lectures on mathematical statistical mechanics

7/24/2019 Lectures on Mathematical Statistical Mechanics

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ISSN 0070-7414

Sgrbhinn Institiuid Ard-Leinn Bhaile Atha Cliath

Sraith. A. Uimh 30

Communications of the Dublin Institute for Advanced Studies

Series A (Theoretical Physics), No. 30

LECTURES ON

MATHEMATICAL

STATISTICAL MECHANICS

By

S. Adams

DUBLIN

Institiuid Ard-Leinn Bhaile Atha Cliath

Dublin Institute for Advanced Studies

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Contents

1 Introduction 1

2 Ergodic theory 22.1 Microscopic dynamics and time averages . . . . . . . . . . . . 22.2 Boltzmanns heuristics and ergodic hypothesis . . . . . . . . . 82.3 Formal Response: Birkhoff and von Neumann ergodic theories 92.4 Microcanonical measure . . . . . . . . . . . . . . . . . . . . . 13

3 Entropy 163.1 Probabilistic view on Boltzmanns entropy . . . . . . . . . . . 163.2 Shannons entropy . . . . . . . . . . . . . . . . . . . . . . . . 17

4 The Gibbs ensembles 204.1 The canonical Gibbs ensemble . . . . . . . . . . . . . . . . . . 204.2 The Gibbs paradox . . . . . . . . . . . . . . . . . . . . . . . . 264.3 The grandcanonical ensemble . . . . . . . . . . . . . . . . . . 274.4 The orthodicity problem . . . . . . . . . . . . . . . . . . . . 31

5 The Thermodynamic limit 335.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Thermodynamic function: Free energy . . . . . . . . . . . . . 375.3 Equivalence of ensembles . . . . . . . . . . . . . . . . . . . . . 42

6 Gibbs measures 446.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 The one-dimensional Ising model . . . . . . . . . . . . . . . . 476.3 Symmetry and symmetry breaking . . . . . . . . . . . . . . . 516.4 The Ising ferromagnet in two dimensions . . . . . . . . . . . . 526.5 Extreme Gibbs measures . . . . . . . . . . . . . . . . . . . . . 576.6 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.7 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 A variational characterisation of Gibbs measures 62

8 Large deviations theory 688.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708.3 Some results for Gibbs measures . . . . . . . . . . . . . . . . . 72

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9 Models 73

9.1 Lattice Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . 749.2 Magnetic Models . . . . . . . . . . . . . . . . . . . . . . . . . 759.3 Curie-Weiss model . . . . . . . . . . . . . . . . . . . . . . . . 779.4 Continuous Ising model . . . . . . . . . . . . . . . . . . . . . . 84

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Preface

In these notes we give an introduction to mathematical statistical mechanics,based on the six lectures given at theMax Planck institute for Mathematics inthe SciencesFebruary/March 2006. The material covers more than what hasbeen said in the lectures, in particular examples and some proofs are workedout as well the Curie-Weiss model is discussed in section 9.3. The coursepartially grew out of lectures given for final year students at the UniversityCollege Dublin in spring 2004. Parts of the notes are inspired from notes ofJoe Pule at University College Dublin.

The aim is to motivate the theory of Gibbs measures starting from basic

principles in classical mechanics. The first part covers Sections 1 to 5 andgives a route from physics to the mathematical concepts of Gibbs ensemblesand the thermodynamic limit. The Sections 6 to 8 develop a mathematicaltheory for Gibbs measures. In Subsection 6.4 we give a proof of the ex-istence of phase transitions for the two-dimensional Ising model via Peierlsarguments. Translation invariant Gibbs measures are characterised by a vari-ational principle, which we outline in Section 7. Section 8 gives a quick intro-duction to the theory of large deviations, and Section 9 covers some modelsof statistical mechanics. The part about Gibbs measures is an excerpt ofparts of the book by Georgii ([Geo88]). In these notes we do not discuss

Boltzmanns equation, nor fluctuations theory nor quantum mechanics.

Some comments on the literature. More detailed hints are found through-out the notes. The books [Tho88] and [Tho79] are suitable for people, whowant to learn more about the physics behind the theory. A standard ref-erence in physics is still the book [Hua87]. The route from microphysics tomacrophysics is well written in [Bal91] and [Bal92]. The old book [Kur60] isnice for starting with classical mechanics developing axiomatics for statisticalmechanics. The following books have a higher level with special emphasison the mathematics. The first one is [Khi49], where the setup for the micro-canonical measures is given in detail (although not in used modern manner).The standard reference for mathematical statistical mechanics is the book[Rue69] by Ruelle. Further developments are in [Rue78] and [Isr79]. Thebook [Min00] contains notes for a lecture and presents in detail the two-dimensional Ising model and the Pirogov-Sinai theory, the latter we do notstudy here. A nice overview of deep questions in statistical mechanics gives[Gal99], whereas [Ell85] and [Geo79],[Geo88] have their emphasis on proba-bility theory and large deviation theory. The book [EL02] gives a very niceintroduction to the philosophical background as well as the basic skeleton ofstatistical mechanics.

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I hope these lectures will motivate further reading and perhaps even fur-

ther research in this interesting field of mathematical physics and stochastics.Many thanks to Thomas Blesgen for reading the manuscript. In particular Ithank Tony Dorlas, who gave valuable comments and improvements.

Leipzig, Easter 2006 Stefan Adams

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1 Introduction

The aim of equilibrium Statistical Mechanics is to derive all the equilibriumproperties of a macroscopic system from the dynamics of its constituent par-ticles. Thus its aim is not only to derive the general laws of thermodynamicsbut also the thermodynamic functions of a given system. Mathematical Sta-tistical Mechanics has originated from the desire to obtain a mathematicalunderstanding of a class of physical systems of the following nature:

The system is an assembly of identical subsystems.The number of subsystems is large (N1023, Avogardos number 6.023 1023, e.g. 1cm3 of hydrogen contains about 2.7

1019 molecules/atoms).

The interactions between the subsystems are such as to produce a thermo-dynamic behaviour of the system.

Thermodynamic behaviour is phenomenological and refers to a macroscopicdescription of the system. Now macroscopic description is operationallydefined in the way that subsystems are considered as small and not individ-ually observed.

Thermodynamic behaviour:

(1) Equilibrium states are defined operationally. A state of an isolated systemtends to an equilibrium state as time tends to +(approach to equilibrium).(2) An equilibrium state of a system consists of one or more macroscopicallyhomogeneous regions called phases.(3) Equilibrium states can be parametrised by a finite number of thermody-namic parameters (e.g. temperature, volume, density, etc) which determinethe thermodynamic functions (e.g. free energy, pressure, entropy, magneti-sation, etc).

It is believed that the thermodynamic functions depend piecewise ana-lytical (or smoothly) on the parameters and that singularities correspond tochanges in the phase structure of the system (phase transitions). Classical

thermodynamics consists of laws governing the dependence of the thermo-dynamic functions on the experimental accessible parameters. These lawsare derived from experiments with macroscopic systems and thus are notderived from a microscopic description. Basic principles are the zeroth, firstand second law as well as the equation of state for the ideal gas.

The art of the mathematical physicist consists in finding a mathematicaljustification for the statements (1)-(3) from a microscopic description. A mi-croscopic complete information is not accessible and is not of interest. Hence,despite the determinism of the dynamical laws for the subsystems, random-

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ness comes into play due to lack of knowledge (macroscopic description). The

large number of subsystems is replaced in a mathematical idealisation by in-finitely many subsystems such that the extensive quantities are scaled to stayfinite in that limit. Stochastic limit procedures as the law of large numbers,central limit theorems and large deviations principles will provide appropri-ate tools. In these notes we will give a glimpse of the basic concepts. Inthe second chapter we are concerned mainly with the Mechanics in the nameStatistical Mechanics. Here we motivate the basic concepts of ensembles viaHamilton equations of motions as done by Boltzmann and Gibbs.

2 Ergodic theory2.1 Microscopic dynamics and time averages

We consider in the following N identical classical particles moving in Rd, d1, or in a finite box Rd. We idealise these particles as point masseshaving the massm. All spatial positions and momenta of the single particlesare elements in the phase space

=R

d Rd)N or =

Rd)N. (2.1)Specify, at a given instant of time, the values of positions and momenta of the

Nparticles. Hence, one has to specify 2dNcoordinate values that determinea single point in the phase space respectively . Each single point inthe phase space corresponds to a microscopic stateof the given system ofNparticles. Now the question arises whether the 2dN dimensional continuumof microscopic states is reasonable. Going back to Boltzmann [Bol84] it seemsthat at that time the 2dN dimensional continuum was not really deeplyaccepted ([Bol74], p. 169):

Therefore if we wish to get a picture of the continuum in words,we first have to imagine a large, but finite number of particles

with certain properties and investigate the behaviour of the en-sembles of such particles. Certain properties of the ensemble mayapproach a definite limit as we allow the number of particles evermore to increase and their size ever more to decrease. Of theseproperties one can then assert that they apply to a continuum,and in my opinion this is the only non-contradictory definition ofa continuum with certain properties...

and likewise the phase space itself is really thought of as divided into a fi-nite number of very small cellsof essentially equal dimensions, each of which

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determines the position and momentum of each particle with a maximum

precision. Here the maximum precision that the most perfect measurementapparatus can possibly provide is meant. Thus, for any position and momen-tum coordinates

p(j)i q(j)

i h, for i= 1, . . . , N , j = 1, . . . , d ,with h = 6, 621034 Js being Plancks constant. Microscopic states ofthe system of Nparticles should thus be represented by phase space cellsconsisting of points in R2dN (positions and momenta together) with givencentres and cell volumes hdN. In principle all what follows should be for-mulated within this cell picture, in particular when one is interested in an

approach to the quantum case. However, in this lectures we will stick to themathematical idealisation of the 2dN continuum of the microscopic states,because all important properties remain nearly unchanged upon going overto the phase cell picture.

Let two functions W: Rd R and V: R+ R be given. The energy ofthe system ofNparticles is a function of the positions and momenta of thesingle particles, and it is called HamiltonianorHamilton function. It isof the form

H(q, p) =N

i=1

p2i2m

+ W(qi)

+

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1ln the identity in Rn, the Hamilton vector field is given as

v : , xJH(x)with x= (q, p) R2n and

H(x) =H

q1, . . . ,

H

qN,H

p1. . . ,

H

pN

.

With the vector field v we associate the differential equations

d

dtx(t) =v(x(t)), (2.4)

where x(t) denotes a single microstate of the system for any time t

R. For

each point x there is one and only one function x : R such thatx(0) =x and dx(t)

dt =v(x(t)) for any t R. For any t Rwe define a phase

space mapt : , xt(x) =x(t).

From the uniqueness property we get that ={t : t R} is a one-parametergroup which is called a Hamiltonian flow. Hamiltonian flows have thefollowing property.

Lemma 2.1 Let be a Hamiltonian flow with Hamiltonian H, then anyfunctionF =f H ofHis invariant under:

F t=Ffor allt R.Proof. The proof follows from the chain rule andx,Jx= 0. Recall thatifG: Rn Rm then G(x) is the m n matrix

Gixj

(x)

i=1,...,mj=1,...,n

for x Rn.

d

dtF(t(x)) =F

(t(x))dtdt

(x)

=f(H(t(x))) H(t(x)) dt

dt(x)

=f(H(t(x))) (H(t(x)))T dtdt

(x)

=f(H(t(x)))H(t(x)), dtdt

(x)=f(H(t(x)))H(t(x)) , JH(t(x)).

The following theorem is the well known Liouvilles Theorem.

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Theorem 2.2 (Liouvilles Theorem) The Jacobian

|det t(x)

|of a Hamil-

tonian flow is constant and equal to 1.

Proof. Let M(t) and A(t) be linear mappings such that

dM(t)

dt =A(t)M(t)

for all t0, then

det M(t) = det M(0)exp

t

0

traceA(s)ds

.

Nowdt(x)

dt = (v t)(x).

Thusdt(x)

dt =v (t(x))t(x)

so that det t(x) = expt0

tracev (s(x))dsbecause0(x) is the identity mapon . Now

v(x) =JH(x) and v(x) =J H(x),where H(x) = 2Hxixj is theHessianofHat x. Since H is twice continu-ously differentiable, H(x) is symmetric. Thus

trace (JH(x)) = trace((JH(x))T) = trace ((H(x))TJT)

= trace (H(x)(J)) =trace (JH(x)).

Therefore trace (JH(x)) = 0 and det(t(x)) = 1.

From Lemma 2.1 and Theorem 2.2 it follows that a probability measure

on the phase space , i.e., an element of the setP(, B) of probabilitymeasure on with Borel algebraB, whose Radon-Nikodym density withrespect to the Lebesgue measure is a function of the HamiltonianHalone isstationary with respect to the Hamiltonian flow ={t : t R}.

Corollary 2.3 Let P(, B) with density= F H for some functionF: R R be given, ie., (A) =

A(x)dx for anyA B. Then

1t = for anyt R.

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Proof. We have

(A) =

1lA(x)(x)dx=

1lA(t(x))(t(x))| det t(x)|dx

=

1l1t A(x)(x)dx= (1t A).

For such a stationary probability measure one gets a unitary group of timeevolution operators in an appropriate Hilbert space as follows.

Theorem 2.4 (Koopmans Lemma:) Let1be a subset of the phase space invariant under the flow , i.e., t1

1 for all t

R. Let

P(1, B1 ) be a probability measure on1 stationary under the flow thatis 1t = for all t R. Define Utf = ft for any t R and anyfunctionf L2(1, ), then{Ut : t R} is a unitary group of operators inthe Hilbert spaceL2(1, ).

Proof. Since UtUt = U0 = I, Ut is invertible and thus we have to proveonly that Ut preserves inner products.

Utf, Utg=

(f t)(x)(g t(x)(dx) =

(f g)(t(x))(dx)

= (f g)(x)( 1t )(dx) = (f g)(x)(dx) =f, g.

Remark 2.5 (Boundary behaviour) If the particles move inside a finitevolume boxRd according to the equations (2.3) respectively (2.4); theseequations of motions do not hold when one of the particles reaches the bound-ary of. Therefore it is necessary to add to theses equations some rules ofreflection of the particles from the inner boundary of the domain. For ex-ample, we can consider the elastic reflection condition: the angle of incidence

is equal to the angle of reflection. Formally, such a rule can be specified by aboundary potentialVbc.

We discuss briefly Boltzmanns Proposal for calculating measured valuesofobservables. Observables are bounded continuous functions on the phasespace. Let 1 be a subset of phase space invariant under the flow , i.e.t11for allt R. Suppose thatf : R is an observable and supposethat the system is in 1 so that it never leaves 1. Boltzmann proposed thatwhen we make a measurement it is not sharp in time but takes place over

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a period of time which is long compared to say times between collisions.

Therefore we can represent the observed value by

f(x) = limT

1

2T

TT

dt(f t)(x).

Suppose that is a probability measure on 1 invariant with respect to tthen

1

f(x)(dx) = limT

1

2T

TT

dt

1

(f t)(x)(dx)

= limT

1

2T T

Tdt1 f(x)( 1t )(dx)

= limT

1

2T

TT

dt

1

f(x)(dx)

=

1

f(x)(dx).

Assume now that the observed value is independent of where the system isat t = 0 in 1, i.e. iff(x) = f for a constant f, then

1f(x)(dx) =

1f (dx) = f .

Therefore

f=

1

f(x)(dx).

We have made two assumptions in this argument:

(1) limT

1

2T

TT

dt(f t)(x) exists.

(2) f(x) is constant on 1.

Statement (1) has been proved by Birkhoff (Birkhoffs pointwise ergodic the-orem (see section 2.3 below)): f(x) exists almost everywhere. We shall provea weaker version of this, Von Neumanns ergodic theorem.Statement (2) is more difficult and we shall discuss it later.

The continuity of the Hamiltonian flow entails that for each f Cb(Rd) andeachx the function

fx : R R,fx(t) =f(x(t)) (2.5)

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is a continuous function of the time t. For any time dependent function

we define the following limit

:= limT

1

2T

TT

dt(t) (2.6)

as the time average.

2.2 Boltzmanns heuristics and ergodic hypothesis

Boltzmanns argument (1896-1898) for the introduction of ensembles in sta-tistical mechanics can be broken down into steps which were unfortunately

entangled.

1. Step: Find a set of time dependent functions which admit an invari-ant mean like (2.6).

2. Step: Find a reasonable set of observables, which ensure that the timeaverage of each observable over an orbit in the phase space is indepen-dent of the orbit.

LetCb(R) be the set of all bounded continuous functions on the real line R,equip it with the supremum norm|||| = suptR |(t)|, and define

M= Cb(R) : = limT

12T

TT

dt(t) exists . (2.7)Lemma 2.6 There exist positive linear functionals :Cb(R) R, nor-malised to 1, and invariant under time translations, i.e., such that

(i) ()0 for all M,(ii) linear,

(iii) (1) = 1,

(iv) (s) =() for alls R, wheres(t) =(t s), t , s R,such that () = limT 12T

TTdt(t) for all Cb(R) where this limit

exists.

Our time evolution

(t, x) R t(x) =x(t)is continuous, and thus we can substitutefxin (2.5) for in the above result.This allows to define averages on observables as follows.

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Lemma 2.7 For every f

Cb() and every x

, there exists a time-

invariant meanx given by

x :Cb() R, fx(f) =(fx),withx depending only on the orbit{x(t) : t R, x(0) =x}.For any E R+ let

E={(q, p) : :H(q, p) =E}denote the energy surface for the energy value Efor a given Hamiltonian

Hfor the system ofN particles.

The strict ergodicity hypothesis

The energy surface contains exactly one orbit, i.e. for everyxandE0

{x(t) : t R, x(0) =x}= E.

There is a more realistic mathematical version of this conjecture.

The ergodicity hypothesis

Each orbit in the phase space is dense on its energy surface, i.e.{x(t) : t R, x(0) =x} is a dense subset ofE.

2.3 Formal Response: Birkhoff and von Neumann er-godic theories

We present in this section briefly the important results in the field of ergodictheory initiated by the ergodic hypothesis. For that we introduce the notionof a classical dynamical system.

Notation 2.8 (Classical dynamical system) A classical dynamical sys-tem is a quadruple (, F, ; ) consisting of a probability space (, F, ),whereF is aalgebra on, and a one-parameter (additive) groupT (RorZ) and a group of actions, :T , (t, x)t(x), of the groupTon the phase space, such that the following holds.

(a) fT:T R, (t, x) f(t(x)) is measurable for any measurablef: R,

(b) t s=t+s for alls, t T,

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(c) (t(A)) =(A) for allt

T andA

F.

Theorem 2.9 (Birkhoff) Let(, F, ; ) be a classical dynamical system.For everyfL1(, F, ), let

Tx (f) = 1

2T

TT

dtf(t(x)).

Then there exists an eventAf F with(Af) = 1 such that(i) x(f) = limT Tx (f) exists for allxAf,

(ii) t(x)

(f) =x

(f) for all(t, x)R

A

f,

(iii)

(dx)x(f) =

(dx)f(x).

Proof. [Bir31] or in the book [AA68].

Note that in Birkhoffs Theorem one has convergence almost surely. Thereexists a weaker version, the following ergodic theorem of von Neumann. Werestrict in the following to classical dynamical system withT = R, the realtime. Let

H= L2(

1,F

, ) and defineUtf=f

tfor anyf

H. Then by

Koopmans lemma Ut is unitary for any t R.Theorem 2.10 (Von Neumanns Mean Ergodic Theorem) Let

M={f H :Utf=f t R},then for anyg H,

gT := 1

T

T0

dtUtg

converges to P g asT , whereP is the orthogonal projection ontoM.For the proof of this theorem we need the following discrete version.

Theorem 2.11 LetH be a Hilbert space and letU:H Hbe an unitaryoperator. LetN ={f H :U f=f}= ker(U I) then

limN

1

N

N1n=0

Ung= Qg

whereQ is the orthogonal projection ontoN.

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Proof of Theorem 2.10. Let U=U1 and g= 1

0dtUtf, then

Ung=

10

dtUn+tf=

n+1n

dtUtf

and thusN1n=0

Ung=

N0

dtUtf.

Therefore 1N

N0

dtUtfconverges as N . ForT R+, by writing T =N+ r where 0r


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Then

[ker (I U)][Range (I U)]= Range (I U).IfgRange(I U), then g= (I U)h for some h. Therefore

1

N

N1n=0

Ung= 1

N{h U h + U h U2h + U2h U3h + . . .+ UN1h UNh}

= 1

N{h UNh}.

Thus

1NN1

n=0 Ung 2hN 0 as N .Approximating elements of Range(I U) by elements of Range(I U), wehave that 1

N

N1n=0 U

ng0 =P g for all

g[ker (I U)] = Range(I U).Ifgker (I U), then

1

N

N1

n=0Ung= g = P g.

Definition 2.12 (Ergodicity) Let = (t)tR be a flow on 1 and aprobability measure on1 which is stationary with respect to . is said tobe ergodic if for every measurable setF 1 such thatt(F) = F for allt R, we have(F) = 0 or(F) = 1.Theorem 2.13 (Ergodic flows) = (t)tR is ergodic if and only if theonly functions inL2(1, )which satisfyft=fare the constant functions.

Proof. Below a.s.(almost surely) means that the statement is true excepton a set of zero measure. Suppose that the only invariant functions are theconstant functions. Ift(F) = F for all t then 1lF is an invariant functionand so 1lF is constant a.s. which means that 1lF(x) = 0 a.s. or 1lF(x) = 1 a.s.Therefore (F) = 0 or (F) = 1l.Conversely suppose t is ergodic and f t = f. LetF ={x| f(x) < a}.Then tF =F since

t(F) ={t(x)| f(x)< a}={t(x)| f(t(x)< a}= F.

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Therefore (F) = 0 or (F) = 1. Thusf(x) < a a.s. orf(x)

a a.s. for

everya R. Leta0= sup{a| f(x)a a.s.}.

Then, ifa > a0, ({x| f(x)a}) = 0, and ifa < a0, ({x| f(x)< a}) = 0.Let (an) and (bn) be sequences converging to a0 such that an > a0 > bn.Then

{x| f(x)=a0}=n{x| f(x)an} {x| f(x)< bn}.Thus

({x| f(x)=a0})

n(({x| f(x)an}) + ({x| f(x)< bn})) = 0,

and so f(x) =a0 a.s.

If we can prove that a system is ergodic, then there is no problem in applyingBoltzmanns prescription for the time average. For an ergodic system, by theabove theorem,Mis the one-dimensional space of constant functions so that

P g=1, g1 =1

g(x)(dx).

Therefore

limT1

2T TT dtUtg= 1 g(x)(dx).Remark 2.14 However proving ergodicity has turned out to be the most dif-ficult part of the programme. There is only one example for which ergodicityhas been claimed to be proved and that is for a system of hard rods (Sinai).This concerns finite systems. In the thermodynamic limit (see Chapter 5)ergodicity should hold, but we do not discuss this problem.

2.4 Microcanonical measure

Suppose that we consider a system with Hamiltonian H and suppose alsothat we fix the energy of the system to be exactly E. We would like todevise a probability measure on the points of with energy Esuch that themeasure is stationary with respect to the Hamiltonian flow.Note that the energy surface E is closed since E = H

1({E}) and H isassumed to be continuous. Clearlyt(E) = EsinceHt = H . Let A()denote the algebra of continuous functions R with compact support.The following Riesz-Markov theorem identifies positive linear functionals onA() with positive measures on .

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Theorem 2.15 (Riesz-Markov) Ifl :

A()

Ris linear and for any pos-

itivef A() it holds l(f)0, then there is a unique Borel measure on such that

l(f) =

f(x)(dx).

Now define a linear functional lEonA() by

lE(f) = lim0

1

[E,E+]

f(x)dx,

where [E,E+] ={x| H(x) [E, E+ ]} is the energy-shell of thickness .By the Riesz-Markov theorem there is a unique Borel measure Eon suchthat

lE(f) =

f(x)E(dx)

with the properties:

(i) E is concentrated on E.

If suppf E = then for small enough since suppf and [E,E+]are closed [E,E+] suppf= and

[E,E+]f(x)dx=

f(x)1l[E,E+] (x)dx= 0.

(ii) E is stationary with respect to t.

Since the Lebesgue measure is stationary,

lE(f t) = lim0

1

[E,E+]

(f t)(x)dx= lim0

1

t([E,E+])

f(x)dx

= lim0

1

[E,E+]

f(x)dx= lE(f).

Definition 2.16 (Microcanonical measure) If(E) := E() 0 is the inverse temperature and H the Hamiltonian of the system.

The mathematical justification to replace the microcanonical ensemble bythe canonical or grandcanonical Gibbs ensemble goes under the name equiv-alence of ensembles, which we will discuss in Subsection 5.3. In this sectionwe introduce first the canonical Gibbs ensemble. Then we study the so-calledGibbs paradox concerning the correct counting for a system of indistinguish-able identical particles. It follows the definition of the grandcanonical Gibbsensemble. In the last subsection we relate all the introduced Gibbs ensem-bles to classical thermodynamics. This leads to the orthodicity problem,namely the question whether the laws of thermodynamics are derived fromthe ensembles averages in the thermodynamic limit.

4.1 The canonical Gibbs ensemble

We define the canonical Gibbs ensemble for a finite volume box Rdand a fixed number N N of particles with Hamiltonian H(N) having ap-propriate boundary conditions (like elastic ones as for the microcanonicalensemble or periodic ones). In the following we denote the Borel--algebraon the phase space byB. The universal Boltzmann constant isk = kB = 1.3806505 1023 joule/kelvin. In the following T denotes tem-perature measured in Kelvin.

Definition 4.1 Call the parameter =

1

kT > 0 the inverse temperature.Thecanonical Gibbs ensemble for parameteris the probability measure,N P(, B) having the density

,N(x) =eH

(N) (x)

Z(, N) , x, (4.10)

with respect to the Lebesgue measure. Here

Z(, N) =

dx eH(N) (x) (4.11)

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is the normalisation and is called partition function (Zustandsssume).

Gibbs introduced this canonical measure as a matter of simplicity: he wantedthe measure with density to describe an equilibrium, i.e., to be invari-ant under the time evolution, so the most immediate candidates were to befunctions of the energy. Moreover, he proposed that the most simple caseconceivable is to take the log linear in the energy. The following theoremwas one of his justifications of the utility of the definition of the canonicalensemble.

Theorem 4.2 Let1, 2 Rd, 1 2 be an aggregate phase space0and

0 P(

0,B0

) with Lebesgue density0

be given. Define the reducedprobability measures (or marginals) i P(i, Bi), i= 1, 2, as

1(A) =

A2

1lA(x1)0(x1, x2)dx2 forA B1,

2(B) =

1B

1lA(x2)0(x1, x2)dx1 forB B2

with the Lebesgue densities

1(x1) = 0 0(x1, x2)dx2 and 2(x2) = 0 0(x1, x2)dx1.Then the entropies

Si =ki

i(x)log i(x)dx with i= 0, 1, 2,

satisfy the inequalityS0S1+ S2

with equalityS0=S1+ S2 if and only if0 = 12.

Proof. The proof is given with straightforward calculation and the use ofJensens inequality for the convex function f(x) =x log x+ 1 x. Gibbs himself recognised the condition for equality as a condition for inde-pendence. He claimed that with the notations of Theorem 4.2 in the specialcase is the canonical ensemble density and the Hamiltonian is of the formH0 =H1+ H2 with H1 (respectively H2) is independent of 2 (respectively1), the reduced densities (marginal densities) 1 and 2 are independent,i.e.,0 = 12, and are themselves canonical ensemble densities.In [Gib02] he writes:

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-a property which enormously simplifies the discussion, and is the

foundation of extremely important relations to thermodynamics.

Indeed, it follows from this that the temperatures are all equal, i.e., T1 =T2=T0.

Remark 4.3 (Lagrange multipliers) We note that the inverse tempera-ture in the canonical Gibbs ensemble can be seen as the Lagrange multipli-cator for the extremum problem for the entropy under the constraint that themean energy is fixed with a parameter like , see further [Jay89], [Bal91],[Bal92], [EL02].

Theorem 4.4 (Maximum Principle for the entropy) Let > 0, and N N be given. The canonical Gibbs ensemble ,N, where > 0 isdetermined by

dx,N(x) =U, maximises the entropy

S() =k

(x)log (x)dx

for any P(, B) having a Lebesgue density subject to the constraint

U= (x)H(N) (x)dx. (4.12)Moreover, the values of the temperatureTand the partition functionZ(, N)are uniquely determined from the condition

U=

log Z(, N) with = 1

kT.

Proof. We give only a rough sketch of the proof. We use that

a log a

b log b

(a

b)(1 + log a) a, b

(0,

).

Let P(, B) with Lebesgue density . Put a= ,N(x) and b= (x)for any x and recall that ,N is the density of the canonical Gibbsensemble. Then

,N(x)log ,N(x) (x)log (x)(,N(x) (x))(1 log Z(, N)

H(N) (x)).

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Integrating with respect to Lebesgue we get

S(,N) S() k

((1 log Z(, N) H(N) (x)),N(x)dx

(1 log Z(, N) H(N) (x))(x))dx

=k {(1 log Z(, N) U) (1 log Z(, N) U)}= 0.

ThereforeS(,N)S().

Note that the entropy for the canonical ensemble is given by

S(,N) =k

(log Z(, N) + H(N)

(x)),N(x)dx

=k log Z(, N) + k

H(N) (x),N(x)dx.

To prove the second assertion, note that

log Z(, N) =

,N(x)H(N)

(x)dx,

2log Z=

,N(x)H(N) (x)

,N(x)H(N)

(x)dx2dx0.

Thermodynamic functions

For the canonical ensemble the relevant thermodynamical variables are thetemperatureT (or = (kT)1) and the volume Vof the region R. Wehave already defined the entropy Sof the canonical ensemble by

S(, N) =k log Z(, N) + 1

TE,N

(H(N) ),

where U = E,N(H(N) ) =

H(N) (x),N(x)dx, the expectation of H,

sometimes denoted also byH(N) . We define the Helmholtz Free EnergybyA = UT S, and we shall call the Helmholtz Free Energy simply thefreeenergy from now on. We have

A= U T S(, N) = E,N(H(N)

) T(k log Z(, N) + 1

TE,N

(H(N) ))

= 1

log Z(, N).

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By analogy with Thermodynamics we define the absolute pressure Pof the

system by

P =

A

V

T

.

The other thermodynamic functions can be defined as usual:The Gibbs Potential,G= U+ P V T S=A + P V,The Heat Capacity at Constant Volume, CV =

UT

V

.

Note that

S=

A

T

V

is also satisfied. The thermodynamic functions can all be calculated from A.Therefore all calculations in the canonical ensemble begin with the calcula-tion of the partition function Z(, N).To make the free energy density finite in the thermodynamic limit we redefinethe canonical partition function by introducing correct Boltzmanncounting

Z(, N) = 1

(n/d)!

eH(N) (x)dx=

1

N!

eH(x)dx, (4.13)

see the following Subsection 4.2 for a justification of this correct Boltzmanncounting.

Example 4.5 (The ideal gas in the canonical ensemble) Consider a non-interacting gas ofNidentical particles of massm ind dimensions, containedin a box Rd of volumeV. The Hamiltonian for this system is

H(x) = 1

2m

Ni=1

p2i , x= (q, p).

We have for the partition functionZ(, N)

Z(, N) = 1

N! hNd eH(x)dx= 1

N! hN dVNRd e

p2

2m dpN

= 1

N! hNdVN

R

ep2

2m dp

N d=

1

N!VN

2m

h2

12

N d

= 1

N!

V

d

N,

(4.14)

where

=

h2

2m

12

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is called thethermal wavelengthbecause it is of the order of the de Broglie

wavelength of a particle of massmwith energy 1. Thefree energyA(, N)is given by

A(, N) = 1

log Z(, N) = 1

(log N! + Nd log Nlog V) .

Thus the pressure is given by

P(, N) =A(, N)

V

T

= N

V =

kT N

V .

Let aN(, v) be the free energy per particle considered as a function of thespecific densityv, that is,

aN(, v) = 1

NAN(, N),

whereNis a sequence of boxes with volumevNand letpN(, v) =PN(, N)be the corresponding pressure. Then

pN(, v) =aN(, v)

v

T

.

For the ideal gas we get then

aN(, v) = 1

1

N log N! + d log log v log N

,

leading to

a(, v) := limN

aN(, v) = 1

(d log log v 1) ,

since

limN

1

N log N! log N

=1.

Ifp(, v) := limNpN(, v), one gets

p(, v) =a(, v)

v

T

,

and thus p(, v) = 1v

. We can also define the free energy density as afunction of the particle density, i.e.,

fl(, ) = 1

VlAl(,Vl),

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wherel is a sequence of boxes with volumeVl with liml

Vl =

and

f(, ) = liml

fl(, ).

The pressurep(, ) then satisfies

p(, ) =

f(, )

T

f(, ).

Clearlyf(, ) = a(, 1

). For the ideal gas we get

f(, ) =

(d log + log

1)

and therefore

p(, ) =

.

Finally we want to check the relative dispersion of the energy in the canonicalensemble. LetH(N) = E,N(H

(N)

). Then

(H(N) H(N) )2H(N) 2

=2log Z(, N)

(log Z(, N))2.

This gives for the ideal gas(H(N) H(N) )2

H(N) = (

1

2dN)

12 =O(N

12 ).

4.2 The Gibbs paradox

The Gibbs paradox illustrates an essential correction of the counting withinthe microcanonical and the canonical ensemble. Gibbs 1902 was not awareof the fact that the partition function needed a re-definition, for instance a

redefinition in (4.13) in case of the canonical ensemble. The ideal gas sufficesto illustrate the main issue of that paradox. Recall the entropy of the idealgas in the canonical ensemble

S(, N) =kNlog(V T d2 ) +

d

2kN(1 + log(2m)) , 1 =kT, (4.15)

where V =|| is the volume of the box Rd. Now, make the followingGedanken-experiment. Consider two vessels having volume Vi containingNi, i = 1, 2, particles separated by a thin wall. Suppose further that both

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vessels are in equilibrium having the same temperature and pressure. Now

imagine that the wall between the two vessels is gently removed. The ag-gregate vessel is now filled with a gas that is still in equilibrium at the sametemperature and pressure. Denote by S1 and S2 the entropy on each side ofthe wall. Since the corresponding canonical Gibbs ensembles are indepen-dent of one another, the entropy S12 of the aggregate vessel - before the wallis removed - is exactly S1+ S2. However an easy calculation gives us

S12 (S1+ S2) =k((N1+ N2)log(V1+ V2) N1log V1 N2log V2)=k

N1log

V1V1+ V2

+ N2log V2

V1+ V2> 0.

(4.16)

This shows that the informational (Shannon) entropy has increased, whilewe expected the thermodynamic entropy to remain constant, since the wallbetween the two vessels is immaterial from a thermodynamical point of view.This is the Gibbs paradox.We have indeed lost information in the course of removing the wall. Imaginethe gas before removing the wall consists of yellow molecules in one vesseland of blue molecules in the other. After removal of the wall we get auniform greenish mixture throughout the aggregate vessel. Before we knewwith probability 1 that a blue molecule was initially in the vessel where wehad put it, after removal of the wall we only know that it is in that part of

the aggregate vessel with probability N1N1N2 .The Gibbs paradox is resolved in classical statistical mechanics with an adhoc ansatz. Namely, instead of the canonical partition function Z(, N)one takes 1

N!Z(, N) and instead of the microcanonical partition function

(E, N) one takes 1N!

(E, N). This is called the correct Boltzmanncounting. The appearance of the factorial can be justified in quantummechanics. It has something to do with the in-distinguishability of iden-tical particles. A state describing a system of identical particles should beinvariant under any permutation of the labels identifying the single parti-cle variables. However, this very interesting issue goes beyond the scope of

this lecture, and we will therefore assume it from now on. In Subsection 5.1we give another justification by computing the partition function and theentropy in the microcanonical ensemble.

4.3 The grandcanonical ensemble

We give a brief introduction to the grandcanonical Gibbs ensemble. Onecan argue that the canonical ensemble is more physical since in experimentswe never consider an isolated system and we never measure the total energy

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but we deal with systems with a given temperature. Similarly we like not to

specify the number of particles but the average number of particles. In thegrandcanonical ensemble the system can have any number of particles withthe average number determined by external sources. The grandcanonicalGibbs ensemble is obtained if the canonical ensemble is put in a particle-bath, meaning that the particle number is no longer fixed, only the mean ofthe particle number is determined by a parameter. This was similarly donein the canonical ensemble for the energy, where one considers a heat-bath.The phase space for exactly Nparticles in box Rd can be written as

,N={( Rd) : ={(q, pq) :q

}, Card(

) =N}, (4.17)

where, the set of positions occupied by the particles, is a locally finitesubset of , and pq is the momentum of the particle at positions q. If thenumber of the particles is not fixed, then the phase space is

={( Rd) : ={(q, pq) :q}, Card() finite}. (4.18)A counting variable on is a random variable N on for any Borel set defined by N() = Card ( ) for any .Definition 4.6 (Grandcanonical ensemble) Let Rd, >0 andR. Define the phase space =N=0,N, where,N= ( Rd)2N is thephase space in forNparticles, and equip it with the-algebra

B generated

by the counting variables. The probability measure , P(, B) suchthat the restrictions,

,N

onto ,Nhave the densities

(N),(x) =Z(, )1e(H

(N) (x)N) , N N,

whereH(N) is the Hamiltonian forN particles in, and partition function

Z(, ) =

N=0

,N

e(H(N) (x)N)dx (4.19)

is called thegrandcanonical ensemble in for the inverse temperature

and thechemical potential .

Instead of the chemical potential sometimes the fugacity or activity e

is used for the grandcanonical ensemble. Observables are now sequencesf = (f0, f1, . . .) with f0 R and fN: ,N R, N N, are functions onthe N-particle phase spaces. Hence, the expectation in the grandcanonicalensemble is written as

E,(f) =

1

Z(, )

N=0

eNZ(, N)

,N

fN(x),N(dx). (4.20)

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If

Ndenotes the particle number observable we get that

E,(N) = 1

log Z(, ).

For the grandcanonical measure we have a Principle of Maximum En-tropyvery similar to those for the other two ensembles. We maximise theentropy subject to the constraint that the mean energy E,

(H) and the

mean particle number E,(N) are fixed, where H= (H(0) , H(1) , . . .) is the

sequence of Hamiltonians for each possible number of particles.

Theorem 4.7 (Principle of Maximum Entropy) LetP

be a probabilitymeasure on such that its restriction to ,N, denoted byPN, is absolutelycontinuous with respect to the Lebesgue measure, that is

PN(A) =

A

N(x)dx for anyA B(N) .

Define the entropy of the probability measureP to be

S(P) =k0log 0 k

N=1

,N

N(x)log(N!N(x))dx.

Then the grandcanonical ensemble/measure , , where and are deter-mined byE,

(H) =EandE,(N) =N0, N0N,maximises the entropy

among the absolutely continuous probability measures on with mean en-ergyEand mean particle numberN0.

Proof. As in the two previous cases we usea log a b log b(a b)(1 +log a) and so

a log ta b log tb(a b)(1 + log a + log t).

Let ,N (x) =eNeHN(x)

N!Z(, ) and put a = (N),(x), b = N(x) and t = N!.

Then, writing Z for Z(, ),

(N),(x)log(N!(N)

,(x)) N(x)log(N!N(x))((N),(x) N(x))(1 log Z H(N) (x) + N).

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Integrating with respect to the Lebesgue measure on ,Nand summing over

Nwe get

S(,) S() k

N=1

,N

(1 log Z H(N) (x) + N)(N),(x)dx

+ (1 log Z)(0),

N=1

,N

(1 log Z H(N) (x) + N)N(x))dx

(1 log Z)0

=k {(1 log Z E+ N0) (1 log Z E+ N0)}= 0.Therefore

S(,)S().Note that the entropy for the grandcanonical ensemble is given by

S(,) =k log Z(, ) + kE,(H) kE, (N).

Thermodynamic Functions:

We shall write Z for Z(, ) and we suppress for a while some obvious

sub-indices and arguments. We have already defined the entropy Sby

S=k log Z+ 1

T(E,

(H) E, (N)),

and as before we define the internal energy of the systemUbyU= E,(H).

We then define the Helmholtz Free Energy as before by A = U T S, andwe shall call the Helmholtz Free Energy simply the free energy from now on.We have

A=U T S= E, (H) T(k log Z+ 1

T(E,

(H) E, (N)))

= 1log Z E, (N) .

In analogy with thermodynamics we should define the absolute pressure Pof the system by

P =

A

V

T

with the constraintE,

(N) = constant.

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This constraint means that is a function ofV and . Therefore

P = 1

V

log Z E, (N)

+

1

V log Z=

1

V log Z.

It is argued that 1

V log Zshould be independent ofV for largeVand therefore

we can write

P = 1

V log Z=

1

V

V

1

V log Z

=

1

V log Z+

V

V

1

V log Z

1

V log Z.

Therefore we define the pressure by the equation

P = 1

V log Z.

This definition can be justified a posterioriwhen we consider the equivalenceof ensembles, see Subsection 5.3. The other thermodynamic functions canbe defined as usual:The Gibbs Potential G= U+ P V T S=A + P V,The heat capacity at constant volume, CV =

UT

V

. Note

S=ATVis also satisfied.

All the thermodynamic functions can be calculated from Z = Z(, ).Therefore all calculations in the grandcanonical ensemble begin with thecalculation of the partition function Z=Z(, ).

4.4 The orthodicity problem

We refer to one of the main aims of statistical mechanics, namely to derive

the known laws of classical thermodynamics from the ensemble theory. Thefollowing question is called the Orthodicity Problem.

Which set Eof statistical ensembles or probability measures has the propertythat, as an element E changes infinitesimally within the setE, thecorresponding infinitesimal variations dUand dV ofU and V are related tothe pressure Pand to the average kinetic energy per particle,

Tkin=E(Tkin)

N , Tkin=

1

2m

Ni=1

p2i ,

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such that the differential

dU+ PdVTkin

is an exact differential at least in the thermodynamic limit. This will thenprovide the second law of thermodynamics. Let us provide a heuristic checkfor the canonical ensemble. Here,

E,N

(Tkin) = 1

Z(, N)

,N

Tkin(x)eH(N) (x)dx,

and U =Z(, N). The pressure in the canonical ensemble can becalculated as

P(,N) =

Q

N

Z(, N)

p>0

eH(N) (x)

1

2mp2

a

A

dq2 dqNdp1 dpNN!

,

where the sum goes over all small cubes Q adjacent to the boundary of thebox with volume V by a side with area a while A =

Q a is the total

area of the container surface andq1 is the centre ofQ. Letn(Q, v)dv, wherev = 1

2mp is the velocity, be the density of particles with normal velocityv

that are about to collide with the external walls ofQ. Particles will cede amomentum 2mv=p in normal direction to the wall at the moment of their

collision (mvmv due to elastic boundary conditions). ThenQ

v>0

dvn(Q, v)(2mv)va

A

is the momentum transferred per unit time and surface area to the wall.Gaussian calculation gives then after a couple of steps that due toF(, N) =1 log Z(, N) and S(E; N) = (U F(, N)) we have that T =(k)1 = 2

dkTkin

N , and that

TdS

= d(F

+ T S

) +pdV = dU+pdV,

with p = 1 V

log Z(, N). Details can be found in [Gal99], where alsoreferences are provided for rigorous proofs for the orthodicity in the canonicalensemble. The orthodicity problem is more difficult in the microcanonicalensemble. The heuristic approach goes similar. However, for a rigorous proofof the orthodicity one needs here a proof that the expectation of the kineticenergy in the microcanonical ensemble satisfies

E(Tkin) = E(Tkin)(1 + N) , >0,

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with N

0 as N

(thermodynamic limit). The last requirement

would be easy for independent velocity, but this is not the case here dueto the microcanonical energy constraint, and therefore this refers not to anapplication of the usual law of large numbers. A rigorous proof concerningany fluctuations and moments of the kinetic energy in the microcanonicalensemble is in preparation [AL06].

5 The Thermodynamic limit

In this section we introduce the concept of taking the thermodynamic limit,give a simple example in the microcanonical ensemble and prove in Subsec-tion 5.2 the existence of the thermodynamic limit for the specific free energyin the canonical Gibbs ensemble for a given class of interactions. In the lastsubsection we briefly discuss the equivalence of ensembles and the thermo-dynamic limit at the level of states/measures.

5.1 Definition

Let us callstateof a physical system an expectation value functional on theobservable quantities for this system. The averages, i.e., the expectation val-ues with respect to the Gibbs ensembles, are such states. We shall say that

the systems for which the expectation with respect to the Gibbs ensemblesis taken are finite systems (e.g. finitely many particles in a region withfinite volume), but we may also consider the corresponding infinite sys-temswhich contain an infinity of subsystems and extend throughout Rd orZ

d for lattice systems. Thus the discussion in the introduction leads us to as-sume that the ensemble expectation for finite systems approach in some sensestates/measures of the corresponding infinite system. Besides the existenceof such limit states/measures one is also interested in proving that these areindependent on the choice of the ensemble leading to the question ofequiv-alence of ensembles. One of the main problems of equilibrium statistical

mechanics is to study the infinite systems equilibrium states/measures andtheir relation to the interactions which give rise to them. In Section 6 we in-troduce the mathematical concept of Gibbs measures which appear as naturalcandidates for equilibrium states/measures. We turn down one level in ourstudy and consider the problem of determination of the thermodynamic func-tions from statistical mechanics in the thermodynamic limit. We introducedearlier for each Gibbs ensemble a partition function, which is the total massof the measure defining the Gibbs ensemble. The logarithm of the partitionfunction divided by the volume of the region containing the system has a

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limit when this systems becomes large (

R

d), and this limit is identified

with a thermodynamic function. Any singularities for these thermodynamicfunctions in the thermodynamic limit may correspond to phase transitions(see [Min00],[Gal99] and [EL02] for details on theses singularities).

Taking the thermodynamic limitthus involves letting tend to infin-ity, i.e., approaching Rd or Zd respectively. We have to specify how tendsto infinity. Roughly speaking we consider the following notion.

Notation 5.1 (Thermodynamic limit) A sequence(n)nNof boxesnR

d is acofinal sequence approachingRd if the following holds,

(i) n

Rd asn

,

(ii) If hn ={x Rd : dist(x, ) h} denotes the set of points withdistance less or equalh to the boundary of, the limit

limn

|hn||| = 0

exists.

The thermodynamic limit consists thus in letting n for a cofinal se-quence (n)n

N of boxes with the following additional requirements for the

microcanonical and the canonical ensemble:

Microcanonical ensemble: There are energy densitiesn(0, ), givenasn =

En|n| withEn asn , and particle densitiesn (0, ),

given as n = Nnn

with Nn as n , such that n andn(0, ) asn .Canonical ensemble: There are particle densities n (0, ), given asn=

Nnn

withNn asn , such thatn(0, ) asn .

In some models one needs more assumptions on the cofinal sequence of

boxes, for details see [Rue69] and [Isr79].

We check the thermodynamic limit of the following simple model in themicrocanonical ensemble, which will give also another justification of thecorrect Boltzmann counting.

Ideal gas in the microcanonical ensemble

Consider a non-interacting gas of N identical particles of mass m in d di-mensions, contained in the box of volume V =||. The gradient of the

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Hamiltonian for this system is

H(x) = 12m

ni=1

p2i = 1

m(0, . . . , 0, p1, . . . , pn) , x,

where as usual n= N d. We have

|H(x)|2 = 1m2

ni=1

p2i = 2

mH(x) , x,

where| | denotes the norm in R2n. Therefore on the energy surface E,

|H

|= (2E

m)

12 . Let S(r) be the hyper sphere of radius r in dimensions,

that is S(r) ={x| x R, |x|= r}. Let S(r) be the surface area ofS(r).Then

S(r) =cr1 for some constant c.

For the non-interacting gas, we have E= N Sn((2mE) 12 ) and

(E) =m

2E

12

E

d=m

2E

12

VNSn((2mE)12 )

=m

2E

12

VNcN d(2mE)12(Nd1) =mVNcNd(2mE)

12

N d1.

The entropySis given by

exp(S/k) =(E) =mVNcNd(2mE)12

Nd1

and therefore

(2mE)12 Nd1 =

exp(S/k)VN

mcdN.

Thus, the internal energy follows as

U(S, V) =E= 1

2m

exp

2Sk(N d2)

V

2N(Nd2)

(mcN d)2

(Nd2)

,

and the temperature as the partial derivative of the internal energy withrespect to the entropy is

T =U

S

V

= 2

k(Nd 2) U= 2U

kN d(1 2Nd

) 2U

kN d

for large N. This gives for large Nthe following relations

U d2

NkT,

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CV

= UTV d2 N k.P =

UV

S

= 2

d(1 2Nd

)

U

V 2U

dV NkT

V .

The previous relation is the empirical ideal gas law in whichk is Boltzmannsconstant. We can therefore identifyk in the definition of the entropy withBoltzmanns constant.We need to calculate c. We have via a standard trick

2 =

ex2

dx

=

Re|x|

2

dx=

0

S(r)er2 dr

=c0

r1er2

dr=c

20

t21etdt= c

2

2

.

This gives c= 2

2

( 2), where is the Gamma-Function, defined as

(x) =

0

tx1etdt.

Note that ifn N then (n) = (n 1)!. The behaviour of (x) for largepositive xis given by Stirlings formula

(x) 2xx 12 ex.

This gives limx( 1xlog (x) log x) =1.We now have for the entropy ofthe non-interacting gas in a box of volume V

S(E, N) =k log

mVN 2

Nd2

( N d2

)(2mE)

12 Nd1

.

Let v be the specific volumeand the energy per particle, that is

v= VN

and = EN

.

Let sN(, v) be the entropy per particle considered as a function ofand v,

sN(, v) = 1

NSN(N,N),

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where N is a sequence of boxes with volume vN. Then

sN(, v) = 1

Nk log

mvNNN

2Nd

2

( N d2

)(2mN)

12 N d1

k

log v+

d + 2

2 log N+

d

2log(4m) 1

N log

Nd2

k

log v+ log N+

d

2log(4m) +

d

2d

2log(

d

2)

k log N.

We expect sN(, v) to be finite for large N. Gibbs (see Section 4.2) postu-

lated that we have made an error in calculating (E), the number of statesof the gas with energy E. We must divide (E) by N!. It is not possi-ble to understand this classically since in classical mechanics particles aredistinguishable. The reason is inherently quantum mechanical. In quan-tum mechanics particles are indistinguishable. We also divide(E) byhdN

where h is Plancks constant. This makes classical and quantum statisticalmechanics compatible for high temperatures.We therefore redefine the microcanonical entropy of the system to be

S(E, N) =k log

(E, N)

(n/d)! = k log

(E, N)

N! , (5.21)

where we put Plancks constant h= 1. Then

sN(, v)k

log v+ log N+d

2log(4m) +

d

2d

2log(

d

2) d log h

1N

log(N!)

k

d + 2

2 + log v+

d

2log d

2log

h4m

d

2log(

d

2)

as N .

5.2 Thermodynamic function: Free energy

We shall prove that the canonical free energy density exists in the thermody-namic limit for very general interactions. We consider a general interactinggas ofNidentical particles of mass m in d dimensions, contained in the box of volume ofV with elastic boundary conditions. The Hamiltonian for thissystem is

H(N) = 1

2m

Ni=1

p2i + U(r1, . . . , rN).

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We have for the partition function Z(, N)

Z(, N) = 1

N!hNd

eH(N) (x)dx=

1

N!

Rd

ep2

2m dp

NN

eU(q)dq

= 1

N!

2m

h2

12

N d N

eU(q)dq= 1

N!

1

dN

N

eU(q)dq.

We shall assume that the interaction potential is given by apair interactionpotential : R R, that is

U(q1, . . . , q N) = 1i 0 such that(|q|)0 if|q|> R, qRd.

(ii) The pair potential isstable if there existsB 0 such that for allq1, r2, . . . , q N inR

d

1i


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Figure 1: typical form of

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n-1

(1)

n-1

(2)

Rn

n-1

(4)

n-1

(3)

R

Figure 2: n contains 4 cubes of side length Ln1

Proof. Note that limn Nn/Vn = . Note also that Nn = 2dNn1 andLn = 2Ln1+ R. Because of the last equation n contains 2d cubes of sideLn1 with a corridor of width R between them. Denote these by

(i)n1, i =

1, . . . , 2d and let

UN(q1, . . . , q N) =

1i


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Since n

Nnn

Zn 1Nn!dNn

n

dq1 . . . dqNneUNn(q1,...,qNn)

= Nn!

(Nn1!)2d

1

Nn!

1

dNn

n

dq1 . . . dqNneUNn (q1,...qNn).

Since is tempered, we get for q1, . . . , q Nn nUNn(q1, . . . , q Nn)UNn1 (q1, . . . , q Nn1 ) . . .

+ UNn1 (q(2d1)Nn1+1, . . . , q 2dNn1 ).

Thus

Zn 1(Nn1!))2

d

1

dNn

Nn1n1

dq1 . . . dqNn1 eUNn1(q1,...,qNn1)

2d= (Zn1)2

d

.

Therefore

gn= 1

Nnlog Zn =

1

2dNn1log Zn 1

2dNn1log(Zn1)2

d

=gn1.

The sequence (gn)nN is increasing. To prove that gnconverges it is sufficientto show that gn is bounded from above. Since the pair potential is stable

we haveUNn(q1, . . . , q Nn) BNn

and therefore

Zn= 1

Nn!dNn

Nnn

dq1 . . . dqNneUNn(q1,...,qNn)

1Nn!dNn

Nnn

dq1 . . . dqNneBNn Vn

Nn

Nn!dNneB Nn.

Thus

gn

log Vn

1

Nnlog Nn!

d log + B =

logNnVn + log Nn

1Nn

log Nn! d log + B log + 2 d log + B

for large n since

limn

NnVn

= and limn

log Nn 1

Nnlog Nn!

= 1.

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5.3 Equivalence of ensembles

The equivalence of the Gibbs ensemble is the key problem of equilibriumstatistical mechanics. It goes back to Gibbs 1902, who conjectured that boththe canonical and the grandcanonical Gibbs ensemble are equivalent withthe microcanonical ensemble. The main difficulty to answer the question ofequivalence lies in the precise definition of the notion equivalence. Nowadaysthe term can have three different meanings, each on a different level of infor-mation. We briefly introduce these concepts, but refer for details to one ofthe following research articles ([Ada01], [Geo95], [Geo93]).

Equivalence at the level of thermodynamic functions

Under general assumptions on the interaction potential, e.g. stability andtemperedness as in Subsection 5.2, one is able to prove the following thermo-dynamic limits of the thermodynamic functions given by the three Gibbs en-sembles. Let (n)nNbe any cofinal sequence of boxes with corresponding se-quences of energy densities (n)nNand particle densities (n)nN. Then thereis a closed packing density (cp) (0, ) and an energy density ()(0, )such that the following limits exist under some additional requirements (de-tails are in [Rue69]) depending on the specific model chosen.

Grandcanonical Gibbs ensemble Let > 0 be the inverse temperatureand R the chemical potential. Then the function p(, ), defined by

p(, ) = limn

1

|n| log Zn(, ),

is called the pressure.Canonical Gibbs ensemble Let > 0 be the inverse temperature and(n)nN with n (0, (cp)) a sequence of particle densities n = Nn|n| .Then the function f(, ), defined by

f(, ) = limn 1|n| log Zn(, n|n|),

is called the free energy.Microcanonical Gibbs ensemble Let (n)nN with n(0, (cp)) bea sequence of particle densities n =

Nn|n| , and let (n)nN with n

((), ) be a sequence of energy densities n= En|n| . Then the function

s(, ) = limn

1

|n| log n(n|n|), n|n|) is called entropy.

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Now equivalence at the level of thermodynamic functions is given if all three

thermodynamic functions in the thermodynamic limit are related to eachother by a Legendre-Fenchel transform for specific regions of parameter re-gions of,, and . Roughly speaking this kind of equivalence is mainlygiven in absence of phase transitions, i.e., only for those parameters wherethere are no singularities. For details see the monograph [Rue69], where thefollowing transforms are established.

p(, ) = sup(cp),>()

{s(, ) }

f(, ) = inf>()

{ 1s(, )}p(, ) = sup

(cp){ f(, )}.

Equivalence at the level of canonical and microcanonical Gibbsmeasures

In Section 6 we introduce the concept of Gibbs measures. A similar conceptcan be formulated for canonical Gibbs measures (see [Geo79] for details)as well as for microcanonical Gibbs measures (see [Tho74] and [AGL78] fordetails). The idea behind these concepts is roughly speaking to conditionoutside any finite region on particle density and energy density events.

Equivalence at the level of canonical and microcanonical Gibbs measures isthen given if the microcanonical and canonical Gibbs measures are certainconvex combinations of Gibbs measures (see details in [Tho74], [AGL78] and[Geo79]).

Equivalence at the level of states/measures

At the level of states/measures one is interested in any weak (in the senseof probability measures, i.e., weak--topology) limit points (accumulationpoints) of the Gibbs ensembles. To define a consistent limiting procedure we

need an appropriate phase or configuration space for the infinite systems inthe thermodynamic limit. We consider here only continuous systems whosephase space (configuration space) for any finite region and finitely manyparticles is . In Section 6 we introduce the corresponding configurationspace for lattice systems. Define

={(Rd Rd) : ={(q, pq) :q}},

where , the set of occupied positions, is a locally finite subset ofRd, andpqis the momentum of the particle at position q. LetBdenote the-algebra of

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this set generated by counting variables (see [Geo95] for details). Then each

Gibbs ensemble can be extended trivially to a probability on (, B) just byputting the whole mass on a subset. Therefore it makes sense to considerall weak limit points in the thermodynamic limit. If the limit points are notunique, i.e., there are several accumulation points, one considers the wholeset of accumulation points closed appropriately as the set of equilibriumstates/measure or Gibbs measures.Equivalence at the level of states/measures is given if all accumulation pointsof the different Gibbs ensembles belong to the same set of equilibrium pointsor the same set of Gibbs measure ([Geo93],[Geo95],[Ada01]).

In the next section we develop the mathematical theory for Gibbs mea-

sures without any limiting procedure.

6 Gibbs measures

In this section we introduce the mathematical concept of Gibbs measures,which are natural candidates to be the equilibrium measures for infinite sys-tems, i.e., for systems after taking the thermodynamic limit. We will restrictour study from now on to lattice systems, i.e., the phase space is given as theset of functions (configurations) on some countable discrete set with valuesin a finite set, called the state space.

6.1 Definition

Let Zd the square lattice for dimensions d 1 and let Ebe any finite set.Define := EZ

d

={ = (i)iZd : i E} the set of configurations withvalues in the state space E. LetE be the power set of E, and define thealgebraF=EZd such that (, F) is a measurable space. Denote the setof all probability measures on (, F) byP(, F).Definition 6.1 (Random field) Let P(, F). Any family (i)iZd ofrandom variables which is defined on the probability space(, F, )and whichtakes values in(E, E) is called arandom field.

If one considers the canonical setup, where i : Eare the projectionsfor any i Zd, a random field is synonymous with a probability measure P(, F). LetS ={ Zd :||


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If we return to our physical intuition we are interested in random fields for

which the so-called spin variablesi exhibit a particular type of dependence.We employ a similar dependence structure as for Markov chains, where thedependence is expressed as a condition on past events. This approach wasintroduced by Dobrushin ([Dob68a],[Dob68b],[Dob68c]) and Lanford and Ru-elle ([LR69]). Here, we condition on the complement of any finite set Zd.To prescribe these conditional distributions of all finite collections of variableswe define the -algebras

T=FZd\ for any S. (6.22)

The intersection of all these -algebras is denoted byT =STand calledthe tail--algebraor tail-field.The dependence structure will be described by some functions linking therandom variables and expressing the energy for a given dependence structure.

Definition 6.2 Aninteraction potentialis a family = (A)ASof func-tionsA : R such that the following holds.

(i) A isFA-measurable for allA S,(ii) For any Sand any configuration the expression

H() = AS,A=

A() (6.23)

exists. The term exp(H()) is called theBoltzmann factor forsome parameter >0, where is the inverse temperature.

Example 6.3 (Pair potential) Let A = 0 whenever|A| > 2 and letJ: Zd Zd R, :E E Rand : E Rsymmetric and measurable.Then a general pair interaction potential is given by

A() = J(i, j)(i, j) if A={i, j}, i=j,J(i, i)(i) if A={i},0 if |A|> 2

for.

We combine configurations outside and inside of any finite set of randomvariable as follows. Let and, S, be given. ThenZd\with (Zd\) =and Zd\(Zd\) =Zd\. With this notation we candefine a nearest-neighbour Hamiltonian with given boundary condition.

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Example 6.4 (Hamiltonian with boundary) Let

S and

and

the functionsJ, and as in Example 6.3 be given. Then

H() =1

2

i,j,ij=1

J(i, j)(i, j) +

i,jc,ij=1

(i, j) +i

J(i, i)(i)

denotes a Hamiltonian in with nearest-neighbour interaction and with con-figurational boundary condition , wherex, y = maxi{1,...,d} |xi yi|forx, yZd. Instead of a given configurational boundary condition one canmodel the free boundary condition and the periodic boundary condition aswell.

In the following we fix a probability measure P(E, E) on the state spaceand call it the reference or a priori measure. Later we may also con-sider the Lebesgue measure as reference measure. Choosing a probabilitymeasure as a reference measure for finite sets gives just a constant fromnormalisation.

Definition 6.5 (Gibbs measure)

(i) Let, S, >0 the inverse temperature and be an interac-tion potential. Define for any eventA

F(A|) =Z()1

(d)1lA(Zd\)exp

H(Zd\)

(6.24)with normalisation or partition function

Z() =

(d)exp

H(Zd\)

.

Then (|) is called the Gibbs distribution in with boundarycondition Zd\, with interaction potential , inverse temperature

and reference measure.

(ii) A probability measure P(, F)is aGibbs measurefor the inter-action potential and inverse temperature if

(A|F) =(A|) a.s. for all A F, S, (6.25)where is the Gibbs distribution for the parameter (6.24). The setof Gibbs measures for inverse temperaturewith interaction potential is denoted byG(, ).

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(iii) An interaction potential is said to exhibit a first-order phase transi-

tion if|G(, )|> 1 for some >0.If the interaction potential is known we may skip the explicit appearanceof the interaction potential and write insteadG() for the set of Gibbs mea-sure with inverse temperature . However, the parameter can ever beincorporated in the interaction potential .

Remark 6.6

(i) (A|) isT-measurable for any eventA F.

(ii) The equation (6.25) is called theDLR-equation orDLR-conditionin honour of R. Dobrushin, O. Lanford and D. Ruelle.

6.2 The one-dimensional Ising model

In this subsection we study the one-dimensional Ising model. If the statespace E is finite, then one can show a one-to-one correspondence betweenthe set of all positive transition matrices and a suitable class of nearest-neighbour interaction potentials such that the setG() of Gibbs measuresis the singleton with the Markov chain distribution. This is essentially for a

geometric reason in dimension one, the condition on the boundary is a two-sided Markov chain, for details see [Geo88]. A simple one-dimensional modelwhich shows this equivalence was suggested 1920 by W. Lenz ([Len20]), andits investigation by E. Ising ([Isi24]) was a first and important step towardsa mathematical theory of phase transitions. Ising discovered that this modelfails to exhibit a phase transition and he conjectured that this will hold alsoin the multidimensional case. Nowadays we know that this is not true. InSubsection 6.4 we discuss the multidimensional case.

Let E ={1, +1} be the state space and consider the lattice Z. Ateach site the spin can be downwards, i.e.,

1, or be upwards, i.e., +1. The

nearest-neighbour interaction is modelled by a constantJ, called thecouplingconstant, through the expression Jij for any i, j Zwith|i j|= 1:J > 0: Any two adjacent spins have minimal energy if and only if theyare aligned in that they have the same sign. This interaction is thereforeferromagnetic.

J < 0: Any two adjacent spins prefer to point in opposite directions. Thusthis is a model of an antiferromagnet.

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h

R: A constant h describes the action of an external field (directed

upwards when h >0).Hence the nearest-neighbour interaction potential J,h = (J,hA )ASreads

J,hA () =

Jii+1 , if A={i, i + 1},

h , if A={i},0, else

for . (6.26)

We employ periodic boundary conditions, i.e., for Z finite and with ={1, . . . , ||} we set ||+1 = 1 for any . The Hamiltonian in with periodic boundary conditions reads

H(per) () =J||i=1

ii+1 h||i=1

i. (6.27)

The partition function depends on the inverse temperature > 0, the cou-pling constant Jand the external field h R, and is given by

Z(,J,h) =

1=1

||=1

exp

H(per) ()

. (6.28)

We compute this by the one-dimensional version oftransfer matrix formalism

introduced by [KW41] for the two-dimensional Ising model. More detailsabout this formalism and further investigations on lattice systems can befound in [BL99a] and [BL99b]. Crucial is the identity

Z(,J,h) =

1=1

||=1

V1,2 V2,3 V||1,||V||,1

with

Vii+1 = exp1

2hi+ J ii+1+

1

2hi+1

for any and i = 1, . . . , ||. Hence, Z(,J,h) = TraceV|| with thesymmetric matrix

V=

e(J+h) eJ

eJ e(Jh)

that has the eigenvalues

= eJ cosh(h)

e2J sinh2(h) + e2J 1

2. (6.29)

This gives finallyZ(,J,h) =

||+ +

|| . (6.30)

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This is a smooth expression in the external field parameter h and the inverse

temperature ; it rules out the appearance of a discontinuous isothermalmagnetisation: so far, no phase transition. The thermodynamic limit of thefree energy per volume is

f(,J,h) = limZ

1

|| log Z(,J,h) =1

log +, (6.31)

because ||1 log Z(,J,h) = log + 1||

1+( +

)||

. The magnetisation

in the canonical ensemble is given as the partial derivative of the specific freeenergy per volume,

m(,J,h) =hf(, J, h) = sinh(h)sinh2(h) + e4J

.

This is symmetric, m(,J, 0) = 0 and limh m(, J, h) =1 and for allh= 0 we have |m(,J,h)|>|m(, 0, h)| saying that the absolute value of themagnetisation is increased by the non-vanishing coupling constant J. ThesetG(, ) of Gibbs measures contains only one element, called J,h, see[Geo88] for the explicit construction of this measure as the correspondingMarkov chain distribution, here we outline only the main steps.1.) The nearest-neighbour interaction J,h in (6.26) defines in the usual waythe Gibbs distributions J,h (|) for any finite Z and any boundarycondition . Define the function g : E3 (0, ) by

J,h{i} (i = y|) =g(i1, y , i+1), yE, i Z, . (6.32)We compute

g(x,y,z) = ey(h+Jx+Jz )/2cosh((h + Jx + Jz)) for x, y,zE. (6.33)Fix any aE. Then the matrix

Q= g(a,x,y)g(a,a,y)

x,yE

(6.34)

is positive. By the well-known theorem of Perron and Frobenius we havea unique positive eigenvalue q such that there is a strictly positive righteigenvectorr corresponding to q.2.) The matrix PJ,h, defined as

PJ,h=

Q(x, y)r(y)

qr(x)

x,yE

(6.35)

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is uniquely determined by the matrixQ and therefore byg in (6.33). Clearly,

PJ,h is stochastic. We then let J,h P() denote (the distribution of) theunique stationary Markov chain with transition matrix PJ,h. It is uniquelydefined by

J,h(i=x0, i+1=x1, . . . , i+n=xn) =PJ,h(x0)n

i=1

PJ,h(xi1, xi), (6.36)

where i Z, nN, x0, . . . , xnE, and PJ,h satisfies PJ,hPJ,h= PJ,h.The expectation at each site is given by

EJ,h(i) = e4J + sinh2(h) 12 sinh(h) , i Z.In the low temperature limitone is interested in the behaviour of the setG(, ) of Gibbs measures as . A configuration is called aground state of the interaction potential if for each site i Z the pair(i, i+1) is a minimal point of the function

:{1, +1}2 R; (x, y)(x, y) =Jxy+12

h(x+ y).

Note that the interaction potential = (A)AS with

A= (i, i+1), ifA={i, i + 1}0, otherwiseis equivalent to the given nearest neighbour interaction potential . Wedenote the constant configuration with only upward-spins by+(respectivelythe constant configuration with only downward-spins by ). The Diracmeasure on these constant configurations is denoted by + respectively .Then, for h >0, we get that

J,h+ weakly in sense of probability measures as ,

and hence+is the unique ground state of the nearest-neighbour interactionpotential . Similarly, for h


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6.3 Symmetry and symmetry breaking

Before we study the two-dimensional Ising model, we briefly discuss the roleof symmetries for Gibbs measures and their connections with phase transi-tions. As is seen by the spontaneous magnetisation below the Curie tem-perature, the spin system takes one of several possible equilibrium stateseach of which is characterised by a well-defined direction of magnetisation.In particular, these equilibrium states fail to be preserved by the spin re-versal (spin-flip) transformation. Thus breaking of symmetries has someconnection with the occurrence of phase transitions.Let Tdenote the set of transformations

: , (i1 i)iZd,where : Zd Zd is any bijection of the lattice Zd, and the i :EE, iZ

d, are invertible measurable transformations ofEwith measurable inverses.Each T is a composition of a spatial transformation and the spintransformations i, i Zd, which act separately at distinct sites of the squarelattice Zd.

Example 6.7 (Spatial shifts) Denote by = (i)iZd the group of allspatial transformations or spatial shifts or shift transformationsj :

, (i)

iZd

(i

j)

iZd.

Example 6.8 (Spin-flip transformation) Let the state spaceEbe a sym-metric Borel set ofR and define thespin-flip transformation

: , ()iZd(i)iZd.Notation 6.9 The set of all translation invariant probability measures onis denoted byP(, F) ={ P(, F) : = 1i for any i Zd}.The set of all translation invariant Gibbs measures for the interaction poten-tial and inverse temperatureis denoted byG(, ) ={ G(, ) : =

1

i for any i Zd

}.Definition 6.10 (Symmetry breaking) A symmetry T is said to bebroken if there exists some G(, ) such that()= for some.

A direct consequence of symmetry breaking is that|G(, )| > 1, i.e.,when there is a symmetry breaking the interaction potential exhibit a phasetransition. There are models where all possible symmetries are broken aswell as models where only a subset of symmetries is broken. A first exam-ple is the one-dimensional inhomogeneous Ising model, which is probably the

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simplest model showing symmetry breaking. The one-dimensional inhomoge-

neous Ising model on the lattice Nhas the inhomogeneous nearest-neighbourinteraction potential = (A)AS defined for a sequence (Jn)nN of realnumbersJn>0 for all n Nsuch that

nNe

2Jn 0. Thesimplest spatial shift invariant model which exhibits a phase transition is

the two-dimensional Ising model, which we will study in the next subsec-tion 6.4. This model breaks the spin-flip symmetry while the shift-invarianceis preserved. Another example of symmetry breaking is the discrete two-dimensional Gaussian model by Shlosman ([Shl83]). Here the spatial shiftinvariance is broken. More information can be found in [Geo88] or [GHM00].

6.4 The Ising ferromagnet in two dimensions

Let E={1, 1} be the state space and define the nearest-neighbour inter-action potential = (A)AS as

A =ij , ifA={i, j}, |i j|= 1

0, otherwise .

The interaction potential is invariant under the spin flip transformation and the shift-transformationsi, i Zd. Let+, be the Dirac measures forthe constant configurations + and . The interaction potentialtakes its minimum at + and , hence + and are ground states forthe system. The ground state generacy implies a phase transition if+, are stable in the sense that the set of Gibbs measureG(, ) is attracted byeach of the measures + and for . Let d denote the Levy metriccompatible with weak convergence in the sense of probability measures.Theorem 6.11 (Phase transition) Under the above assumptions it holds

lim

d(G(, ), +) = lim

d(G(, ), ) = 0.

For sufficiently largethere exist two shift-invariant Gibbs measure+,

G(, ) with(+) = and(0) = E(0)< 0 <

+(0) = E+

(0).

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Remark 6.12

(i) +(0) is the mean magnetisation. Thus: The two-dimensional Isingferromagnet admits an equilibrium state/measure of positive magnetisa-tion although there is no action of an external field. This phenomenonis calledspontaneous magnetisation.

(ii)G(, )> 1+(0)> 0 goes back to [LL72]. Moreover, the Grif-fiths inequality implies that the magnetisation+(0)is a non-negativenon-decreasing function of. Moreover there is a critical inverse tem-peraturec such that

G(, )

= 1 when < c and

G(, )

>1

when > c. The value ofc is

c=1

2sinh1 1 =

1

2log

1 +

2)

and the magnetisation forc is

+(0) =

1 (sinh 2)4 18 .(iii) For the same model in three dimensions one has again+,

G(, ),

but there also exist non-shift-invariant Gibbs measures ([Dob73]).

Proof of Theorem 6.11. Let Z

2 be a centred cube. Denote by

B={{i, j} Z2 :|i j|= 1, {i, j} =}the set of all nearest-neighbour bondswhich emanate from sites in . Eachbond b ={i, j} B should be visualised as a line segment between iand j. This line segment crosses a unique dual line segment between twonearest-neighbour sitesu, v in the dual cube (shift by 1

2 in the canonical

directions). The associate setb={u, v}is called the dual bondofb, and wewrite

B={b : b B}={{u, v} :|u v|= 1}

for the set of all dual bonds. Note

b={u+ :|u (i +j)/2|= 12}.

A setc Bis called a circuit of lengthl ifc ={{u(k1), u(k)} : 1kl}forsome (u(0), . . . , u(l)) with u(l) = u(0),|{u(1), . . . , u(l)}| = l and{u(k1), u(k)} B

, 1kl. A circuitc surrounds a site a if for all paths (i(0), . . . , i(n))

in Z2 withi(0) =a and i(n) / and{i(m1), i(m)} Bfor all 1mn thereexits a m N with{i(m1), i(m)} c. We denote the set of circuits whichsurround abyCa. We need a first lemma.

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Lemma 6.13 For alla

and l

1 we have{cCa :|c|= l}l3l1.

Proof. Each cCa of length l contains at least one of the l dual bonds

{a + (k 1, 0), a + (k, 0)} k= 1, . . . , l ,

which cross the horizontal half-axis from a to the right for example. Theremaining l 1 dual bonds are successively added, at each step there are atmost 3 possible choices.

The ingenious idea of Peierls ([Pei36]) was to look at circuits which occur ina configuration. For each we let

B() ={b : b={i, j} B, i=j}

denote the set of all dual bonds in B which cross a bond between spins ofopposite sign. A circuit c with c B() is called a contour for . We let outside of be constant. As in Figure 3 we put outside + -spins. If a sitea is occupied by a minus spin then ais surrounded by a contour for .

The idea for the proof of Theorem 6.11 is as follows. Fix+ boundary

condition outside of . Then the minus spins in form (with high proba-bility) small islands in an ocean of plus spins. Then in the limit Z2 oneobtains a+ G() which is close to the delta measure +for sufficientlylarge. As+ and are distinct, so are

+ and

when is large. HenceG()>1 when is large. We turn to the details. The next lemma just

ensures the existence of a contour for positive boundary conditions and oneminus spin in . We just cite it without any proof.

Lemma 6.14 Let withi= +1 for allic anda=1 for somea. Then there exists a contour for which surroundsa.

Now we are at the heart of the Peierls argument, which is formulated in thenext lemma.

Lemma 6.15 Supposec B is a circuit. Then

(c B()|)e2|c|

for all >0 and for all.

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Proof. Note that for all

we have

H() =

{i,j}Bij =|B|

{i,j}B

(1 ij)

=|B| 2|{{i, j} B : =j}|=|B| 2|B|.Now we define two disjoint sets of configurations which we need later for anestimation.

A1={ : Zd\=Zd\, c B()}A2={ : Zd\=Zd\, c B() =}.

There is a mapping c :

with

(c)i = , ifi is surrounded by c,

, otherwise ,

which flips all spins in the interior of the circuit c. Moreover, for all{i, j} B we have

(c)i(c)j =

ij , if{i, j} /cij , if{i, j}c ,

resulting in B(c)B() = c (this was the motivation behind the defi-nition of mappings), where denotes the symmetric difference of sets. Inparticular we get that c is a bijection from A2 to A1, and we have

H() H(c) = 2|B()| 2|B(c)|=2|c|.Now we can estimate with the help of the set of events A1, A2

(c B()|)

A1 exp(H())A2 exp(H())

=

A2 exp(H(c))

A2 exp(H())= exp(2|c|).

Now we finish our proof of Theorem 6.11. For >0 define

r() = 1 l1

l(3e2)l,

where denotes the minimum, and note that r() 0 as . Thepreceding lemmas yield

(a|+)cCa

(c B()|+)cCa

e2|c|l1

l3l2l,

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and thus (a

|+)

r() for all a

Z

2, > 0 and

Z

2. Choose

N= [N, N]2 Z2 and define the approximating measures

+N= 1

|N|

iNN+i(|+).

AsP(, F) is compact, the sequence (+N)NNhas a cluster point+and onecan even show that + G(, ) ([Geo88]). Our estimation above givesthen +(a=1)r(), and in particular one can show that

lim

+= + and lim

d(G(), +) = 0.

Note = (+). Hence,

+(0=1|+) =(0= +1|).Ifis so large that +(0 =1)r()< 13 , then

+(0 =1) =(0= +1)0 is a convex set, i.e., if, G(, ) and 0< s


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is a tail event in

Tfor any cofinal sequence (n)n

N with n

Z

d asn

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