Local Moderate and Precise Large Deviations via Cluster Expansions

Abstract

We consider a system of classical particles confined in a box $\varLambda \subset {\mathbb {R}}^d$ with zero boundary conditions interacting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature–low density regime we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates comparing with the infinite volume versions of the above are also provided.

Appendices

Canonical Cluster Expansion

Here we give some details about the cluster expansion needed for the proof of (5.17). We follow the results from [14] to which we refer for a more detailed description.

An abstract polymer model $({\mathcal {V}},{\mathbb {G}}_{{\mathcal {V}}},\omega )$ consists of (i) a set of polymers ${\mathcal {V}}:=\{V_1,\ldots ,V_{|{\mathcal {V}}|}\}$, (ii) a binary symmetric relation $\sim$ of compatibility on ${\mathcal {V}}\times {\mathcal {V}}$ such that for all $i\ne j$, $V_i\sim V_j$ if and only if $V_i\cap V_j=\emptyset$, (iii) a graph ${\mathbb {G}}_{{\mathcal {V}}}\equiv (V({\mathbb {G}}_{{\mathcal {V}}}),E({\mathbb {G}}_{{\mathcal {V}}}))$ such that the vertex set $V({\mathbb {G}}_{{\mathcal {V}}})={\mathcal {V}}$ and an edge $\{i,j\}\in E({\mathbb {G}}_{{\mathcal {V}}})$ if and only if $V_i\not \sim V_j$ and (iv) a weight function $\omega :{\mathcal {V}}\rightarrow {\mathbb {C}}$. Then defining for all $N\in {\mathbb {N}}$

$$\begin{aligned} {\mathcal {V}}\equiv {\mathcal {V}}(N):=\{V\,:\,V\subset \{1,\ldots ,N\},\,|V|\ge 2\} \end{aligned}$$

(A.1)

and

$$\begin{aligned} \omega \equiv \omega _{\varLambda }(V):=\sum _{g\in {\mathcal {C}}_V}\int _{\varLambda ^{|V|}}\prod _{i\in V(g)}\frac{dq_i}{|\varLambda |}\prod _{\{i,j\}\in E(g)}f_{i,j} \end{aligned}$$

(A.2)

with $f_{i,j}:=e^{-\beta V(q_i,q_j)}-1$, we have that thanks to [14]

$$\begin{aligned} \frac{N}{|\varLambda |}\sum _{n\ge 1}\frac{1}{n+1}P_{N,|\varLambda |}(n)B_{\varLambda ,\beta }(n)=\frac{1}{|\varLambda |}\sum _{I\in {\mathcal {I}}}c_I\omega _{\varLambda }^I, \end{aligned}$$

(A.3)

where

$$\begin{aligned} c_I=\frac{1}{I!}\sum _{G\subset {\mathcal {G}}_I}(-1)^{|E(G)|}=\frac{1}{I!}\frac{\partial ^{\sum _{V}I(V)}\log Z_{{\mathcal {V}}(N),\omega _{\varLambda }}}{\partial ^{I(V_1)}\omega _{\varLambda }(V_1)\cdot \cdot \cdot \partial ^{I(V_n)}\omega _{\varLambda }(V_n)}\bigg |_{\omega _{\varLambda }(V)=0} \end{aligned}$$

(A.4)

with

$$\begin{aligned} Z_{{\mathcal {V}}(N),\omega _{\varLambda }}:=\sum _{\{V_1,\ldots ,V_n\}_{\sim }}\prod _{i=1}^n\omega _{\varLambda }(V_i)=\int _{\varLambda ^N}\prod _{i=1}^N\frac{dq_i}{|\varLambda |}\,e^{-\beta H^{{\mathbf {0}}}_{\varLambda }({\mathbf {q}})}. \end{aligned}$$

(A.5)

The second sum in (A.3) is over the set ${\mathcal {I}}$ of multi-indices $I:{\mathcal {V}}(N)\rightarrow \{0,1,\ldots \}$, $\omega _{\varLambda }^{I}=\prod _{V}\omega _{\varLambda }(V)^{I(V)}$, and, denoting $\mathrm {supp}I:=\{V\in {\mathcal {V}}(N)\,:\, I(V)>0\}$, ${\mathcal {G}}_{I}$ is the graph with $\sum _{V\in \mathrm {supp}I}I(V)$ vertices induced from ${\mathcal {G}}_{\mathrm {supp}I}\subset {\mathbb {G}}_{{\mathcal {V}}(N)}$ by replacing each vertex V by the complete graph on I(V) vertices.

Furthermore, the sum in (A.4) is over all connected graphs G of ${\mathcal {G}}_I$ spanning the whole set of vertices of ${\mathcal {G}}_I$ and $I!=\prod _{V\in \mathrm {supp}I}I(V)!$.

Denoting now with $[N]\equiv \{1,\ldots ,N\}$ and $A(I):=\bigcup _{V\in \mathrm {supp}I}V\subset [N]$, from (A.3) we have that

$$\begin{aligned} B_{\beta ,\varLambda }(n)=\frac{|\varLambda |^n}{n!}\sum _{I\,:\,A(I)=[n+1]}c_I\omega _{\varLambda }^I. \end{aligned}$$

(A.6)

We want to recall that $B_{\varLambda ,\beta }(n)$ is the finite volume version of the irreducible Mayer’s coefficient $\beta _n$ in the sense that

$$\begin{aligned} \lim _{\varLambda \rightarrow {\mathbb {R}}^d}B_{\varLambda ,\beta }(n)=\beta _n, \end{aligned}$$

(A.7)

where

$$\begin{aligned} \beta _n:=\frac{1}{n!}\sum _{\begin{array}{c} g\in {\mathcal {B}}_{n+1}\\ V(g)\ni 1 \end{array}}\int _{({\mathbb {R}}^d)^n}\prod _{\{i,j\}\in E(g)}(e^{-\beta V(x_i-x_j)}-1)dx_2\cdot \cdot \cdot dx_n,\,\,x_1\equiv 0. \end{aligned}$$

(A.8)

Here ${\mathcal {B}}_{n+1}$ is the set of the 2-connected graphs and, for all $g\in {\mathcal {B}}_{n+1}$, we denote with V(g) the set of its vertices.

Using this formalization, thanks to [15] and also assuming in order to simplify the calculation that

Assumption 3

$V:\mathbb {R}^d\rightarrow {\mathbb {R}}\cup \{\infty \}$ has compact support $R<l$, i.e.

$$\begin{aligned} V(x_i-x_j)=0\,\,\,\mathrm {if}\,\,\,|x_i-x_j|>R, \end{aligned}$$

(A.9)

for all $x_i,x_j\in {\mathbb {R}}^d$, we have:

Lemma 5

Let $\rho ^*_{\varLambda }$ as in (4.23) such that condition $(\star )$ holds and $\rho _0$ as in (4.21). It results:

$$\begin{aligned} \beta \left| f^{(m)}_{\beta }(\rho _0)-{\mathcal {F}}_{\varLambda ,\beta ,{\mathbf {0}}}^{(m)}(\rho ^*_{\varLambda })\right| \lesssim \frac{|\partial \varLambda |}{|\varLambda |}, \end{aligned}$$

(A.10)

for all $m\ge 0$.

Proof

Let us consider for first the case $m\ge 1$. We define

$$\begin{aligned} \omega ^{(m)}_{\varLambda }(V):=2\rho _{\varLambda }^{1-m}{{|V|+1}\atopwithdelims ()m}\sum _{g\in {\mathcal {C}}_{V}}\int _{\varLambda ^{|V|}}\prod _{i=1}^{|V|}\frac{dq_i}{|\varLambda |}\prod _{\{i,j\}\in E(g)} f_{i,j} \prod _{i=1}^{|V|}F_{q_i}(\epsilon ) \end{aligned}$$

(A.11)

where

$$\begin{aligned} F_{q}(\epsilon ):=(1-\epsilon ){\mathbf {1}}_{\{d(q,\varLambda ^c)< R|V|\}}+\epsilon \,{\mathbf {1}}_{\{d(q,\varLambda ^c)\ge R|V|\}}, \end{aligned}$$

(A.12)

Then, calling $n=|V|$ and using the same estimates of (4.18)–(4.20) of [15] we obtain

$$\begin{aligned} |\omega _{\varLambda }^{(m)}(V)|\le \rho _{\varLambda }^{1-m}\frac{2}{m!}\frac{e^{2\beta B}(n+1)^m}{n}\left[ e^{2\beta B}C(\beta ,R)\rho _{\varLambda }\right] ^{n-1} \end{aligned}$$

(A.13)

far from the boundary ($\epsilon =1$), and

$$\begin{aligned} |\omega _{\varLambda }^{(m)}(V)|\le \rho _{\varLambda }^{1-m}\frac{2}{m!}\frac{e^{2\beta B}(n+1)^m}{n}\left[ e^{2\beta B}C(\beta ,R)\rho _{\varLambda }\right] ^{n-1}\frac{dR}{l} \end{aligned}$$

(A.14)

near the boundary ($\epsilon =0$). Noting that $|\omega ^{(m)}_{\varLambda }(V)|$ is an upper bound for the $n^{th}$ term of $\mathcal {F}^{int,(m)}_{\varLambda ,\beta ,\mathbf {0}}$ (see (6.7) and (6.8)) and revisiting sections 4, 5 and 6 of [15] we have

$$\begin{aligned} \beta \bigg |f_{\beta }^{(m)}(\rho ^*_{\varLambda })-{\mathcal {F}}_{\varLambda ,\beta ,{\mathbf {0}}}^{(m)}(\rho ^*_{\varLambda })\bigg |\lesssim \frac{|\partial \varLambda |}{|\varLambda |}. \end{aligned}$$

(A.15)

The conclusion follows (also when $m=0$) from the fact that by construction $|\rho ^*_{\varLambda }-\rho _{0}|\lesssim |\partial \varLambda |/|\varLambda |$ (see also [3]) and from the exponential decay of $F_{\varLambda ,\beta ,N}(n)$ given in (2.25). Indeed from (2.10), (2.23) and (2.21)–(A.8) we have

$$\begin{aligned} \beta f_{\beta }^{(m)}(\rho _0)=\frac{d^m}{d\rho ^m}\rho (\log \rho -1)\bigg |_{\rho =\rho _0}+\sum _{n\ge 1}{n+1\atopwithdelims ()m}\rho _0^{n+1-m}\frac{\beta _n}{n+1}. \end{aligned}$$

(A.16)

The first term gives

$$\begin{aligned}&\frac{d^m}{d\rho ^m}\rho (\log \rho -1)\bigg |_{\rho =\rho _0}\nonumber \\&\quad =(-1)^m\frac{1}{\rho _0^{m-1}}=(-1)^m\left( \frac{1}{(\rho ^*_{\varLambda })^{m-1}}+\frac{\sum _{k=1}^{m-1}{m-1\atopwithdelims ()k}(\rho ^*_{\varLambda })^{m-k}(\rho _0-\rho ^*_{\varLambda })^k}{(\rho _0\rho ^*_{\varLambda })^{m-1}}\right) \qquad \end{aligned}$$

(A.17)

for all $m\ge 2$. In the above formula, the first term cancels with the corresponding in $f_{\beta }^{(m)}(\rho ^*_{\varLambda })$ while the second is bounded by $|\partial \varLambda |/|\varLambda |$. For the second term in (A.16) we have:

$$\begin{aligned} \begin{aligned}&\sum _{n\ge 1}{n+1\atopwithdelims ()m}\rho _0^{n+1-m}\frac{\beta _n}{n+1}=\sum _{n\ge 1}{n+1\atopwithdelims ()m}(\rho _0\pm \rho ^*_{\varLambda })^{n+1-m}\frac{\beta _n}{n+1} \\&\quad \,=\sum _{n\ge 1}{n+1\atopwithdelims ()m}\left[ \sum _{k=0}^{n+1-m}{n+1-m\atopwithdelims ()k}(\rho _0-\rho ^*_{\varLambda })^{n+1-m-k}(\rho ^*_{\varLambda })^k\right] \frac{\beta _n}{n+1} \\&\quad \,=\sum _{n\ge 1}{n+1\atopwithdelims ()m}(\rho ^*_{\varLambda })^{n+1-m}\frac{\beta _n}{n+1}+\sum _{n\ge 1}{n+1\atopwithdelims ()m}\sum _{k=0}^{n -m}\bigg [{n+1-m\atopwithdelims ()k} \\&\qquad \times (\rho _0-\rho ^*_{\varLambda })^{n+1-m-k}(\rho ^*_{\varLambda })^k\bigg ]\frac{\beta _n}{n+1}, \end{aligned} \end{aligned}$$

(A.18)

where

$$\begin{aligned} \begin{aligned}&\sum _{k=0}^{n-m}\left[ {n+1-m\atopwithdelims ()k}(\rho _0-\rho ^*_{\varLambda })^{n+1-m-k}(\rho ^*_{\varLambda })^k\right] \frac{\beta _n}{n+1} \\&\quad \le \frac{\rho ^*_{\varLambda }}{2^m}\frac{|\partial \varLambda |}{|\varLambda |}\sum _{n\ge 1}(n+1)^m\left( \frac{2}{\rho ^*_{\varLambda }}\right) ^n\left| \frac{\rho ^*_{\varLambda }}{n+1}\beta _n\right| \le \frac{\rho ^*_{\varLambda }}{2^m}\frac{|\partial \varLambda |}{|\varLambda |}\sum _{n\ge 1}\left( \frac{2}{\rho ^*_{\varLambda } e^c}\right) ^n\lesssim \frac{|\partial \varLambda |}{|\varLambda |}. \end{aligned}\nonumber \\ \end{aligned}$$

(A.19)

which concludes the proof. $\square$

Remark 6

From (4.21), i.e. the fact that $|\rho _0-{\bar{\rho }}_{\varLambda }|\lesssim |\partial \varLambda |/|\varLambda |$ with ${\bar{\rho }}_{\varLambda }$ given by (2.17), the previous Lemma is also valid if we consider ${\bar{\rho }}_{\varLambda }$ instead of $\rho ^*_{\varLambda }$.

Stirling’s Approximation

We recall Stirling’s formula: for $N\in {\mathbb {N}}$ large enough

$$\begin{aligned} \sqrt{2\pi N}\left( \frac{N}{e}\right) ^N\le N!\le e^{1/12N}\sqrt{2\pi N}\left( \frac{N}{e}\right) ^N. \end{aligned}$$

(B.1)

Using (2.6), (2.23) and (2.26), for $\rho _{\varLambda }=N/|\varLambda |\in (0,1)$ we have:

$$\begin{aligned} \beta [f_{\varLambda ,\beta ,{\mathbf {0}}}(N)-{\mathcal {F}}_{\varLambda ,\beta ,{\mathbf {0}}}(\rho _{\varLambda })]= & {} -\frac{1}{|\varLambda |}\log \frac{|\varLambda |^{\rho _{\varLambda }|\varLambda |}}{(\rho _{\varLambda }|\varLambda |)!}-\rho _{\varLambda }(\log \rho _{\varLambda }-1)\nonumber \\= & {} \frac{1}{|\varLambda |}\log \left[ (\rho _{\varLambda }|\varLambda |)!\left( \frac{e}{\rho _{\varLambda }|\varLambda |}\right) ^{\rho _{\varLambda }|\varLambda |}\right] \nonumber \\=: & {} \rho _{\varLambda }B_{|\varLambda |}(\rho _{\varLambda }|\varLambda |) =: S_{|\varLambda |}(\rho _{\varLambda }). \end{aligned}$$

(B.2)

Thus, from (B.1), we get

$$\begin{aligned} \frac{\log \sqrt{2\pi \rho _{\varLambda }|\varLambda |}}{|\varLambda |}\le S_{|\varLambda |}(\rho _{\varLambda })\le \frac{\log \sqrt{2\pi \rho _{\varLambda }|\varLambda |}}{|\varLambda |}+\frac{1}{12\rho _{\varLambda }|\varLambda |^2}. \end{aligned}$$

(B.3)

We can generalize the first quantity defined in (B.2) as follows

$$\begin{aligned} B(x)=\frac{1}{x}\log \left[ \varGamma (x+1)\left( \frac{x}{e}\right) ^{-x}\right] \end{aligned}$$

(B.4)

for all $x\in {\mathbb {R}}^+$ and where $\varGamma (\cdot )$ is the gamma function $\varGamma (x):=\int _{0}^{\infty }t^{x-1}e^{-t}dt$ with the following properties [9]:

$$\begin{aligned} \varGamma (N+1)=N! \end{aligned}$$

(B.5)

for all $N\in {\mathbb {N}}$,

$$\begin{aligned} \sqrt{\frac{2\pi }{x}}\left( \frac{x}{e}\right) ^{x}\le \varGamma (x)\le \sqrt{\frac{2\pi }{x}}\left( \frac{x}{e}\right) ^{x}e^{\frac{1}{12x}} \end{aligned}$$

(B.6)

for all $x\in {\mathbb {R}}^+$. Moreover,

$$\begin{aligned} \psi (x):=\frac{d}{dx}\log (\varGamma (x))=\frac{\varGamma '(x)}{\varGamma (x)} =\log x-\frac{1}{12x}-p(x), \end{aligned}$$

(B.7)

for all $x\in {\mathbb {R}}^+$ and where $0\le p(x)\le 1/12x^2$. Then we have:

$$\begin{aligned} \frac{d}{dx}B(x)= & {} \frac{\psi (x+1)-\log x}{x}-\frac{B(x)}{x}\nonumber \\= & {} \frac{1}{x}\left[ \log \left( 1+\frac{1}{x}\right) +\frac{1}{12(x+1)}+p(x+1)\right] -\frac{B(x)}{x}, \end{aligned}$$

(B.8)

so that, denoting with $B'(\rho _{\varLambda }|\varLambda |)=d/dx B(x)|_{x=\rho _{\varLambda }|\varLambda |}$ we have

$$\begin{aligned} S'_{|\varLambda |}(\rho _{\varLambda })&= B(\rho _{\varLambda }|\varLambda |)+\rho _{\varLambda }|\varLambda |B'(\rho _{\varLambda }|\varLambda |)=\frac{13\rho _{\varLambda }|\varLambda |+12}{12\rho _{\varLambda }|\varLambda |(\rho _{\varLambda }|\varLambda |+1)}\nonumber \\&\quad +\sum _{n\ge 2}\frac{(-1)^{n+1}}{n}\left( \frac{1}{\rho _{\varLambda }|\varLambda |}\right) ^n+p(\rho _{\varLambda }|\varLambda |+1)\,\lesssim \,\frac{1}{|\varLambda |}. \end{aligned}$$

(B.9)