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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 808276, 14 pages
http://dx.doi.org/10.1155/2011/808276
Research Article

A Direct Method for the Analyticity of the Pressure for Certain Classical Unbounded Models

King Fahd University of Petroleum and Minerals, P.O. Box 419, Dhahran 31261, Saudi Arabia

Received 24 November 2010; Accepted 20 January 2011

Academic Editor: Giorgio Kaniadakis

Copyright © 2011 Assane Lo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The goal of this paper is to provide estimates leading to a direct proof of the analyticity of the pressure for certain classical unbounded models. We use our new formula (Lo, 2007) to establish the analyticity of the pressure in the thermodynamic limit for a wide class of classical unbounded models in statistical mechanics.

1. Introduction

This paper is a continuation of [1] on the analyticity of the pressure. It attempts to study a direct method for the analyticity of the pressure for certain classical unbounded spin systems. The paper presents a simple hypothesis, on a -estimate of the moments of the source term to show that it does yield analyticity in the infinite volume limit.

The study of the analyticity of the pressure is very important in Statistical Mechanics. In fact the analytic behavior of the pressure is the classical thermodynamic indicator for the absence or existence of phase transition [219].

Because the th-derivatives of the pressure are commonly represented in terms of the truncated functions, most of the techniques available so far for proving analyticity of the pressure take advantage of a sufficiently rapid decay of correlations and cluster expansion methods or Brascamp-Lieb inequality [1, 5, 2035].

In this paper, we propose a new method for proving the analyticity of the pressure for a wide class of classical unbounded models. The method is based on a powerful representation of the th-derivatives of the pressure by means of the Witten Laplacians [36] given by These operators are in some sense deformations of the standard Laplace Beltrami operator. They are, respectively, equivalent to Indeed, and the map is unitary. More precisely, we will use the formula where

This formula, due to Helffer and Sjöstrand [29, 37], is a stronger and more flexible version of the Brascamp-Lieb inequality [20]. It allowed us in [1] to obtain an exact formula for the th-derivatives of the pressure. In this paper, we will use this exact formula to show that a simpler assumption on the source term similar to the weak decay used in [3, 11] will guarantee the analyticity of the pressure in the infinite volume limit for a wide class of classical unbounded models.

We will consider classical unbounded systems, where each component is located at a site of a crystal lattice and is described by a continuous real parameter . A particular configuration of the total system will be characterized by an element of the product space .

The will denote the Hamiltonian which assigns to each configuration a potential energy . The probability measure that describes the equilibrium of the system is then given by the Gibbs measure

The is a normalization constant.

We are eventually interested in the behavior of the system in the thermodynamic limit, that is,

Assume that is finite, and consider a Hamiltonian of the phase space satisfying the assumptions of the following theorem.

Theorem 1.1 (see [30]). Let be a finite domain in . If   satisfies  the following:(1), (2)for some , any with is bounded on ,(3)for , for some , (4) for some ,then for any -function satisfying where with some and some , there exists a unique -function solution of

Remark 1.2. This theorem was established by Johnsen [30]. A detailed proof of this theorem in the convex case that includes the regularity theory may also be found in [38]. The function spaces to be considered are the Sobolev spaces defined by where These are subspaces of the well-known Sobolev spaces . By regularity arguments, one may prove that the solution of (1.10) belongs to each for all .

2. The Analyticity of the Pressure

2.1. Preliminaries

We first recall the context over which the formula for the th-derivative of the pressure was derived in [1].

Let be a finite domain in , and consider as above the Hamiltonian of the phase space satisfying the assumptions of Theorem 1.1.

Let be a smooth function on satisfying

Let where , and is a thermodynamic parameter.

The finite volume pressure is defined by Denote that

Because of the assumptions made on , one may find such that, for all , satisfies all the assumptions of Theorem 1.1. Thus, each is associated with a unique -solution of the equation Hence, where . Notice that the map is well defined [1] and that is a family of smooth solutions on corresponding to the family of potential We proved in [1] that is a smooth function of by means of regularity arguments. The following proposition proved in [1] gives an exact formula for the th-derivatives of the pressure.

Proposition 2.1 (see [1]). If where and satisfies the assumptions of Theorem 1.1,  then there exist such that, for all , the th-derivative of the finite volume pressure is given by the formula where

This formula gives a direction towards proving the analyticity of the pressure in the thermodynamic limit. In fact one only needs to provide a suitable estimate for .

Remark 2.2. Though formula (2.12) was derived in [1] for models of Kac type, it is clear from the proof that it remains valid for Hamiltonians for which the Helffer-Sjöstrand formula holds. In [30], Johnsen proved that this formula remains valid for a wide class of none convex Hamiltonians.

3. An Estimate for the Coefficients

In this section we propose to provide an estimate that establishes the analyticity of the pressure in the infinite volume limit.

Recall that if is an infinitely differentiable function defined on an open set , then is real analytic if for every compact set there exists a constant such that for every and every nonnegative integer the following estimate holds:

We propose to establish the above estimate for the -derivatives of the pressure. First we have the following convolution formula.

Proposition 3.1. Under the assumptions and notations of Proposition 2.1, one has

Proof. First observe that Setting yields Now dividing by , summing over , and noticing that on the right-hand side one obtains a telescoping sum yield

Next, we need the following lemma.

Lemma 3.2. Let and be two sequences of real numbers such that for some positive constant . Then

Proof. Let be the sequence defined recursively by It is clear that We need to prove that We have
By induction assume that the result is true if is replaced by . We have

Proposition 3.3. In addition to the assumptions of Proposition 2.1 assume that for the thermodynamic limit of exists, and where is a positive constant and is a positive real variable function satisfying Then the infinite volume pressure is analytic at .

Proof. First choose large enough so that . We then have
Let
We have , and by Proposition 3.1 from which we have Now applying Lemma 3.2 we get where Now using the fact that, for we have This shows that the Taylor series of the infinite volume pressure at has a nonvanishing radius of convergence.
Next, we propose to prove that the pressure is equal to its Taylor series in a neighborhood of .
By (3.16), the power series has nonvanishing radius of convergence . Put
Inside the interval of convergence of , the convolution formula of Proposition 3.1 gives This implies that Equivalently or Thus Now using Proposition 2.1, we obtain Now adding on both sides of this above equality, we get Because has bounded derivatives, we have Thus by permuting sum and integral we obtain

4. Comparison with Known Results

In [11], Lebowitz derived some regularity properties of the infinite volume pressure by assuming that the truncated functions have a weak decay of the type

where is a rapidly decreasing function independent of . However, he only obtained infinite differentiability rather than analyticity. The obstacle from getting analyticity is that, when is rapidly or exponentially decaying, the bounds obtained increase too fast with . In [3], Duneau et al. considered stronger decay assumptions of the truncated functions and showed that they do yield analyticity.

We showed in this paper that if the decay assumption is made on the th moments of for instance, then an even weaker assumption would imply analyticity.

Let us also mention that, even though our results concern unbounded models whose Hamiltonians satisfy the assumptions of Theorem 1.1, it could be useful for the study of certain bounded models. Indeed, it has been shown in [32] that the investigation of the critical behavior of the two-dimensional Kac models may be reduced in the mean-field approximation to the study of unbounded models of the type discussed above.

It is also clear that if the thermodynamic limit exists, then the assumption is equivalent to saying that the partition function is analytic at . Thus, Proposition 3.3 provides a simple and direct proof of the analyticity of the pressure from the analyticity of the partition function. Recall that even in the grand canonical ensemble, where the partition function is directly given as a power series, the classical proofs of the analyticity of the pressure that are available in the literature involve in general cluster expansions, sometimes with complicated renormalization arguments.

Acknowledgment

The support of King Fahd University is deeply acknowledged. The author also would like to express his appreciation to Professor Haru Pinson.

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