Cell Gas Free Energy as an Approximation of the Continuous Model

A continuous infinite system of point particles interacting via two-body strong superstable potential is considered in the framework of cell gas (CG) model of classical statistical mechanics. We consider free energy of this model as an approximation of the correspondent value of the continuous system. It converges to the free energy of the conventional continuous gas if the parameter of approximation 0 → a for any values of an inverse temperature 0 > β and volume per particle 0 > v .

in some interval of change of density ρ , pressure p does not depend on density or a specific volume of 1 v ρ = (see, for example [1]). Taking into account the well-known thermodynamics formulas it means that free energy of the system depends on specific volume v linearly in the indicated interval of change of density. This result was obtained as early as the end of 60th for the lattice gas model in the articles of F. A. Berezin and Ya. G. Sinai [2] for unpositive potentials and R. L. Dobrushin [3] for more general potentials of interaction.
However, the lattice gas is some kind of "toy" model which is very far from the real continuous system. The model of cell-type gas, which actually is the model of the continuous system of point particles and differs from the standard model of gas only by determination of the phase (configuration) space, was offered in recent work [4] of one of the authors of this article.
Cell gas in d  is a continuous gas but its space of configurations is arranged so that for a given partition a ∆ of d  into elementary hiper-cubes d ∆ ⊂  with a rib 0 a > there is no more then one point particle in each cell (cube). These particles move in d  and interact via two-body strong superstable potential φ . According to the results of articles [5] [6] and [7] the correlation functions and the pressure of cell gas system tend to the corresponding values of conventional continuous gas at 0 a → . Within the framework of the grand canonical ensemble this result followed from a convenient representation of the corresponding quantities by Poisson integrals on the configuration space of the system. In this short paper we establish a similar result for the free energy of the system. This result requires more hard work as the corresponding representation in the canonical ensemble less convenient for mathematical calculations.
Why do we need this result? In the article [4] it was shown that it was possible to introduce an approximation of the interaction potential in such a way that the cell gas model grows into the model of the lattice gas on the lattice d a , and at 0 a → both models coincide with a model which describes the continuous statistical system. Therefore we consider the result of this article as the first modest step to realization of the Dobrushin's way [3] to solve the phase transition problem in continuum.

Configuration Space
The restriction of σ λ to ( ) B Λ Γ we also denote by σ λ . For more detailed structure of the configuration spaces Γ , 0 Γ , Λ Γ and measures on them see e.g. [8] (see also latest review [9]). Let 0 a > be arbitrary. Following [10] for each d r a ∈  we define an elementary cube with an edge a and a center r , if a cube ∆ is considered to be arbitrary and there is no reason to emphasize that it is centered at the concrete point . Without loss of generality we consider Λ in the form of a large cube and only that and subsets , X Y ⊂ Λ which are union of cubes ( ) a r ∆ and corresponding partition: Then for any X ⊆ Λ which is a union of cubes and is called cell gas system of particles. For detail structure of this model see [4].

Definition of the System
We consider a general type of two-body interaction potential (A): Assumption on the interaction potential. Potential φ is continuous on The potentials of this type are strong superstable. Definition 2.2. Interaction is called strong superstable (SSS), if there exist 0 0 a > , and constants ≥ , and 2 m ≥ such that for any 0 0 a a < ≤ and any 0 γ ∈ Γ an interaction energy of particles satisfy the following inequality: Remark 2.1. Superstable interactions were introduced by D. Ruelle (see [11] or [12], Ch. 3.2.9 and [10]). Y. M. Park (see [14]) was the first, who used the condition (12) with 2 m > for the proof of bounds for exponent of local number operator of quantum systems of interacting Bose gas. We have changed the definition of strong superstability including the case 2 m = , but with the constants which depends on parameter a (see, e.g., [13] [4]). SSS potentials include all interaction potentials which are nonintegrable in the initial point.
One of the most popular example which is used in molecular physics is Lenard-Jonson potential: where constants 0, 0 C D > > . In this article we consider the potentials of this type. The typical behavior of such potentials is shown in Figure 1.
See for the proof [13].

Partition Functions, Free Energy and Pressure
The main physical characteristics of the system are determined by thermodynamic potentials that associated with small and grand partition functions by the following formulas: 1) free energy where limit is done in such a way that volume per particle ( ) N v σ Λ → , 1 kT β = , and small partition function where z is activity of the system and holds for all positive , v β and

The Proof of the Main Results
The proof of the Theorem 2.1 is the same as the corresponding proof of such theorem for ( ) , f v β in [15]. The only remark to the proof is that the construction of auxiliary partitions into cubes in [15] should be agreed with the partition a ∆ . To prove the Theorem 2.2 we insert the unite Separating the first term of the expansion which corresponds to the value X = ∅ we can rewrite (30) in the form: To estimate the second term in (33) we split the energy ( ) U γ in every term of the sum in (32): and use SSS inequality (12). Then We denote the integral in (32) (after estimating (37)) by the letter X I and rewrite an expression for ! X X . There are at least two variables from the configuration . Denote the number of variables that are in cubes 1 . Among all 2 N terms which appear in the right side of (39)) does not vanish only those terms in which the integration is performed with respect to the variables { } To estimate the ratio of the partition functions in (40) we use the following lemma. Lemma 1 Suppose that the interaction potential φ satisfies the assumptions A (see (9), (10)). Then there and chose the Λ sufficiently large and a sufficiently small to satisfy the following inequality: . □ Now, the proof of the Theorem 2 follows from the trivial estimates of the combinatorial sums in (40). Let for simplicity 2 m = in SSS assumption (12).