[Paper Review] Cluster expansions and correlation functions

Daniel Ueltschi|ArXiv.org|Apr 1, 2003

Random Matrices and Applications7 references3 citations

TL;DR

This paper presents a unified cluster expansion framework applicable to both continuous and discrete systems, extending the Kotecký-Preiss criterion for convergence. It provides explicit expressions and estimates for correlation functions, validated in classical/quantum gases and lattice polymer models, establishing analyticity of thermodynamic pressure under a condition on activity and interaction strength.

ABSTRACT

A cluster expansion is proposed, that applies to both continuous and discrete systems. The assumption for its convergence involves an extension of the neat Kotecky-Preiss criterion. Expressions and estimates for correlation functions are also presented. The results are applied to systems of interacting classical and quantum particles, and to a lattice polymer model.

Motivation & Objective

• To develop a general cluster expansion method valid for both continuous and discrete systems in statistical mechanics.
• To extend the Kotecký-Preiss convergence criterion to cover continuous systems, including classical and quantum particle models.
• To derive explicit expressions and rigorous bounds for correlation functions using the cluster expansion framework.
• To demonstrate the method's applicability to classical and quantum gases and lattice polymer models.
• To establish analyticity of the thermodynamic pressure under a condition involving activity and interaction potential.

Proposed method

• The cluster expansion is formulated using a complex measure space $({ m d} u, { m d} u)$ and a symmetric function $\zeta(A, A')$ encoding pair interactions.
• A key convergence criterion is introduced: $ \int |\mu|({\rm d}A') |\zeta(A, A')| e^{a(A')} \leq a(A) $, with $ a(A) $ a non-negative function and $ \int e^{a(A)} |\mu|({\rm d}A) < \infty $.
• The partition function is expressed as $ Z = \exp\left\{ \sum_{n \geq 1} \int \varphi(A_1, \dots, A_n) \prod_{i=1}^n {\rm d}\mu(A_i) \right\} $, where $ \varphi $ is a combinatorial function over connected graphs.
• Correlation functions are derived via the cluster expansion, with bounds established using the same convergence criterion.
• The method is applied to quantum systems by mapping paths to polymers, using the Feynman-Kac representation and path integrals with $ \omega \in \Omega_{xx}^{\ell\beta} $.
• For the quantum gas, the interaction is encoded in $ \zeta({\boldsymbol{\omega}}, {\boldsymbol{\omega}}') = \exp\left\{ -\sum_{m,n} \int_0^\beta U(\omega(m\beta + t) - \omega'(n\beta + t)) {\rm d}t \right\} - 1 $.

Experimental results

Research questions

• RQ1 Can a single cluster expansion framework be constructed that unifies treatment of continuous and discrete systems in statistical mechanics?
• RQ2 Does the Kotecký-Preiss convergence criterion extend to continuous systems, such as quantum gases and classical particle systems?
• RQ3 What explicit expressions and bounds can be derived for correlation functions using the cluster expansion?
• RQ4 Under what conditions is the thermodynamic pressure analytic in the activity $ z $ and inverse temperature $ \beta $?
• RQ5 Can the method be applied to lattice polymer models and quantum many-body systems with non-trivial interactions?

Key findings

• The cluster expansion converges absolutely under the extended Kotecký-Preiss condition: $ \int |\mu|({\rm d}A') |\zeta(A, A')| e^{a(A')} \leq a(A) $, with $ \int e^{a(A)} |\mu|({\rm d}A) < \infty $.
• The partition function is expressed as $ Z = \exp\left\{ \sum_{n \geq 1} \int \varphi(A_1, \dots, A_n) \prod_{i=1}^n {\rm d}\mu(A_i) \right\} $, with $ \varphi $ defined via connected graphs.
• For the quantum gas, the criterion is satisfied if $ \frac{\beta}{(2\pi\beta)^{d/2}} \int U(x) {\rm d}x \sum_{\ell \geq 1} \ell^{-d/2} \leq -\log z $, ensuring analyticity of pressure.
• The method yields bounds on correlation functions, such as $ 1 + \sum_{n \geq 2} n \int |\mu|({\rm d}A_2) \cdots |\varphi(A_1, \dots, A_n)| \leq e^{a(A_1)} $.
• The framework applies to classical and quantum gases and lattice polymers, with explicit expressions derived for the interaction terms and measures.
• No condensation occurs in the range where the convergence criterion holds, implying analyticity of thermodynamic functions.