Convergence of Cluster and Virial expansions for Repulsive Classical Gases

Abstract

We study the convergence of cluster and virial expansions for systems of particles subject to positive two-body interactions. Our results strengthen and generalize existing lower bounds on the radii of convergence and on the value of the pressure. Our treatment of the cluster coefficients is based on expressing the truncated weights in terms of trees and partition schemes, and generalize to soft repulsions previous approaches for models with hard exclusions. Our main theorem holds in a very general framework that does not require translation invariance and is applicable to models in general measure spaces. Our virial results, stated only for homogeneous single-space systems, rely on an approach due to Ramawadh and Tate. The virial coefficients are computed using Lagrange inversion techniques but only at the level of formal power series, thereby yielding diagrammatic expressions in terms of trees, rather than the doubly connected diagrams traditionally used. We obtain a new criterion that strengthens, for repulsive interactions, the best criterion previously available (proposed by Groeneveld and proven by Ramawadh and Tate). We illustrate our results with a few applications showing noticeable improvements in the lower bound of convergence radii.