Revisiting Groeneveld's approach to the virial expansion

TL;DR

This paper extends Groeneveld's convergence criterion for the virial expansion to inhomogeneous systems, providing bounds for correlation functions using graph generating functionals and recurrence relations.

Contribution

It generalizes Groeneveld's criterion to inhomogeneous systems and introduces new recurrence relations for graph weights related to correlation functions.

Findings

Proves a generalized convergence criterion for the virial expansion.

Derives bounds for density expansions of correlation functions.

Develops recurrence relations based on graph weights and Kirkwood-Salsburg equations.

Abstract

A generalized version of Groeneveld's convergence criterion for the virial expansion and generating functionals for weighted $2$-connected graphs is proven. The criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions $\rho_s$ (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions. The proof is based on recurrence relations for graph weights related to the Kirkwood-Salsburg integral equation for correlation functions. The proof does not use an inversion of the density-activity expansion, however a Moebius inversion on the lattice of set partitions enters the derivation of the recurrence relations.