and the Debye frequency D = cef f kD where

312=+ 3 .(2.148)c3ef fc3lctThe internal energy is obtained again in the same way as for the electromagnetic radiation apartfrom the limit on the frequency integration,Z DU (T )=d u(, T ) .(2.149)V0We consider first the limit of small temperatures kB T kB D = ~D (D : Debye temperature).The internal energy is given by Z (kB T )4 3x3 2 (kB T )43 4 kB T T 3U (T ) = V4dx x=V=NA(2.150)(2~)3 c3ef fe 15D10~3 c3ef f0|{z}= 4 /15and correspondingly the low-temperature heat capacity is 312 4TCV =NA kB,5D43

(2.151)

D( )

D( )

D

Debye

real materials

Figure 2.10: Density of states of phonons. Left panel: Debye model; right panel: more realisticspectrum. Note that the low frequency part in both cases follows an 2 law and leads to the T 3 behavior of the heat capacity at low temperature. This is a consequence of the linear dispersionwhich is almost independent of the lattice structure and coupling.

the famous Debye law. On the other hand, at high temperatures (T D ) we use111 ~ +.~ 212e~ 1

(2.152)

This yields for the internal energy3VU (T ) = 2 32 cef f

Z

D

d

0

~2 4~ 3 kB T +212kB T2

(= 3NA kB T

3 ~D11+8 kB T20

~DkB T

+ (2.153)

2 )+

and leads to the heat capacity

CV = 3NA kB

21 D120 T 2

+ .

(2.154)

In the high-temperature limit the heat capacity approaches the value of the equipartition lawfor 3NA harmonic oscillators (Fig.2.11).The Debye temperature lies around room temperature usually. However, there also notableexception such as lead (Pb) with D = 88K or diamond with D = 1860K.

2.9

Diatomic molecules

We now investigate the problem of the diatomic moleculesP (made out of N atoms) which arebound by a two-particle interaction potential V (r) =ri ~rj |).i 0), or the chemical potential is positive (negative). The compressibility for = 0 is then given by

1V1+ (4.32)T 2kB T2which is enhanced (reduced) for attractive (repulsive) interaction V , as we would expect simplyby noticing that it is easier to add particles, if they attract each other than when they repeleach other.At low temperatures we obtain for the particle number with = 0,

L11+ V >0,LeV /2 sinh(V /2) 25N = 1 + q(4.33)22V1+esinh (V /2)LV 0 yields a partial filling optimizing the free energy for a given chemicalpotential. On the other hand, V < 0 tends towards complete filling. The compressibility75

remains in both cases finite,

T =

55/21 kB T (1 + 5)2

14kB T

V >0,(4.34)V 0 showingthat repulsive interaction yields an incompressible (crystalline) state of particles (one particleevery second site). On the other hand, attractive interaction yields a diverging compressibility,indicating that the gas tends to get strongly compressible, i.e. it goes towards a liquid phasewith much reduced effective volume per particle.We may consider this also from the point of view of the fluctuation-dissipation theorem whichconnects the response function T with the fluctuations of the particle number,hN 2 i hN i2 =

kB T hN i2T (T ) LeV /2 .L

(4.36)

The message of this relation is that for the repulsive case the particle number fluctuations gorapidly to zero for kT < |V |, as expected, if the system solidifies with the particle densityn = N/L = 1/2. On the other hand, for attractive interaction the fluctuations strongly increasein the same low-temperature regime for the given density (n = 1/2) at finite temperature,indicating that particles can be further condensed.

4.2.2

Correlation function

The transfer matrix method is also very convenient to obtain the correlation function,l = hni ni+l i hni ihni+l i .

(4.37)

We consider firsthni i =

1Z

Xn1 ,...,nL

Pn1 ,n2 Pn2 ,n3 Pni1 ,ni ni Pni ,ni+1 PnL ,n1(4.38)

o1 n1 n Lo= tr P i1 w P Li = tr wPZZwith w defined as wn,n0 = nn,n0 . Let us transform this to the basis in which P is diagonalthrough the unitary transformation

0

0+ 0w00 w01101UPU ==PandUwU ==w0 .(4.39)000 w10w1176

This leads tohni i =

0 L + w 0 L1 n 0 0 o w00+11 0.tr wP = w00LZL++

(4.40)

In the same way we treat nowo1 n l Ll o1 n 0 0 l 0 0 Ll o1 n P l w P Lil = tr wP wP= tr w (P ) w (P )hni ni+l i = tr P i wZZZ00l Lll Ll02 Lw0 200 L+ + w01 w10 (+ + + ) + w 11 =LL+ +

2w0 00

+

00w01w10

+

l.(4.41)

The correlation function is given by00l = w01w10

+

l

= n2 el/ [sign( /+ )]l

(4.42)

where we identify l=0 with n2 and the correlation length=

1ln |+ / |

.

(4.43)

For V > 0 the correlation function shows alternating sign due to the trend to charge densitymodulation. In the special case = V we obtain1l = (tanh(V /4))l2

and

=

1.ln | coth(V /4)|

(4.44)

In the zero-temperature limit diverges, but is finite for any T > 0.It is important that both the thermodynamics as well as the correlation functions are determinedthrough the eigenvalues of the transfer matrix P .

4.3

Long-range order versus disorder

We find in the one-dimensional spin chain no phase transition to any long-range ordered state atfinite temperatures. The term long-range order is easiest understood through the correlationfunction. We consider here again the example of a spin chain. The correlation function has theproperty, l = h ~s i ~s i+l i h ~s i i h ~s i+l i l 0(4.45)which is true for a spin system with or without order. By rewritinglim h ~s i ~s i+l i = h ~s i i h ~s i+l i ,

l

(4.46)

we see that the left hand side is finite only with a finite mean value of h ~s i i. Note that for theclassical spin chain only at T = 0 long-range order is realized. All spins are parallel for J < 0(ferromagnet) or alternating for J > 0 (antiferromagnet) such thatJ 0assuming that the spin align parallel to the z-axis (z is the unit vector along the z-axis). Naturally the limit in Eq.(4.46) is then finite. 4 Also the lattice gas in one dimension does not showlong-range order except at T = 0.4

Previously we encountered long-range order in a Bose-Einstein condensate looking at the correlation functionin Eqs.(3.103) and (3.110). Extending the correlation function to~ ) = hb ( ~r ) b ( ~r + R~ )i h b ( ~r )ih b ( ~r + R~ )i ,g( R

77

(4.48)

Interestingly, in the case of quantum systems quantum phase transitions can destroy longrange order even at T = 0. An important example is the antiferromagnetic spin chain with aHeisenberg Hamiltonian. In contrast the ferromagnetic Heisenberg chain has an ordered groundstate with all spins aligned, because the state with all spins parallel is an eigenstate of the totalP +1 ~spin operator, ~Stot = Ni=1 ~si and Stot commutes with the Hamiltonian, since it has full spinrotation symmetry as in the classical case (N + 1: number of spins on the chain),|FM i = | i

zStot|FM i = (N + 1)~s|FM i .

(4.50)

In case of the antiferromagnetic chain the (classical) state with alternating spins | cAF i =| i, is not an eigenstate of the quantum Heisenberg Hamiltonian and consequently notthe ground state of the quantum antiferromagnetic chain.Our discussion of the melting transition in Sect. 3.7.3 is suitable to shed some light on theproblem of quantum fluctuation. Also a crystal lattice of atoms is a long-range ordered state.Considering the melting transition based on the Lindemann criterion for the displacement fields,we tackle the stability of the lattice from the ordered side. Thus, we use again the languageof the elastic medium as in Sect.3.7.3 and rewrite the fluctuation of the displacement field inEq.(3.150),

Z~~cl kdD k 12h ~u ( ~r ) i =coth(4.51)m(2)D ~k2with D, the dimension. For D = 1 we findhu2 i =

~2n cl

Z

kD

dk0

1cothk

~cl k2

T 0

~2n cl

Z

kD

dk0

1.k

(4.52)

This integral diverges even at T = 0 and, thus, exceeds the Lindemann criterion in Eq.(3.151),Lm hu2 i/a2 . At T = 0 only quantum fluctuations can destroy long range order. We callthis also quantum melting of the lattice. For D 2 we find a converging integral at T = 0.Nevertheless, also here quantum melting is possible, if the lattice is soft enough, i.e. theelastic modulus is small enough. Note that He is such a case, as we will discuss later. At zerotemperature He is solid only under pressure which leads to an increase of the elastic modulus ,i.e. under pressure the lattice becomes stiffer.At finite temperature also in two-dimensions an ordered atomic lattice is not stable,

Z kD~~cl k2h ~u ( ~r ) i =dk coth(4.53)2m cl 02which diverges at the lower integral boundary due to the fact that for k 0 we expandcoth(~cl k/2) 2kB T /~cl k yielding a logarithmic divergence. That two-dimensional are rathersubtle can be seen for the case of graphene, a single layer of graphite (honeycomb lattice ofcarbon) which is stable.

~ ) = 0, since with the Bogolyubov approximation,we find that always lim R( R~ gb ( ~r ) b ( ~r + R~ )i = lim h b ( ~r )ih b ( ~r + R~ )i = lim 0 ( ~r )0 ( ~r + R~ ) = n0lim h

~ R

~ R

~ R

for T < Tc and a constant phase of 0 ( ~r ).

78

(4.49)

Chapter 5

Phase transitionsPhase transitions in macroscopic systems are ubiquitous in nature and represent a highly important topic in statistical physics and thermodynamics. Phase transitions define a change of stateof a system upon changing external parameters. In many cases this kind of change is obvious,e.g. transition between liquid and gas or between paramagnetic and ferromagnetic phase, andin most cases it is accompanied by anomalies in measurable macroscopic quantities.In the previous chapter we have seen a phase transition, the Bose-Einstein condensation. Thistransition is special in the sense that it occurs for non-interacting particles. Generally, phasetransitions require an interaction favoring an ordered phase. Then the phase transition occurs asa competition between the internal energy (or enthalpy) which is lowered by the order and theentropy which at finite temperature favors disorder. The relevant thermodynamic potentials todescribe phase transitions are the Helmholtz free energy F (T, V, N ) and the Gibbs free energyG(T, p, N ),F = U TSandG = H TS .(5.1)These potentials show anomalies (singularities) at the phase transition.

5.1

Ehrenfest classification of phase transitions

The type of singularity in the thermodynamic potential defines the order of the phase transition.According to Ehrenfest classification we call a phase transition occurring at a critical temperatureTc (different phase for T > Tc and T < Tc ) to be of nth order, if the following properties hold: m m m m G G G G=(5.2)=and

mmmTTppm T =TcppT =Tc+T =Tc+

T =Tc

for m n 1, and

nGT n

p

T =Tc+

6=

nGT n

p

andT =Tc

nGpn

T =Tc+

6=

nGpn

(5.3)T =Tc

The same definition is used for the free energy. In practice this classification is rarely usedbeyond n = 2.n = 1: A discontinuity is found in the entropy and in the volume:

GGS=andV =T pp T

(5.4)

The discontinuity of the entropy is experimentally the latent heat. The change in volume isconnected with the difference in the density of the substance. A well-known example is the

79

transition between the liquid and the gas phase, for which the former is much denser than thelatter and accordingly takes a much smaller volume.n = 2: The discontinuities are not in the first derivatives but in the second derivatives of theHelmholtz free energy or Gibbs free energy, i.e. in the response functions. Some such quantitiesare the heat capacity, the compressibility or the thermal expansion coefficient:

2 2 G G1 2G1(5.5)Cp = T, T = , =22T pV p TV T pAs we will see later, second order phase transitions are usually connected with spontaneoussymmetry breaking and can be associated with the continuous growth of an order parameter.Such transitions show also interesting fluctuation features which lead to the so-called criticalphenomena and universal behavior at the phase transition.Ehrenfest relations: Interesting relations between various discontinuities at the phase transitionexist. They are generally known at Ehrenfest relations. We consider first a first-order transitionsuch as the gas-liquid transition. The phase boundary line in the p-T -phase diagram describesthe relation between temperature and vapor pressure in the case of liquid-gas transition. Forthe differentials of the Gibbs free energy in the two phases, the following equality holds:

dGl = dGg

Sl dT + Vl dp = Sg dT + Vg dp .

(5.6)

This allows us to get from the vapor pressure curve (p(T ) at the phase boundary in the p-T plane) the relationSg SldpL==(5.7)dTVg VlT V

where L = T (Sg Sl ) is the latent heat and V = Vg Vl is the change of the volume. Thisrelation is known as the Clausius-Clapeyron equation.If the transition is of second order then the both the entropy and the volume are continuousthrough the transition between two phase A and B:SA (T, p) = SB (T, p)

and VA (T, p) = VB (T, p) ,

(5.8)

which yields the relations through the equality of their differentials,

SASASBSBdSA =dT +dp =dT +dp = dSB ,T pp TT pp T(5.9)

dVA =

VAT

dT +p

We now use the Maxwell relation

and obtain

VAp

Sp

dp =

T

T

=

SBT

VBT

VT

dT +

p

VBp

T

VBT

dp = dVB .T

SAT

= V

p

(5.10)

Cpdpp p == SBdTT V SAp

p

(5.11)

T

and analogously

VAT

dpp p == .VBdTT VAp

p

T

T

Various other relations exist and are of experimental importance.80

(5.12)

5.2

Phase transition in the Ising model

The Ising model is the simplest model of a magnetic system. We consider magnetic momentsor spins with two possible states, si = s (Ising spins). Sitting on a lattice they interact withtheir nearest neighbors (analogously to the spin chain in the Chapter 3). We write the modelHamiltonian asXXH = Jsi sj si H .(5.13)i

hi,ji

P

The sum hi,ji denotes summation over nearest neighbors on the lattice, counting each bondonly once. J is the coupling constant which we assume to be positive. The second termcorresponds to a Zeeman term due to an external magnetic field. The Ising spins are classicalvariables, unlike quantum spins ~s whose different components do not commute with each other.Ising spins represent only one component of a quantum spin.The interaction favors the parallel alignment of all spins such that this model describes a ferromagnetic system. The ferromagnetic phase is characterized by a finite uniform mean valuehsi i = m 6= 0, the magnetization, even in the absence of an external magnetic field.

5.2.1

Mean field approximation

The analysis of many coupled degrees of freedom is in general not simple. For the Ising model wehave exact solutions for the one- and two-dimensional case. For three dimensions only numericalsimulations or approximative calculations are possible. One rather frequently used method isthe so-called mean field approximation. In the lattice model we can consider each spin si asbeing coupled to the reservoir of its neighboring spins. These neighboring spins act then likea fluctuating field on the spin. In case of order they form a net directed field on si , a meanfield. Since this conditions applies to all spins equally, the scheme can be closed by having aself-consistency in the mean field.Let us now tackle the problem in a more systematic way. We rewrite the spin for each sitesi = hsi i + (si hsi i) = m + (si m) = m + si

(5.14)

and insert it into the Hamiltonian, where we approximate hsi i = m uniformly.XXH = J{m + (si m)} {m + (sj m)} si Hi

hi,ji

= J

X

Xm2 + m(si m) + m(sj m) + si sj si H

= J

X

i

hi,ji

i

(5.15)

Xz Xzmsi m2 si H Jsi sj .2i

hi,ji

Here z is the number of nearest neighbors (for a hypercubic lattice in d dimensions z = 2d). Inthe mean field approximation we neglect the last term assuming that it is small. This meansthat the fluctuations around the mean value would be small,Eij =

hsi sj ihsi sj i=1,hsi ihsj im2

(5.16)

to guarantee the validity of this approximation. We will see later that this condition is notsatisfied very near the phase transition and that its violation is the basis for so-called criticalphenomena. We now write the mean field HamiltonianXzHmf = si heff + N J m2withheff = Jzm + H ,(5.17)2i

81

which has the form of an ideal paramagnet in a magnetic field heff . It is easy to calculate thepartition function and the free energy as a function of the parameter m,2 N/2

ZN (T, m, H) = eJzm

{2 cosh(sheff )}N

(5.18)

and

zF (T, H, m) = kB T ln ZN = N J m2 N kB T ln {2 cosh(sheff )} .(5.19)2The equilibrium condition is reached when we find the minimum of F for given T and H. Tothis end we minimize F with respect to m as the only free variable,0=

F= N Jzm N Jzs tanh(sheff ) .m

(5.20)

This equation is equivalent to the self-consistence equation for the mean value of si :P

2si heff1 FeJzm N/2 X PN0 si0 hef fsi =s si ei=1= P= s tanh(sheff ) = si em = hsi i =si heffZNN H T,msi =s e{sj }

(5.21)This is a non-linear equation whose solution determines m and eventually through the freeenergy all thermodynamic properties. 1

5.2.2

Instability of the paramagnetic phase

The disordered phase above a certain critical temperature Tc is called paramagnetic phase. Forthis phase we consider first the magnetic susceptibility (T ) at zero magnetic field, which isobtained from

d2 F dFF m d F dhsi i ==+=(5.26)(T ) = NdH H=0dH 2 H=0dH Hm H H=0dH H H=01Variational approach: Consider the Ising model (5.13) without magnetic field. We now determine the freeenergy on a variational level assuming a distribution of N independent spin with a net magnetization M = N+ N with m = M/N .(N = N+ + N ). The probability that a certain spin is +s or -s is given by w = 12 (1 m)There areN! 1 .(5.22)(M ) = 1(N+M)! 2 (N M ) !2

configurations corresponding to a given M . We may now determine the free energy as F = U T S = hHiM kB T ln (M ) in the following way:U = hHiM = J

X

hsi sj i =

hi,ji

JN zs2 2JN zs2(w+ w+ + w w w+ w w w+ ) = m22

(5.23)

where we use for hsi sj i simply the configurational average for pairs of completely independent spins for givenM , i.e. w+ w+ + w w (w+ w + w+ w ) is the probability that neighboring spins are parallel (antiparallel).In this approach there is no correlation between the neighboring spins. For the entropy term we use Stirlingapproximation and keep only extensive terms,

11T S = kB T ln (M ) N kB T ln 2 (1 + m) ln(1 + m) (1 m) ln(1 m).(5.24)22Thus, we have expressed the free energy by a variational phase represented by independent spins whose variationalparameter is the mean moment m = hsi = s(w+ w ) = sm. We minimize F with respect to m,

2FN kB T1+mJzs m0== Jzs2 N m +ln m = tanh(5.25)m21mkB Twhich corresponds to Eq.(5.21) in the absence of a magnetic field. Thus, this variational approach and mean fieldare equivalent and give the same thermodynamic properties. As we will see, the mean field approach is moreeasily improved. In many cases similar variational calculations of the free energy based on independent degreesof freedom yield thermodynamics equivalent to a mean field approach.

82

where we used the equilibrium condition (5.20). Thus we obtain

dm d(T ) = N= Nstanh [(Jzsm(H) + sH)]

dH H=0dHH=0Ns=kB T

(5.27)

dm sN s2Jzs+s=Jzs(T)+.dH H=0kB TkB T

where we used that for a paramagnet m(H = 0) = 0. This leads to the susceptibility

T = Tc

mT > Tc

T < Tc

mFigure 5.1: Graphical solution of the self-consistence equation (5.21). The crossing points of thestraight line and the step-like function gives the solution. There is only one solution at m = 0for T Tc and three solutions for T < Tc .

(T ) =

N s2kB T Jzs2

(5.28)

which is modified compared to that of the ideal paramagnet. If kB T Jzs2 from above (T )is singular. We define this as the critical temperatureTc =

Jzs2.kB

(5.29)

As the system approaches T = Tc it becomes more and more easy to polarize its spin by a tinymagnetic field. This indicates an instability of the system which we now analyze in terms ofthe self-consistence equation (5.21) in the absence of a magnetic field. Looking at Fig. 5.1 wefind that indeed the critical temperature Tc plays an important role in separating two types ofsolutions of equations (5.21). For T Tc there is one single solution at m = 0 and for T < Tcthere are three solutions including m = 0, m(T ), m(T ). The physically relevant solution isthen the one with finite value of m, as we will show below. It is obvious that below T = Tcthe mean field m grows continuously from zero to a finite value. In order to see which of thesolutions is a minimum of the free energy we expand F in m assuming that m and H are small. 2

mkB T (sheff )2 (sheff )4F (T, H, m) N Jz N kB T ln 2(5.30)2Jz212

83

For H = 0 we find"F (T, H = 0, m) F0 (T ) + N Jz

Tc1T

m21+212s2

TcT

3

#4

m

(5.31)

F0 (T ) + N Jz

2

mTm41+Tc212s2

where for the last step we took into account that our expansion is only valid for T Tc .Moreover, F0 = N kB T ln 2. This form of the free energy expansion is the famous Landautheory of a continuous phase transition.

F

T = Tc

T > TcT < Tc

m

Figure 5.2: Landau free energy: T > Tc : 2nd -order term is positive and minimum of F atm = 0; T = Tc , 2nd vanishes and free energy minimum at m = 0 becomes very shallow; T < Tc :2nd -order term is negative and minimum of F is at finite value of m, bounded by the 4th -orderterm.It is obvious that for T > Tc the minimum lies at m = 0. For T < Tc the coefficient of them2 -term (2nd -order) changes sign and a finite value of m minimizes F (see Fig. 5.2). Theminimization leads to s 3 T < Tcm(T ) =(5.32)0T Tcwith = 1 T /Tc as a short-hand notation. There are two degenerate minima and the systemchooses spontaneously one of the two (spontaneous symmetry breaking).Next we analyze the behavior of the free energy and other thermodynamic quantities aroundthe phase transition. The temperature dependence of the free energy and the entropy is givenbyF (T ) = F0 (T )

3N kB Tc 2( )4

S(T ) =

and

F (T )3N kB = N kB ln 2 ( ) ,T2(5.33)

and eventually we obtain for the heat capacity,CS3N kB==( ) + C0TT2Tc

(5.34)

where C0 is zero in the present approximation for H = 0. While the free energy and the entropyare continuous through the transition, the heat capacity shows a jump indicating the release ofentropy through ordering. Thus, we conclude that this phase transition is of second order.Within mean field approximation the region close to Tc is described by the Landau expansion.However, taking the solution of the complete mean field equations leads to the thermodynamic84

F/NkBTc

-0.5

-0.75

S/NkB

0.75-10.50.25

C Tc / NkBT

021.510.50

0

0.5

1

T / Tc

Figure 5.3: Thermodynamic quantities within mean field theory. Free energy, entropy and heatcapacity.

behavior for the whole temperature range as shown in Fig. 5.3. Note that in this mean fieldapproximation the entropy is N kB ln 2 in the paramagnetic phase, the maximal value the entropycan reach.

5.2.3

Phase diagram

So far we have concentrated on the situation without magnetic field. In this case the phasetransition goes to one of two degenerate ordered phases. Either the moments order to m = +|m|or m = |m|. An applied magnetic field lifts the degeneracy by introducing a bias for one of thetwo states. The order with m parallel to the field is preferred energetically. In a finite field thetransition turns into a crossover, since there is already a moment m for temperatures above Tc .This is also reflected in the thermodynamic properties which show broad features around Tc andnot anymore the sharp transition, e.g. the heat capacity is turned into a broadened anomaly(see Fig. 5.4).1.5

1

1

0.5

0

0.5

0

0.5

1

1.5

0

T / Tc

0.5

1

CTc / NkBT

m(T,H) / s

1.5

01.5

T / Tc

Figure 5.4: Ising system in a finite magnetic field: (left panel) Magnetization as a function oftemperature in a fixed magnetic field (solid line) and in zero field (dashed line); (right panel)heat capacity for a fixed magnetic field. In a magnetic field no sharp transition exists.Next we turn to the behavior of the magnetization m as a function of the magnetic field andtemperature (illustrated in Fig. 5.5 and 5.6). At H = 0 going from high to low temperaturesthe slope of m(H)|H=0 is linear and diverges as we approach Tc . This reflects the divergingsusceptibility as a critical behavior.85

stable

mmetastable

instable

HT > TcT = TcT < Tc

Figure 5.5: Magnetization as a function of magnetic field for different temperatures.

For all temperatures T > Tc m(H) is a single-valued function in Fig. 5.5 . Below Tc , however,m(H) is triply valued as a solution of the self-consistence equation. The part with dm/dH > 0is stable or metastable represent