Convergence of cluster and virial expansions for repulsive classical gases

Roberto Fernández, Nguyen Tong Xuan

Published 2019-09-29Version 1

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. For the virial coefficients we resort to an approach due to Ramawadth and Tate that uses Lagrange inversion techniques only at the level of formal power series and leads to 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 Groenveld and proven by Ramawadth and Tate). We illustrate our results with a few applications showing noticeable improvements in the lower bound of convergence radii.