L2+ε-Esimates on Exponential Decay of Correlations in Equilibrium States of Classical Continuous Systems of Point Particles

We present and prove L2+ε-estimates on exponential decay of correlations in equilibrium states of classical continuous systems of point particles interacting via an exponentially decaying pair potential of interaction, where ε is arbitrary small and positive real number. The obtained estimates exhibit not only the explicit dependence on the distance between the areas of the equilibrium classical systems between which the correlations are estimated but also on the volume of these areas, which can be used in the future for the investigation of the corresponding non-equilibrium and dynamic systems.


Introduction
In the present work, we provide and prove L 2+ε -estimates on exponential decay of correlations in equilibrium states of classical continuous systems of point particles interacting via regular stable pair potential that exponentially decays with distance between particles. The parameter ε can be any (arbitrary small) positive real number. In our estimates, we obtain an explicit dependence of correlations not only on the distance between the regions of an equilibrium classical system of point particles but also on their volumes, that, in the future, can be used for the investigation of the corresponding non-equilibrium and dynamic systems.
The contents of the present work is as follows. In Section 2, we briefly remind the definition and basic properties of the Lebesgue-Poisson measure over the space of locallyfinite configurations and provide some formulae needed for the further consideration. In Section 3, we consider continuous systems of point particles interacting via a pair potential, formulate conditions that this pair potential should satisfy, provide a definition of a limiting Gibbs measure corresponding to equilibrium states of continuous systems of point particles interacting via the pair potential, and construct cluster expansions for the density of the corresponding limiting Gibbs measure with respect to the Lebesgue-Poisson measure over the configuration space. In Section 4, we formulate the result on the existence of the limiting Gibbs measure corresponding to equilibrium states of continuous systems of point particles at a sufficiently high temperature and the sufficiently low activity of the system and prove it by showing the convergence of the corresponding cluster expansions constructed for the density of the limiting Gibbs measure with respect to the Lebesgue-Poisson measure over the corresponding configuration space. Finally, Section 5 is devoted to formulation and proof of the main result of the present work concerning L 2+ε -estimates on the exponential decay of correlations in hight temperature dilute equilibrium states of continuous systems of point particles interacting via a regular stable pair potential that exponentially decays with distance between particles.
To prove the main result of the present paper, that is formulated in Section 5, we use the method of cluster expansions proposed and further developed by M. Duneau, B. Souillard, D. Iagolnitzer, V. Malyshev, and R. Minlos in [1,2,3,4,7].
In application to equilibrium systems of classical statistical mechanics, the method of cluster expansions consists in representing macro-and microscopic observables of an infinite system of interacting particles in a form of the convergent series, each term of which takes into account interaction between particles only withing some their groups or, in other words, clusters. Such presentation allows to investigate various properties of the said macro-and microscopic observables of the equilibrium states of the classical system for those values of the thermodynamic parameters of the classical system for which the series of cluster expansions converge.
In particular, the method of cluster expansions is very convenient for investigation of decay of correlations in equilibrium systems of classical statistical mechanics as it allows to reduce the problem of investigation of correlations in equilibrium systems of infinitely many interacting particles to a set of more simple problems of investigating the correlations in analogous systems of particles interacting only withing some groups of particles, or clusters.
A consideration of such more simple problems for the equilibrium continuous systems of point particles interacting via a regular stable potential that exponentially decays with L 2+ε -Esimates on Exponential Decay of Correlations in Equilibrium States of Classical Continuous Systems of Point Particles the distance between the particles allows to show that some terms in expressions that describe decay of correlations in such systems mutually cancel and, from the remaining terms, one can extract the common factor that exponentially decays with distance. That leads to the final estimates on decay of correlations in such systems of interacting particles. All this technique is detailed in the proof of Lemma 4, the statement of which constitutes the most important estimate leading to the main result of the present paper. As to the main result of this paper, it is formulated in Theorem 2.
It should be noted that similar results concerning decay of correlations for truncated correlation functions of lattice systems was obtained in the classical paper by M. Duneau, B. Souillard, D. Iagolnitzer [3].

A Poisson Measure on the Configuration Space
Let B(R d ) denote a Borel σ-algebra on R d and let L(Λ) denote the system of all bounded Borel subsets of the Borel set Λ ∈ B(R d ).
Definition 1. For any Borel set Λ ∈ B(R d ), the configuration space Γ Λ is defined as the set of all locally finite subsets of the set Λ, that is, where #γ denotes the number of elements of the set γ.
Next, for any Borel sets Λ and Λ ′ from B(R d ) such that Λ ′ ⊂ Λ, we denote by B Λ ′ (Γ Λ ) the σ-algebra on Γ Λ generated by all possible mappings of the form where Λ ′′ ∈ L(Λ ′ ) and configurations γ are taken from the space Γ Λ .

Remark 1.
In what follows, we are going to systematically omit the subindices at Γ and B in case they are equal to R d , that is, we are going to write Γ and B instead of Γ R d and For any Borel set Λ ∈ B(R d ), we will also denote by γ Λ the canonical projection of the configuration γ onto the space Γ Λ , that is, by definition, γ Λ := γ ∩ Λ.

Definition 2. For any Borel set
where

Lemma 1. For any Borel sets
, the following equality holds:

Lemma 2. For any Borel sets
, the following equality holds: ∫ Lemmas 1 and 2 easily follow from Definition 2.
For further consideration, we will also need the following lemma, proof of which can be obtained directly from Definition 2. (See also [7,8].) The summation in the left-hand side of equality (3) is undertaken over all ordered partitions of the configuration γ ∈ Γ Λ onto subsets that are mutually non-intersecting and, in general, can be empty. As to the summation in the left-hand side of equality (4), it is undertaken over all ordered partitions of the configuration γ ∈ Γ Λ onto the non-empty subsets that are mutually non-intersecting.

Limiting Gibbs measures corresponding to continuous systems of point particles
In what follows, we consider classical continuous systems of point particles interacting via a pair potential v that is assumed to be 3) stable, that is, such that satisfies the following stability condition: 4) and regular, that is, such that Remark 2. From the physical point of view, it is natural to assume even more, namely, that the pair potential of interaction v is not only symmetric and translationally invariant but also rotationally invariant, that is, Having in mind exactly such potential of pair interaction of point particles, we, nevertheless, will not use the assumption of rotational invariance of the potential of pair interaction of point particles.
Definition 3. In a finite volume Λ ∈ L(R d ), we define the equilibrium Gibbs states G z,β Λ (·) of continuous systems of point particles interacting via a pair potential v at an inverse temperature β and an activity z as probability measures on the configuration space (Γ Λ , B(Γ Λ )) such that for any con- where and The summation in the right-hand side of (7) is undertaken over all two-element subsets of the configuration γ Λ .
where F Λ0 is an arbitrary bounded Λ 0 -local function, that is, measurable with respect to the σ-algebra B Λ0 .
Remark 3. The stability condition imposed on the pair potential of interaction ensures existence of the Gibbs state G z,β Λ (·) in the finite volume Λ ∈ L(R d ).
It is far not trivial question whether the set of all limiting Gibbs states of a classical continuous system of point particles interacting via a pair potential v is not empty at least for some values of activity z and inverse temperature β.
One of the possible ways to prove that the question above has a positive answer is to construct a cluster expansion for the quantity Λ ∈ L(R d ), and to prove its convergence at Λ ↗ R d .
Let ∑ (γ) denote the sum over all subdivisions of an arbitrary configuration γ onto non-empty subsets, that is, We denote the sets of all graphs, all connected graphs, and all trees 1 whose vertices coincide with the elements of the set γ ∈ Γ (or, in other words, with the points of the configuration γ ∈ Γ) by G(γ), G c (γ), and T(γ), respectively. Let also In the following, we are going to employ the cluster expansions suggested and further developed in [1,2,3,4,7]. Namely, for an arbitrary bounded Borel set Λ ∈ L(R d ), we present the quantity p Λ (γ Λ ) as where k(·) is a function defined on Γ \ {∅} and such that Under most general assumptions on stability and regularity of the pair potential of interaction v, (13) and (14) yield the existence of the limiting Gibbs state that corresponds to a classical continuous system of point particles in equilibrium interacting via the pair potential v at the sufficiently small inverse temperature β and activity z. See, for example, Theorem 1 below.

Remark 4.
For the consideration to follow, it is also convenient to set k(∅) := 0.
and Remark 4 that for any Λ ∈ L(R d ).

Statement 1.
If the formula (13) for the cluster expansion of the quantity p Λ (γ Λ ), Λ ∈ L(R d ), given by (10) holds and ] . (17) Proof. Formulae (10), (8), and (13) yeild that From this, we get the first equality of formula (17), whereas the second equality of formula (17) is an obvious corollary of Lemma 3, Remark 4, definition (12), and the fact that To construct a cluster expansion (13), it suffices to show that for any non-empty finite configuration γ and therefore equality (18) holds with for any γ ∈ Γ \ {∅}. The proof of estimate (14) on the absolute value of the function k(γ) defined by formula (19) can be found in [7] (Lemma 2 from Subsection 4 of Section 4).

Convergence of the Cluster Expansions and Existence of the Limiting Gibbs Measure
Estimate (14) yields (see, for example, [7]) the following theorem of existance of a limiting Gibbs state.
Theorem 1. For a stable regular potential of pair interaction v, sufficiently small inverse temperature β, and activity z such that there exists a limiting Gibbs state G z,β corresponding to the limit (9) of Gibbs states G z,β Λ . Moreover, the density of its restriction onto B Λ ′ (Γ), Λ ′ ∈ L(R d ), can be represented by the formula where and Proof. It is easy to see that for any Λ ′ ∈ B(Λ), Λ ∈ L(R d ), the density of the restriction of the Gibbs state G z,β Λ in finite volume Λ onto B Λ ′ (Γ Λ ) is given by the formula and Then, as it follows from the cluster property of the Lebesgue-Poisson measure, and thus, the expression for f Λ (Λ ′ ) can also be presented as follows: Next, estimate (14) on the absolute value of the function k, definition of the Lebesgue-Poisson measure (2), formula (16), and inequality (15) yield that where α is a function of the parameters z, β, δ and κ. This means that the quantity ∫ It is easy to see that, for an arbitrary configuration γ ∈ Γ, there exists the limit of the quantity Moreover, for an arbitrary configuration γ ∈ Γ lim Λ↗R d Therefore, for λ z Λ ′ -almost all γ Λ ′ ∈ Γ Λ ′ \ {∅}, there exist limits of the quantities f Λ (Λ ′ ), r Λ ′ Λ (γ Λ ′ ), and p Λ ′ Λ (γ Λ ′ ) as Λ ↗ R d and, moreover, these limits equal to f (Λ ′ ), r Λ ′ (γ Λ ′ ), and p Λ ′ (γ Λ ′ ), respectively.

Decay of Correlations
Under an additional assumption that the potential of pair interaction v exponentially decays with distance, from the convergence of the cluster expansions (13), one can get some L p -estimates showing that correlations in limiting Gibbs measures corresponding to equilibrium states of classical continuous systems of point particles interacting via the pair potential v also exponentially decay with distance. In this section, we formulate and prove the corresponding L 2+εestimates, where ε is an arbitrary small but positive real number.
Definition 5. We will say that a symmetric translationinvariant potential of pair interaction v exponentially decays with distance if there exist such positive constants v 0 , r 0 , and κ that where, as before, v(x, For further consideration, we need the following two propositions: where ψ(C) := Ce C . Statement 3. If the potential of pair interaction v exponentially decays with distance, then where Remark 7. If the potential of pair interaction v exponentially decays with distance, then where Ω(d) denotes the volume of d-dimensional sphere with radius 1.
Theorem 2. Suppose that the potential of pair interaction v is stable, regular, and exponentially decays with distance in the sense of Definition 5. Then, for any positive ε and δ, sufficiently small inverse temperature β, and activity z such that where D = 10 α 1 , α 2 are some functions of the parameters z, β, δ, κ, and ⟨ · ⟩ denotes expectation with respect to the probability measure G z,β constructed in Theorem 1.

Remark 8.
As it has already been noted in Remark 7, formula (33) yields formula (20), and so, according to Theorem 1, the limiting Gibbs measure G z,β mentioned in Theorem 2 exists and formulae (22)-(24) for the density of its restriction to the σ-algebra B Λ ′ (Γ), Λ ′ ∈ L(R d ), hold. With the standard technique (see, for example, the proof of Theorem 17.2.2 from [5]), one can get the statement of Theorem 2 from the following L ∞ -estimate on exponential decay of correlations in the limiting Gibbs state G z,β .
Proof. Let us denote by I 12 the following integral: ∫ Then, using formula (21), the first inequality from (22), and the fact that we can write that where, for i = 1, 2, we denoted Next, using formula (24), one can get that