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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 α → 0 for any values of an inverse temperature β ＞ 0 and volume per particle ν ＞ 0.

One of the most important mathematical problem of statistical mechanics is description of the gas-liquid phase transition within the framework of standard model of 2-partical Lenarda-Johnson type intermolecular interaction. The presence of phase transition at some temperature

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 [

Cell gas in

Why do we need this result? In the article [

Let

where

and

By

and hence as a measure

Define the Lebesgue-Poisson measure

The restriction of

Let

We will write

Then for any

and

Definition 2.1. Infinite system of point particles in

For detail structure of this model see [

We consider a general type of two-body interaction potential

(A): Assumption on the interaction potential. Potential

where

The potentials of this type are strong superstable.

Definition 2.2. Interaction is called strong superstable (SSS), if there exist

Remark 2.1. Superstable interactions were introduced by D. Ruelle (see [

One of the most popular example which is used in molecular physics is Lenard-Jonson potential:

where constants

Remark 2.2. For the potentials which are considered in this article (see (9)-(11)) the corresponding con- stants

where

See for the proof [

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

2) pressure

where

The correspondent values for cell gas model are defined by the same formulas but with help of partition functions(see Definition 2.1):

and

where

Remark 2.3. The product of functions

Now, we can formulate the main result of the paper.

Theorem 1 Suppose that the interaction potential

for any

Theorem 2 Suppose that the interaction potential

holds for all positive

The proof of the Theorem 2.1 is the same as the corresponding proof of such theorem for

To prove the Theorem 2.2 we insert the unite

where

and

Separating the first term of the expansion which corresponds to the value

where

The Equation (31) gives:

To estimate the second term in (33) we split the energy

where

and use SSS inequality (12). Then

where

We denote the integral in (32) (after estimating (37)) by the letter

Every set in

to the variables

respect to permutations of variables

To estimate the ratio of the partition functions in (40) we use the following lemma.

Lemma 1 Suppose that the interaction potential

for any

Proof. Let us fix some

with any

with

where

and chose the

Then, taking into account that for

and

we obtain:

Holder’s inequality to (49) with respect to probability measure

Using the property (9) and definition (42) we have:

Using this inequality and taking into account that

Taking into account that

with

Now, the proof of the Theorem 2 follows from the trivial estimates of the combinatorial sums in (40). Let for simplicity

with

It is clear from the Equations (15), (16), (38) that

so, this gives the proof of the main result. □

The main result of the article is presented by the Theorem 2.2. It proves that all thermodynamics properties of the infinite system which is defined by phase space (2.1) and interaction potential (2.9) - (2.11) can be described by the cell gas model, phase space and thermodynamics descriptions which are determined by the formulas (2.7), (2.22) - (2.25). In other words, this model approximates the statistical continuous system of interacting point particles up to any preassigned accuracy. It is needed to mark another surprising fact that the set

We thank the referee for valuable remarks which improved the original version. The authors gratefully acknowledge the financial support of the Ukrainian Scientific Project “Investigation of the spectral characteristics and critical behavior of complex systems of mathematical physics” (2011-2015).