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

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 𝐶^𝑛-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 [2–19].

Because the 𝑛th-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, 20–35].

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 𝑛th-derivatives of the pressure by means of the Witten Laplacians [36] given by𝐖_Φ⁽⁰⁾=||−𝚫+∇Φ²4−𝚫Φ2,𝐖_Φ⁽¹⁾||=−𝚫+∇Φ²4−𝚫Φ2+𝐇𝐞𝐬𝐬Φ. (1.1) These operators are in some sense deformations of the standard Laplace Beltrami operator. They are, respectively, equivalent to𝐴_Φ⁽⁰⁾𝐴∶=−𝚫+∇Φ⋅∇,_Φ(1)∶=−𝚫+∇Φ⋅∇+𝐇𝐞𝐬𝐬Φ. (1.2) Indeed, 𝑊_Φ^(⋅)=𝑒^−Φ/2∘𝐴_Φ^(⋅)∘𝑒^Φ/2, (1.3) and the map 𝑈_Φ∶𝐿²ℝ^Λ⟶𝐿²ℝ^Λ,𝑒^−Φ𝑑𝑥𝑢⟼𝑒^Φ/2𝑢 (1.4) is unitary. More precisely, we will use the formula𝐜𝐨𝐯𝑔,ℎ=𝑍⁻¹𝐴⁽¹⁾⁻¹_Φ∇𝑔⋅∇ℎ𝑒^−Φ(𝑥)𝑑𝑥, (1.5) where 𝑍=𝑒^−Φ(𝑥)𝑑𝑥. (1.6)

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 𝑛th-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 𝑖 of a crystal lattice Λ⊂ℤ^𝑑 and is described by a continuous real parameter 𝑥_𝑖∈ℝ. A particular configuration of the total system will be characterized by an element 𝑥=(𝑥_𝑖)_𝑖∈Λ of the product space ℝ^Λ.

The Φ=Φ^Λ will denote the Hamiltonian which assigns to each configuration 𝑥∈ℝ^Λ a potential energy Φ(𝑥). The probability measure that describes the equilibrium of the system is then given by the Gibbs measure 𝑑𝜇^Λ(𝑥)=𝑍_Λ⁻¹𝑒^−Φ(𝑥)𝑑𝑥. (1.7)

The 𝑍_Λ>0 is a normalization constant.

We are eventually interested in the behavior of the system in the thermodynamic limit, that is, lim_Λ→ℤ^𝑑_.

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

Theorem 1.1 (see [30]). Let Λ be a finite domain in ℤ^𝑑. If Φ  satisfies  the following:(1)lim_|𝑥|→∞|∇Φ(𝑥)|=∞, (2)for some 𝑀, any 𝜕^𝛼Φ with |𝛼|=𝑀 is bounded on ℝ^Λ,(3)for |𝛼|≥1, |𝜕^𝛼Φ(𝑥)|≤𝐶_𝛼(1+|∇Φ(𝑥)|²)^1/2 for some 𝐶_𝛼>0, (4)𝐇𝐞𝐬𝐬Φ≥𝛿 for some 0<𝛿≤1,then for any 𝐶^∞-function 𝑔 satisfying 𝜕^𝛼𝑔≤𝐶_𝛼1+𝑍_Φ^𝑞𝛼, (1.8) where 𝑍_Φ=|∇Φ|2, (1.9)𝛼∈ℕ^|Λ| with some 𝐶_𝛼 and some 𝑞_𝛼>0, there exists a unique 𝐶^∞-function 𝑢 solution of 𝐴_Φ⁽⁰⁾𝑣=𝑔−𝑔_{𝐿2(𝜇Λ)},⟨𝑣⟩_𝐿²_(𝜇Λ)=0. (1.10)

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 𝐵^𝑘_Φ(ℝ^Λ) defined by 𝐵^𝑘_Φℝ^Λ=𝑢∈𝐿²ℝ^Λ∶𝑍^ℓ_Φ𝜕^𝛼𝑢∈𝐿²ℝ^Λ,∀ℓ+|𝛼|≤𝑘, (1.11) where 𝑍_Φ=|∇Φ|2. (1.12) These are subspaces of the well-known Sobolev spaces 𝑊^𝑘,2(ℝ^Λ),𝑘∈ℕ. By regularity arguments, one may prove that the solution of (1.10) belongs to each 𝑒^Φ/2𝐵^𝑘_Φ(ℝ^Λ) for all 𝑘∈ℕ.

2. The Analyticity of the Pressure

2.1. Preliminaries

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

Let Λ be a finite domain in ℤ^𝑑(𝑑≥1), and consider as above the Hamiltonian Φ of the phase space ℝ^Λ satisfying the assumptions of Theorem 1.1.

Let 𝑔 be a smooth function on ℝ^ℤ𝑑 satisfying 𝜕^𝛼∇𝑔≤𝐶_𝛼,∀𝛼∈ℕ^Λ,𝐇𝐞𝐬𝐬𝑔≤𝐶forsomepositiveconstant𝐶. (2.1)

LetΦ^𝑡_Λ(𝑥)=Φ(𝑥)−𝑡𝑔(𝑥), (2.2) where 𝑥=(𝑥_𝑖)_𝑖∈Λ, and 𝑡∈[0,∞) is a thermodynamic parameter.

The finite volume pressure is defined by 𝑃_Λ(𝑡)=1|Λ|log_ℝΛ𝑑𝑥𝑒^{−Φ𝑡Λ(𝑥)}. (2.3) Denote that𝑍=_ℝΛ𝑑𝑥𝑒^{−ΦΛ(𝑥)},𝑍_𝑡=_ℝΛ𝑑𝑥𝑒^{−Φ𝑡Λ(𝑥)},⟨⋅⟩_𝑡,Λ=∫⋅𝑑𝑥𝑒^{−Φ𝑡Λ(𝑥)}𝑍_𝑡. (2.4)

Because of the assumptions made on 𝑔, one may find 𝑇>0 such that, for all 𝑡∈[0,𝑇), Φ^𝑡_Λ(𝑥) satisfies all the assumptions of Theorem 1.1. Thus, each 𝑡∈[0,𝑇) is associated with a unique 𝐶^∞-solution 𝑓(𝑡) of the equation 𝐴_Φ⁽⁰⁾_𝑡Λ𝑓(𝑡)=𝑔−𝑔_𝐿2(𝜇),𝑓(𝑡)_𝐿2(𝜇)=0. (2.5) Hence,𝐴_Φ⁽¹⁾_𝑡Λ𝐯(𝑡)=∇𝑔, (2.6) where 𝐯(𝑡)=∇𝑓(𝑡). Notice that the map 𝑡⟼𝐯(𝑡) (2.7) is well defined [1] and that{𝐯(𝑡)∶𝑡∈[0,𝑇)} (2.8) is a family of smooth solutions on ℝ^Λ corresponding to the family of potentialΦ^𝑡_Λ∶𝑡∈[0,𝑇). (2.9) We proved in [1] that 𝐯 is a smooth function of 𝑡 by means of regularity arguments. The following proposition proved in [1] gives an exact formula for the 𝑛th-derivatives of the pressure.

Proposition 2.1 (see [1]). If 𝑃_Λ(𝑡)=1|Λ|ln_ℝΛ𝑑𝑥𝑒^{−Φ𝑡(𝑥)}, (2.10) where Φ^𝑡(𝑥)=Φ_Λ(𝑥)(𝑥),𝜕−𝑡𝑔_𝛼∇𝑔≤𝐶_𝛼,∀𝛼∈ℕ^|Λ|,𝐇𝐞𝐬𝐬𝑔≤𝐶forsomepositiveconstant𝐶, (2.11) and Φ_Λ(𝑥) satisfies the assumptions of Theorem 1.1,  then there exist 𝑇>0 such that, for all 𝑡∈[0,𝑇), the 𝑛th-derivative of the finite volume pressure 𝑃_Λ(𝑡) is given by the formula 𝑃_Λ^(𝑛)(𝑡)=(𝑛−1)!𝐴_𝑔^𝑛−1𝑔_𝑡,Λ|Λ|,𝑛≥1, (2.12) where 𝐴_𝑔ℎ∶=𝐴⁽¹⁾⁻¹_Φ𝑡Λ∇ℎ⋅∇𝑔. (2.13)

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 ⟨𝐴_𝑔^𝑛−1𝑔⟩_Λ,𝑡.

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 Φ for which the Helffer-Sjöstrand formula 𝐜𝐨𝐯𝑓,𝑔=𝐴⁽¹⁾⁻¹_Φ∇𝑓⋅∇𝑔_Λ (2.14) 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 𝑓 is an infinitely differentiable function defined on an open set 𝐷, then 𝑓 is real analytic if for every compact set 𝐾⊂𝐷 there exists a constant 𝐶 such that for every 𝑥∈𝐾 and every nonnegative integer 𝑛 the following estimate holds:𝜕^𝑛𝜕𝑥^𝑛𝑓(𝑥)≤𝐶^𝑛+1𝑛!. (3.1)

We propose to establish the above estimate for the 𝑛th-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 ^𝑛−1_𝑘=0𝑔^𝑘_Λ,𝑡𝐴_𝑔^{𝑛−𝑘−1}𝑔_Λ,𝑡=1𝑘!()!𝑔𝑛−1^𝑛Λ,𝑡. (3.2)

Proof. First observe that 𝑔^𝑝𝐴_𝑔ℎ_Λ,𝑡=𝑔^𝑝𝐴⁽¹⁾⁻¹_Φ𝑡∇ℎ⋅∇𝑔_Λ,𝑡=1𝐴𝑝+1^{(1)−1Φ𝑡}∇ℎ⋅∇𝑔^𝑝+1_Λ,𝑡=1𝑔𝑝+1𝐜𝐨𝐯^𝑝+1=1,ℎ𝑔𝑝+1^𝑝+1ℎ_Λ,𝑡−𝑔^𝑝+1_Λ,𝑡⟨ℎ⟩_Λ,𝑡,𝑝=0,1,…. (3.3) Setting 𝑘=𝑝+1,ℎ=𝐴_𝑔^{𝑛−𝑘−1}𝑔 (3.4) yields 𝑔^𝑘_Λ,𝑡𝐴_𝑔^{𝑛−𝑘−1}𝑔_Λ,𝑡=𝑔^𝑘𝐴_𝑔^{𝑛−𝑘−1}𝑔_Λ,𝑡−𝑘𝑔^𝑘−1𝐴_𝑔^𝑛−𝑘𝑔_Λ,𝑡. (3.5) Now dividing by 𝑘!, summing over 𝑘, and noticing that on the right-hand side one obtains a telescoping sum yield ^𝑛−1_𝑘=0𝑔^𝑘_Λ,𝑡𝐴_𝑔^{𝑛−𝑘−1}𝑔_Λ,𝑡=1𝑘!()!𝑔𝑛−1^𝑛Λ,𝑡. (3.6)

Next, we need the following lemma.

Lemma 3.2. Let {𝑎_𝑛} and {𝑏_𝑛} be two sequences of real numbers such that 𝑎₀=1,𝑏₀≤𝐶,𝑎_𝑛≤𝐶^𝑛,^𝑛_𝑘=0𝑎_𝑘𝑏_𝑛−𝑘≤(𝑛+1)𝐶^𝑛+1 (3.7) for some positive constant 𝐶. Then 𝑏_𝑛≤𝑛+1+^𝑛−1_𝑘=02^𝑘(𝑛−𝑘)𝐶^𝑛+1,∀𝑛≥1. (3.8)

Proof. Let {𝛼_𝑛} be the sequence defined recursively by 𝛼₀𝛼=1,_𝑛=(𝑛+1)+𝛼_𝑛−1+𝛼_𝑛−2+⋯+𝛼₁+𝛼₀if𝑛≥1. (3.9) It is clear that 𝛼_𝑛=𝑛+1+^𝑛−1_𝑘=02^𝑘(𝑛−𝑘),∀𝑛≥0. (3.10) We need to prove that 𝑏_𝑛≤𝛼_𝑛𝐶^𝑛+1,∀𝑛. (3.11) We have 𝑏₀≤𝐶=𝛼₀𝐶⁰⁺¹. (3.12)
By induction assume that the result is true if 𝑛 is replaced by 𝑛≤𝑛. We have ^𝑛+1_𝑘=0𝑎_𝑘𝑏_{𝑛+1−𝑘}≤(𝑛+2)𝐶^𝑛+2⟺𝑏_𝑛+1+^𝑛+1_𝑘=1𝑎_𝑘𝑏_{𝑛+1−𝑘}≤(𝑛+2)𝐶^𝑛+2⟹𝑏_𝑛+1−^𝑛+1_𝑘=1𝑎_𝑘𝑏_{𝑛+1−𝑘}≤(𝑛+2)𝐶^𝑛+2⟹𝑏_𝑛+1≤(𝑛+2)𝐶^𝑛+2+^𝑛+1_𝑘=1𝑎_𝑘𝑏_{𝑛+1−𝑘}⟹𝑏_𝑛+1≤(𝑛+2)𝐶^𝑛+2+^𝑛+1_𝑘=1𝐶^𝑘𝛼_{𝑛+1−𝑘}𝐶^{𝑛+2−𝑘}⟺𝑏_𝑛+1≤(𝑛+2)+^𝑛+1_𝑘=1𝛼_{𝑛+1−𝑘}𝐶^𝑛+2,𝑏_𝑛≤𝛼_𝑛+1𝐶^𝑛+2. (3.13)

Proposition 3.3. In addition to the assumptions of Proposition 2.1 assume that for 𝑡₀∈[0,𝑇) the thermodynamic limit of 𝑃_Λ(𝑡₀) exists, and 𝑔^𝑛_Λ,𝑡0𝐶≤𝑛!2^𝑛𝑢max_{𝑥1,𝑥2∈Λ}𝑑𝑥₁,𝑥₂,∀𝑛≥1, (3.14) where 𝐶 is a positive constant and 𝑢 is a positive real variable function satisfying 𝑢(𝑠)≤1if𝑠≥𝑀forsome𝑀>0. (3.15) Then the infinite volume pressure lim_Λ→ℤ^𝑑𝑃_Λ(𝑡) is analytic at 𝑡₀.

Proof. First choose Λ large enough so that 𝑢[max_{𝑥1,𝑥2∈Λ}𝑑(𝑥₁,𝑥₂)]≤1. We then have 𝑔^𝑛_Λ,𝑡0𝐶≤𝑛!2^𝑛. (3.16)
Let 𝑎_𝑛=𝑔^𝑛_Λ,𝑡0𝑛!,𝑏_𝑛=1|Λ|𝐴^𝑛_𝑔𝑔_Λ,𝑡0,𝐶₁=𝐶2. (3.17)
We have 𝑎₀=1,|𝑏₀|=(1/|Λ|)|⟨𝑔⟩_Λ,𝑡0|≤𝐶₁,|𝑎_𝑛|≤𝐶^𝑛₁, and by Proposition 3.1^𝑛_𝑘=0𝑎_𝑘𝑏_𝑛−𝑘=1|Λ|(𝑛+1)𝑎_𝑛+1, (3.18) from which we have ^𝑛_𝑘=0𝑎_𝑘𝑏_𝑛−𝑘≤(𝑛+1)𝐶₁^𝑛+1. (3.19) Now applying Lemma 3.2 we get 𝐴^𝑛_𝑔𝑔_Λ,𝑡0≤|Λ|𝛼_𝑛𝐶₁^𝑛+1,∀𝑛≥1, (3.20) where 𝛼_𝑛=𝑛+1+^𝑛_𝑘=02^𝑘(𝑛−𝑘). (3.21) Now using the fact that, for 𝑛≥2𝛼_𝑛−1=𝑛+^𝑛−1_𝑘=02^𝑘(𝑛−1−𝑘)=𝑛1+^𝑛−1_𝑘=02^𝑘(𝑛−1−𝑘)𝑛≤𝑛1+^𝑛−1_𝑘=02^𝑘=𝑛1+1−2^𝑛1−2=𝑛2^𝑛, (3.22) we have 𝐴_𝑔^𝑛−1𝑔_Λ,𝑡0≤|Λ|𝑛2^𝑛𝐶^𝑛₁𝑃,∀𝑛≥1,_Λ(𝑛)𝑡₀≤(𝑛−1)!𝑛2𝐶₁^𝑛=𝑛!𝐶^𝑛,∀𝑛≥1. (3.23) This shows that the Taylor series of the infinite volume pressure at 𝑡₀ has a nonvanishing radius of convergence.
Next, we propose to prove that the pressure is equal to its Taylor series in a neighborhood of 𝑡₀.
By (3.16), the power series ^∞_𝑛=0𝑔^𝑛_Λ,𝑡0𝑛!𝑡−𝑡₀^𝑛 (3.24) has nonvanishing radius of convergence 𝑅_Λ. Put 𝐺(𝑡)=^∞_𝑛=0𝑔^𝑛_Λ,𝑡0𝑛!𝑡−𝑡₀^𝑛,𝐴(𝑡)=^∞_𝑛=0𝐴^𝑛_𝑔𝑔_Λ,𝑡0𝑡−𝑡₀^𝑛. (3.25)
Inside the interval of convergence of 𝐺(𝑡), the convolution formula of Proposition 3.1 gives 𝐺(𝑡)𝐴(𝑡)=𝐺^′(𝑡). (3.26) This implies that 𝐴(𝑡)=𝐺^′(𝑡)𝐺(𝑡)if𝑡≠𝑡₀⟹𝐴(𝑡)𝑑𝑡=ln|𝐺(𝑡)|+const,if𝑡≠𝑡₀. (3.27) Equivalently ^∞_𝑛=0𝐴^𝑛_𝑔𝑔_Λ,𝑡0𝑛+1𝑡−𝑡₀^𝑛+1=ln^∞_𝑛=0𝑔^𝑛_Λ,𝑡0𝑛!𝑡−𝑡₀^𝑛if𝑡≠𝑡₀ (3.28) or ^∞_𝑛=1𝐴_𝑔^𝑛−1𝑔_Λ,𝑡0𝑛𝑡−𝑡₀^𝑛=ln^∞_𝑛=0𝑔^𝑛_Λ,𝑡0𝑛!𝑡−𝑡₀^𝑛if𝑡≠𝑡₀. (3.29) Thus ^∞_𝑛=1𝐴_𝑔^𝑛−1𝑔_Λ,𝑡0|Λ|𝑛𝑡−𝑡₀^𝑛=1|Λ|ln^∞_𝑛=0𝑔^𝑛_Λ,𝑡0𝑛!𝑡−𝑡₀^𝑛,if𝑡≠𝑡₀. (3.30) Now using Proposition 2.1, we obtain ^∞_𝑛=1𝑃_Λ^(𝑛)𝑡₀𝑛!𝑡−𝑡₀^𝑛=1|Λ|ln^∞_𝑛=0𝑔^𝑛_Λ,𝑡0𝑛!𝑡−𝑡₀^𝑛,if𝑡≠𝑡₀. (3.31) Now adding 𝑃_Λ(𝑡₀) on both sides of this above equality, we get ^∞_𝑛=0𝑃_Λ^(𝑛)𝑡₀𝑛!𝑡−𝑡₀^𝑛=1|Λ|ln^∞_𝑛=0𝑍_𝑡𝑔^𝑛_Λ,𝑡0𝑛!𝑡−𝑡₀^𝑛,if𝑡≠𝑡₀=1|Λ|ln^∞_𝑛=0ℝΛ𝑡−𝑡₀^𝑛𝑔𝑛!^𝑛𝑒^{−ΦΛ(𝑥)+𝑡0𝑔(𝑥)}𝑑𝑥,if𝑡≠𝑡₀. (3.32) Because 𝑔 has bounded derivatives, we have _ℝΛ^∞_𝑛=0𝑡−𝑡₀^𝑛𝑔𝑛!^𝑛𝑒^{−ΦΛ(𝑥)+𝑡0𝑔(𝑥)}𝑑𝑥=_ℝΛ𝑒^{|(𝑡−𝑡0)𝑔|−ΦΛ(𝑥)+𝑡0𝑔(𝑥)}𝑑𝑥<∞. (3.33) Thus by permuting sum and integral we obtain ^∞_𝑛=0𝑃_Λ^(𝑛)𝑡₀𝑛!𝑡−𝑡₀^𝑛=1|Λ|ln_ℝΛ𝑒^{−ΦΛ(𝑥)+𝑡𝑔(𝑥)}𝑑𝑥,if𝑡≠𝑡₀=𝑃_Λ(𝑡),if𝑡≠𝑡₀. (3.34)

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 𝑥_𝑖1,…,𝑥_𝑖𝑛^𝑇_Λ≤𝐶^𝑛−1(𝑛−1)!𝑢max_𝑥1,𝑥2𝑑𝑥₁,𝑥₂, (4.1)

where 𝑢 is a rapidly decreasing function independent of Λ. However, he only obtained infinite differentiability rather than analyticity. The obstacle from getting analyticity is that, when 𝑢 is rapidly or exponentially decaying, the bounds obtained increase too fast with 𝑛. 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 𝑛th moments of 𝑔=∑_𝑖∈Λ𝑥_𝑖 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 lim_{Λ→ℤ𝑑ℝΛ}𝑑𝑥𝑒^{−Φ𝑡𝑜(𝑥)} (4.2) exists, then the assumption 𝑔^𝑛_Λ,𝑡0𝐶≤𝑛!2^𝑛 (4.3) is equivalent to saying that the partition function 𝑍_𝑡,Λ=_ℝΛ𝑑𝑥𝑒^{−Φ𝑡(𝑥)} (4.4) is analytic at 𝑡₀. 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.