Multispecies Virial Expansions

TL;DR

This paper investigates the virial expansion for mixtures of infinitely many particle types, proving convergence in small densities and expressing coefficients via two-connected graphs, with applications in combinatorics and branching processes.

Contribution

It introduces a convergence proof for the virial expansion in multi-species mixtures and links virial coefficients to graph-theoretic structures, extending previous single-species results.

Findings

Virial expansion converges absolutely at small densities.

Virial coefficients are expressed through two-connected graphs.

Applications include counting coloured trees and analyzing branching processes.

Abstract

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange-Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs.