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

Authors: Assane Lo
Journal: Advances in Mathematical Physics
 (2011)

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## 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.


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## Body

## 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 Cn-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 thenth-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 thenth-derivatives of the pressure by means of the Witten Laplacians [36] given by(1.1)WΦ(0)=(-Δ+|∇Φ|24-ΔΦ2),WΦ(1)=-Δ+|∇Φ|24-ΔΦ2+HessΦ.
These operators are in some sense deformations of the standard Laplace Beltrami operator. They are, respectively, equivalent to(1.2)AΦ(0):=-Δ+∇Φ⋅∇,AΦ(1):=-Δ+∇Φ⋅∇+HessΦ.
Indeed,
(1.3)WΦ(⋅)=e-Φ/2∘AΦ(⋅)∘eΦ/2,
and the map
(1.4)UΦ:L2(RΛ)⟶L2(RΛ,e-Φdx)u⟼eΦ/2u
is unitary. More precisely, we will use the formula(1.5)cov(g,h)=Z-1∫(AΦ(1)-1∇g⋅∇h)e-Φ(x)dx,
where (1.6)Z=∫e-Φ(x)dx.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 sitei of a crystal lattice Λ⊂ℤd and is described by a continuous real parameter xi∈ℝ. A particular configuration of the total system will be characterized by an element x=(xi)i∈Λ of the product space ℝΛ.TheΦ=ΦΛ will denote the Hamiltonian which assigns to each configuration x∈ℝΛ a potential energy Φ(x). The probability measure that describes the equilibrium of the system is then given by the Gibbs measure (1.7)dμΛ(x)=ZΛ-1e-Φ(x)dx.TheZΛ>0 is a normalization constant.We are eventually interested in the behavior of the system in the thermodynamic limit, that is,limΛ→ℤd.Assume thatΛ is finite, and consider a Hamiltonian Φ of the phase space ℝΛ satisfying the assumptions of the following theorem.Theorem 1.1 (see [30]).
LetΛ be a finite domain in ℤd.  If Φ  satisfies  the following:(1)
lim|x|→∞|∇Φ(x)|=∞,(2)
for someM, any ∂αΦ with |α|=M is bounded on ℝΛ,(3)
for|α|≥1, |∂αΦ(x)|≤Cα(1+|∇Φ(x)|2)1/2 for some Cα>0,(4)
HessΦ≥δ for some 0<δ≤1,then for any  C∞-function  g satisfying
(1.8)|∂αg|≤Cα(1+ZΦ)qα,
where
(1.9)ZΦ=|∇Φ|2,α∈ℕ|Λ| with some  Cα and some  qα>0, there exists a unique  C∞-function  u solution of
(1.10)AΦ(0)v=g-〈g〉L2(μΛ),〈v〉L2(μΛ)=0.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 BΦk(ℝΛ) defined by
(1.11)BΦk(RΛ)={u∈L2(RΛ):ZΦl∂αu∈L2(RΛ),∀l+|α|≤k},
where
(1.12)ZΦ=|∇Φ|2.
These are subspaces of the well-known Sobolev spaces Wk,2(ℝΛ),k∈ℕ. By regularity arguments, one may prove that the solution of (1.10) belongs to each eΦ/2BΦk(ℝΛ) for all k∈ℕ.


## 2. The Analyticity of the Pressure

### 2.1. Preliminaries

We first recall the context over which the formula for thenth-derivative of the pressure was derived in [1].LetΛ be a finite domain in ℤd(d≥1), and consider as above the Hamiltonian Φ of the phase space ℝΛ satisfying the assumptions of Theorem 1.1.Letg be a smooth function on ℝℤd satisfying (2.1)|∂α∇g|≤Cα,∀α∈NΛ,Hessg≤CforsomepositiveconstantC.Let(2.2)ΦΛt(x)=Φ(x)-tg(x),
where x=(xi)i∈Λ, and t∈[0,∞) is a thermodynamic parameter.The finite volume pressure is defined by(2.3)PΛ(t)=1|Λ|log⁡[∫RΛdxe-ΦΛt(x)].
Denote that(2.4)Z=∫RΛdxe-ΦΛ(x),Zt=∫RΛdxe-ΦΛt(x),〈⋅〉t,Λ=∫⋅dxe-ΦΛt(x)Zt.Because of the assumptions made ong, one may find T>0 such that, for all t∈[0,T), ΦΛt(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 (2.5)AΦΛt(0)f(t)=g-〈g〉L2(μ),〈f(t)〉L2(μ)=0.
Hence,(2.6)AΦΛt(1)v(t)=∇g,
where v(t)=∇f(t). Notice that the map (2.7)t⟼v(t)
is well defined [1] and that(2.8){v(t):t∈[0,T)}
is a family of smooth solutions on ℝΛ corresponding to the family of potential(2.9){ΦΛt:t∈[0,T)}.
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(2.10)PΛ(t)=1|Λ|ln⁡[∫RΛdxe-Φt(x)],
where
(2.11)Φt(x)=ΦΛ(x)-tg(x),|∂α∇g|≤Cα,∀α∈N|Λ|,Hessg≤CforsomepositiveconstantC,
and ΦΛ(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  PΛ(t) is given by the formula
(2.12)PΛ(n)(t)=(n-1)!〈Agn-1g〉t,Λ|Λ|,n≥1,
where
(2.13)Agh:=AΦΛt(1)-1∇h⋅∇g.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〈Agn-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
(2.14)cov(f,g)=〈AΦ(1)-1(∇f)⋅∇g〉Λ
holds. In [30], Johnsen proved that this formula remains valid for a wide class of none convex Hamiltonians.


## 2.1. Preliminaries

We first recall the context over which the formula for thenth-derivative of the pressure was derived in [1].LetΛ be a finite domain in ℤd(d≥1), and consider as above the Hamiltonian Φ of the phase space ℝΛ satisfying the assumptions of Theorem 1.1.Letg be a smooth function on ℝℤd satisfying (2.1)|∂α∇g|≤Cα,∀α∈NΛ,Hessg≤CforsomepositiveconstantC.Let(2.2)ΦΛt(x)=Φ(x)-tg(x),
where x=(xi)i∈Λ, and t∈[0,∞) is a thermodynamic parameter.The finite volume pressure is defined by(2.3)PΛ(t)=1|Λ|log⁡[∫RΛdxe-ΦΛt(x)].
Denote that(2.4)Z=∫RΛdxe-ΦΛ(x),Zt=∫RΛdxe-ΦΛt(x),〈⋅〉t,Λ=∫⋅dxe-ΦΛt(x)Zt.Because of the assumptions made ong, one may find T>0 such that, for all t∈[0,T), ΦΛt(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 (2.5)AΦΛt(0)f(t)=g-〈g〉L2(μ),〈f(t)〉L2(μ)=0.
Hence,(2.6)AΦΛt(1)v(t)=∇g,
where v(t)=∇f(t). Notice that the map (2.7)t⟼v(t)
is well defined [1] and that(2.8){v(t):t∈[0,T)}
is a family of smooth solutions on ℝΛ corresponding to the family of potential(2.9){ΦΛt:t∈[0,T)}.
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(2.10)PΛ(t)=1|Λ|ln⁡[∫RΛdxe-Φt(x)],
where
(2.11)Φt(x)=ΦΛ(x)-tg(x),|∂α∇g|≤Cα,∀α∈N|Λ|,Hessg≤CforsomepositiveconstantC,
and ΦΛ(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  PΛ(t) is given by the formula
(2.12)PΛ(n)(t)=(n-1)!〈Agn-1g〉t,Λ|Λ|,n≥1,
where
(2.13)Agh:=AΦΛt(1)-1∇h⋅∇g.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〈Agn-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
(2.14)cov(f,g)=〈AΦ(1)-1(∇f)⋅∇g〉Λ
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 iff 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:(3.1)|∂n∂xnf(x)|≤Cn+1n!.We propose to establish the above estimate for thenth-derivatives of the pressure. First we have the following convolution formula.Proposition 3.1.
Under the assumptions and notations of Proposition2.1, one has
(3.2)∑k=0n-1〈gk〉Λ,t〈Agn-k-1g〉Λ,tk!=1(n-1)!〈gn〉Λ,t.Proof.
First observe that(3.3)〈gpAgh〉Λ,t=〈gpAΦt(1)-1∇h⋅∇g〉Λ,t=1p+1〈AΦt(1)-1∇h⋅∇gp+1〉Λ,t=1p+1cov(gp+1,h)=1p+1[〈gp+1h〉Λ,t-〈gp+1〉Λ,t〈h〉Λ,t],p=0,1,….
Setting
(3.4)k=p+1,h=Agn-k-1g
yields
(3.5)〈gk〉Λ,t〈Agn-k-1g〉Λ,t=〈gkAgn-k-1g〉Λ,t-k〈gk-1Agn-kg〉Λ,t.
Now dividing by k!, summing over k, and noticing that on the right-hand side one obtains a telescoping sum yield
(3.6)∑k=0n-1〈gk〉Λ,t〈Agn-k-1g〉Λ,tk!=1(n-1)!〈gn〉Λ,t.Next, we need the following lemma.Lemma 3.2.
Let{an} and {bn} be two sequences of real numbers such that
(3.7)a0=1,|b0|≤C,|an|≤Cn,|∑k=0nakbn-k|≤(n+1)Cn+1
for some positive constant C. Then
(3.8)|bn|≤[n+1+∑k=0n-12k(n-k)]Cn+1,∀n≥1.Proof.
Let{αn} be the sequence defined recursively by
(3.9)α0=1,αn=(n+1)+αn-1+αn-2+⋯+α1+α0ifn≥1.
It is clear that
(3.10)αn=n+1+∑k=0n-12k(n-k),∀n≥0.
We need to prove that
(3.11)|bn|≤αnCn+1,∀n.
We have
(3.12)|b0|≤C=α0C0+1.
By induction assume that the result is true ifn is replaced by ñ≤n. We have
(3.13)|∑k=0n+1akbn+1-k|≤(n+2)Cn+2⟺|bn+1+∑k=1n+1akbn+1-k|≤(n+2)Cn+2⟹||bn+1|-|∑k=1n+1akbn+1-k||≤(n+2)Cn+2⟹|bn+1|≤(n+2)Cn+2+|∑k=1n+1akbn+1-k|⟹|bn+1|≤(n+2)Cn+2+|∑k=1n+1Ckαn+1-kCn+2-k|⟺|bn+1|≤[(n+2)+∑k=1n+1αn+1-k]Cn+2,|bn|≤αn+1Cn+2.Proposition 3.3.
In addition to the assumptions of Proposition2.1 assume that for t0∈[0,T) the thermodynamic limit of PΛ(t0) exists, and
(3.14)|〈gn〉Λ,t0|≤n!(C2)nu[max⁡x1,x2∈Λd(x1,x2)],∀n≥1,
where C is a positive constant and u is a positive real variable function satisfying
(3.15)u(s)≤1ifs≥MforsomeM>0.
Then the infinite volume pressure limΛ→ℤdPΛ(t) is analytic at t0.Proof.
First chooseΛ large enough so that u[maxx1,x2∈Λd(x1,x2)]≤1. We then have
(3.16)|〈gn〉Λ,t0|≤n!(C2)n.
Let(3.17)an=〈gn〉Λ,t0n!,bn=1|Λ|〈Agng〉Λ,t0,C1=C2.
We havea0=1,|b0|=(1/|Λ|)|〈g〉Λ,t0|≤C1,|an|≤C1n, and by Proposition 3.1(3.18)∑k=0nakbn-k=1|Λ|(n+1)an+1,
from which we have
(3.19)|∑k=0nakbn-k|≤(n+1)C1n+1.
Now applying Lemma 3.2 we get
(3.20)|〈Agng〉Λ,t0|≤|Λ|αnC1n+1,∀n≥1,
where
(3.21)αn=n+1+∑k=0n2k(n-k).
Now using the fact that, for n≥2(3.22)αn-1=n+∑k=0n-12k(n-1-k)=n[1+∑k=0n-12k(n-1-k)n]≤n(1+∑k=0n-12k)=n(1+1-2n1-2)=n2n,
we have
(3.23)|〈Agn-1g〉Λ,t0|≤|Λ|n2nC1n,∀n≥1,|PΛ(n)(t0)|≤(n-1)!n(2C1)n=n!Cn,∀n≥1.
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 oft0.
By (3.16), the power series
(3.24)∑n=0∞〈gn〉Λ,t0n!(t-t0)n
has nonvanishing radius of convergence RΛ. Put
(3.25)G(t)=∑n=0∞〈gn〉Λ,t0n!(t-t0)n,A(t)=∑n=0∞〈Agng〉Λ,t0(t-t0)n.
Inside the interval of convergence ofG(t), the convolution formula of Proposition 3.1 gives
(3.26)G(t)A(t)=G′(t).
This implies that
(3.27)A(t)=G′(t)G(t)ift≠t0⟹∫A(t)dt=ln⁡|G(t)|+const,ift≠t0.
Equivalently
(3.28)∑n=0∞〈Agng〉Λ,t0n+1(t-t0)n+1=ln⁡|∑n=0∞〈gn〉Λ,t0n!(t-t0)n|ift≠t0
or
(3.29)∑n=1∞〈Agn-1g〉Λ,t0n(t-t0)n=ln⁡|∑n=0∞〈gn〉Λ,t0n!(t-t0)n|ift≠t0.
Thus
(3.30)∑n=1∞〈Agn-1g〉Λ,t0|Λ|n(t-t0)n=1|Λ|ln⁡|∑n=0∞〈gn〉Λ,t0n!(t-t0)n|,ift≠t0.
Now using Proposition 2.1, we obtain
(3.31)∑n=1∞PΛ(n)(t0)n!(t-t0)n=1|Λ|ln⁡|∑n=0∞〈gn〉Λ,t0n!(t-t0)n|,ift≠t0.
Now adding PΛ(t0) on both sides of this above equality, we get
(3.32)∑n=0∞PΛ(n)(t0)n!(t-t0)n=1|Λ|ln⁡|∑n=0∞Zt〈gn〉Λ,t0n!(t-t0)n|,ift≠t0=1|Λ|ln⁡|∑n=0∞∫RΛ(t-t0)nn!gne-ΦΛ(x)+t0g(x)dx|,ift≠t0.
Because g has bounded derivatives, we have
(3.33)∫RΛ∑n=0∞|(t-t0)nn!gne-ΦΛ(x)+t0g(x)|dx=∫RΛe|(t-t0)g|-ΦΛ(x)+t0g(x)dx<∞.
Thus by permuting sum and integral we obtain
(3.34)∑n=0∞PΛ(n)(t0)n!(t-t0)n=1|Λ|ln⁡|∫RΛe-ΦΛ(x)+tg(x)dx|,ift≠t0=PΛ(t),ift≠t0.


## 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 (4.1)|〈xi1,…,xin〉ΛT|≤Cn-1(n-1)!u(max⁡x1,x2⁡d(x1,x2)),whereu 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 thenth moments of g=∑i∈Λxi 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 Theorem1.1, 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(4.2)lim⁡Λ→Zd∫RΛdxe-Φto(x)
exists, then the assumption (4.3)|〈gn〉Λ,t0|≤n!(C2)n
is equivalent to saying that the partition function (4.4)Zt,Λ=∫RΛdxe-Φt(x)
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.



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