Convexity in the Theory of Lattice Gases

by Robert B. Israel

Overview

In this book, Robert Israel considers classical and quantum lattice systems in terms of equilibrium statistical mechanics. He is especially concerned with the characterization of translation-invariant equilibrium states by a variational principle and the use of convexity in studying these states.

Arthur Wightman's Introduction gives a general and historical perspective on convexity in statistical mechanics and thermodynamics. Professor Israel then reviews the general framework of the theory of lattice gases. In addition to presenting new and more direct proofs of some known results, he uses a version of a theorem by Bishop and Phelps to obtain existence results for phase transitions. Furthermore, he shows how the Gibbs Phase Rule and the existence of a wide variety of phase transitions follow from the general framework and the theory of convex functions. While the behavior of some of these phase transitions is very "pathological," others exhibit more "reasonable" behavior. As an example, the author considers the isotropic Heisenberg model. Formulating a version of the Gibbs Phase Rule using Hausdorff dimension, he shows that the finite dimensional subspaces satisfying this phase rule are generic.

Originally published in 1979.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9781400868421
Publisher:	Princeton University Press
Publication date:	03/08/2015
Series:	Princeton Series in Physics , #1729
Sold by:	Barnes & Noble
Format:	eBook
Pages:	258
File size:	8 MB

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Convexity in the Theory of Lattice Gases

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By Robert B. Israel

PRINCETON UNIVERSITY PRESS

Copyright © 1979 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08216-5

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CHAPTER 1

INTERACTIONS

1.1. Classical lattice systems

The Ising model is a simple example of a statistical-mechanical system. At each site of the lattice Zv we assume there is a "spin" which can be either "up" (+1) or "down" (-1). Thus for each subset Λ of Zv We have the space ΩΛ = {-1,+ 1}Λ of configurations in Λ. With the product topology this forms a compact metric space, even if Λ is infinite. The full configuration space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will he denoted simply by Ω, and σi(ω) will denote the spin at site i ε Zv in the configuration ω. The Hamiltonian for the system in a finite subset Λ of Zv will be some real-valued function HΛ on ΩΛ. At inverse temperature β we obtain the partition function (in a canonical ensemble)

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thermal averages

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for functions A on ΩΛ.

For more general classical lattice systems, we will allow the "spin" at each site to vary over a compact metric space Ω0. The configuration space corresponding to Λ [subset] Zv is then ΩΛ = (Ω0)Λ, again with the product topology. Again ΩZv will be denoted byΩ. The Hamiltonian for a finite subset Λ of Zv will be a continuous real-valued function HΛ on ΩΛ. To obtain a partition function and thermal averages we need an "a priori" measure to take the place of the summations in (1) and (2). It will be convenient to use a probability measure for this purpose. This measure should describe the state of a noninteracting system, and so the spins at different sites should be independent and identically distributed under this measure. Thus we will take a probability measure μ0 on Ω0, and the a priori measure for Λ [subset] Zv will be the product measure μΛ0 on ΩΛ; expectation values under μΛ0 will frequently be denoted 0,Λ Whenever there is no danger of confusion we will simply use μ0 and 0 in place of μΛ0 and 0,Λ. We also use the natural maps iXY:ΩX -> ΩY (for Y [subset] X [subset] Zv) without warning; thus we will tacitly identify a function F on ΩY with the function F [??] iXY on ΩX. We will always assume that μ0 is supported on all of Ω0 i.e. every non-empty open subset has nonzero μ0-measure.

In many cases the choice of μ0 will be suggested by some symmetry of Ω0. For example, if Ω0 is a finite set we will take μ0 to be normalized counting measure; in the classical Heisenberg model Ω0 is a sphere, so we will take μ0 to be normalized surface measure. In other cases the choice will not be so obvious; we venture no opinion on how Nature chooses an a priori measure.

The partition function and thermal averages for our system in a finite subset Λ of Zv are now

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that these are analytic functions of β and of any parameter entering linearly in the Hamiltonian. Thus the finite system has no phase transitions; to find such interesting phenomena we must pass to an infinite-volume limit. From the physical point of view this limit should provide a description of the properties of the system "in bulk," removing all surface effects.

Out Hamiltonians will arise from interactions. For the Ising model an interaction is usually taken as a real function φ on finite nonempty subsets of Zv, and the associated Hamiltonians are

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now instead of speaking of spins we could use a different formulation of the same model: the "lattice gas." Here the configuration space for a subset Λ of Zv is taken to be the set P(Λ)of subsets of Λ. This can be identified with ΩΛ by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the lattice gas formulation the interaction is again a real function φ on finite non-empty subsets of Zv, but the Hamiltonian in a finite subset Λ of Zv is the function on P(Λ) given by

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Under the identification of P(Λ) with ΩΛ this becomes the function on ΩΛ given by

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We will use a more general formulation which includes both spin and lattice gas interactions as special cases. This will have several advantages:

i) it can be applied to an arbitrary Ω0 and μ0 where there are no obvious choices to replace the σX or ρX above;

ii) it provides a simple way of expressing certain interactions we will use in Chapters II and III;

iii) it provides a space of interactions with a norm naturally connected to the norm topology on states (probability measures on the configuration space).

In our formulation, an interaction Φ assigns to each nonempty finite subset X of Zv a real-valued continuous function Φ(X) onΩX. We will always assume our interactions are translation-invariant, i.e., they satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where τi is the natural map from the space C(ΩX) of continuous functions on ΩX to C(ΩX+i). The Hamiltonian for a finite subset Λ of Zv is then the function HΦΛ on ΩΛ given by

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the Ising model, with Ω0 = {-1,+1}, a "spin language" interaction associated to a Hamiltonian as in (5) is identified with one of our interactions having each Φ(X) a real multiple of σX, while for a "lattice gas language" interaction as in (7) we would take each Φ(X) to be a multiple of ρX.

An interaction Φ has finite range if Φ(X) = 0 for all X of sufficiently large diameter. The linear space of finite-range interactions will be denoted B0 . Banach spaces of interactions are obtained by completing B0 in various norms. We denote by B the space of interactions with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [absolute value of x] is the number of sites in X. [??] will denote the space of interactions with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These are separable real Banach spaces with B0 [subset] [??] [subset] B, B0 being dense in both [??] and B.

For real-valued A [member of] C(ΩX) we define an interaction by ΨXA by

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if Y is not a translate of X .

These will form a particularly important class of interactions.

1.2. The pressure*

We will call the function

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the pressure for the finite region Λ [subset] Zv. This terminology is really appropriate for the lattice gas, where the ensemble considered is grand canonical; in the canonical ensemble it would be more correct to speak of "minus the free energy per site," but we will ignore this distinction. Some of the importance of PΛ arises from what its directional derivatives tell us about the thermal averages of observables. Suppose A [member of] C(ΩX) is real-valued, and let ΨXA be the interaction defined as in

(9). Then since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is the thermal average of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at inverse temperature β = 1 for the Hamiltonian HΦΛ. Since β can be absorbed into the interaction, we will generally take β = 1 for simplicity. If we were only interested in a finite system, we could take X = Λ and obtain the thermal average of A itself; however, we are interested in the infinite system, where a purely local perturbation (leaving out all the other translates of A) would have negligible effect. Our hope is that as Λ increases we obtain the thermal average of A in a translation-invariant state of the infinite system.

Our first task is to study the infinite-volume limit of PΛ(Φ). After some preliminary inequalities, we will consider a sequence of (v-dimensional) cubes Ca, and then show that the same limit is obtained for any sequence Λn tending to infinity in the sense of van Hove (defined below), which means roughly that the boundary of Λ becomes negligible in comparison to Λ itself.

Lemma 1.2.1 (Jensen's Inequality). For any probability measure μ and real random variable f [member of] L1(μ),

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. We can assume f is bounded below, since both sides of (12) are the limits as N -> ∞ of the expressions obtained by substituting max (f,-N) for f. Since adding a constant to f multiplies both sides by a positive constant, we can assume f ≥ 0. Now by Holder's Inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 1.2.2. For any probability measure μ and real functions F, g [member of] L∞ (μ)

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] almost everywhere,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking logarithms yields (13). Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also a probability measure, Jensen's Inequality yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By interchanging f and g we also obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can now obtain some useful estimates on PΛ. Note that

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus by Lemma 1.2.2 we have

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore if we can show that PΛ(Φ)tends to a limit when Φ [member of] B0the usual type of "3[member of]" argument will extend this to B.

Theorem 1.2.3. For positive integers a, let Cabe a cube of side a. Then for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists. P is a convex function on B with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. As remarked above, we need only to prove this in β0 and then use (17) to extend it to B. Let Φ [member of] B0 and let d be a positive integer such that Φ(X) = 0 whenever X has diameter at least d. Given a, let b = n(d + a) + c with n ≥ 1, 0 ≤ c < d + a. We partition the cube Cb as shown in Figure 1A: the unshaded region Λ' is formed by nv cubes of side a, separated by corridors of width d. Since these cubes of side a do not interact with each other, HΦΛ is the sum of nv independent copies of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (functions on the configurations in disjoint cubes, and thus independent with respect to μ0) Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Dividing through by [absolute value of Cb] = bv we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As b -> ∞, n(a + d)/b -> 1, and so for each a

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now taking a -> ∞, the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] form a Cauchy sequence with limit P(Φ). The estimate (17) carries over to the limit to yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Convexity of PΛ is obtained by Hölder's Inequality:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For 0 ≤ t ≤ 1, and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This inequality also carries over to the limit, yielding convexity of P.

For each positive integer a we partition Zv into a family Ca of Cubes of the form {i: nja ≤ ij< (nj + 1, ..., v)} For integers n1, ..., nv. For each finite subset Λ of Zv we define N+a(Λ) as the number of cubes in Ca which intersect Λ. A sequence of finite subsets Λn of Zv will be said to converge to infinity in the sense of van Hove if for all a, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 1.2.4. If Λn -> ∞ (van Hove) and Φ [member of] B, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By the usual "3[member of]" argument it suffices to prove this for Φ [member of] B0.

We again take d so that Φ(X) = 0 when diam (X) ≥ d. Fix a and n. There are N[??]a+d (Λn) cubes in Ca+d contained in Λn, and by removing corridors of width d we are left with the unshaded region Λ' of Figure 1B, consisting of N-a+d(Λn) cubes of side a. As in the proof Theorem 1.2.3 we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. And so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now taking [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So far we have been using "free boundary conditions": the Hamiltonian HΦΛ describes a situation where there is nothing outside Λ to interact with the spins in Λ. For interactions in the space [??] we may also consider other types of boundary conditions. We will find that the infinite-volume pressure P(Φ) is independent of which boundary conditions are used. There are two main types of boundary conditions to consider: external fields (which might more accurately be called "external configurations") and periodic boundary conditions.

An external configuration for the finite subset Λ of Zv is a member τ of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Λc denotes the complement of Λ in Zv). The Hamiltonian corresponding to the external configuration [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the function on ΩΛ

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This sum converges if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The Corresponding pressure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be denoted τPΛ(Φ).

(Continues...)

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Table of Contents

• Frontmatter, pg. i
• CONTENTS, pg. vii
• INTRODUCTION. Convexity and the Notion of Equilibrium State in Thermodynamics and Statistical Mechanics, pg. ix
• I. Interactions, pg. 1
• II. Tangent Functionals and the Variational Principle, pg. 32
• III. DLR Equations and KMS Conditions, pg. 55
• IV. Decomposition of States, pg. 83
• V. Approximation by Tangent Functionals: Existence of Phase Transitions, pg. 112
• VI. The Gibbs Phase Rule, pg. 130
• APPENDIX Α. Hausdorff Measure and Dimension, pg. 143
• APPENDIX B. Classical Hard-Core Continuous Systems, pg. 153
• BIBLIOGRAPHY, pg. 163
• INDEX, pg. 166