Abstract cluster expansion with applications to statistical mechanical systems

Abstract

We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3124770͔ I. INTRODUCTIONThe method of cluster expansions was introduced in the 1930s in statistical mechanics in order to study gases of classical interacting particles. Its main achievement, from the point of view of physics, may be the derivation of the van der Waals equation of state for weakly interacting systems. The method was made rigorous by mathematical physicists in the 1960s, see Ref. 23 and references therein.The method split afterwards. One branch involves continuous systems, with applications to classical systems, 19,17,5 quantum systems, 10,11,21 or quantum field theory. 12,16,2,6 The other branch involves polymer systems, i.e., discrete systems with additional internal structure. 13,7,3,18,25,9,14 An important step forward was the article of Kotecký and Preiss with its simplified setting and its elegant condition for the convergence of the cluster expansion. 15 The methods for proving the convergence are diverse. Let us mention the study of KirkwoodSalsburg equations that involves correlation functions ͑see Ref. 23 and references therein͒, the algebraic approach of Ruelle, 23 combinatorial approaches using tree identities, 19,5,2,6 and inductions for discrete systems. 7,3,18 Important and useful surveys were written by Brydges, 4 Pfister, 20 and Abdesselam and Rivasseau. 1 Recent articles have been devoted to combinatorial aspects 25,8,14 and to weakening the assumptions. 9,8,22 The method of cluster expansions applies when the objects do not interact much; this is the case when they are far apart ͑low density͒ or when interactions are weak. An extension of the criterion of Ref. 15 that takes into account these two aspects was proposed in Ref. 26; it applies to both discrete and continuous systems.All abstract ͑i.e., general͒ approaches involve restrictions that correspond to repulsive interactions. Yet the old results for classical and quantum systems only assume stable interactions, that may include an attractive part. The aim of the present article is to propose a general approach that applies to discrete and continuous systems with repulsive or stable interactions. Our proof is split into several independent steps and this helps clarify the situation.The setting and the results are presented in Sec. II. We consider applications to classical systems of particles in Sec. III, to polymer systems in Sec. IV, and to the quantum gas in Sec. V. A fundamental tree estimate is derived in Sec. VI, and the theorems of Sec. II are proved in Sec. VII. II. CLUSTER EXPANSIONSWe consider a set X whose elements may represent widely different objects-in the three applications considered in this article, an element x X represents ͑i͒ the position of a classical particle, ͑ii͒ a polymer, i.e., a connected set of Z d , and ͑iii͒ a closed Brownian br...

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