Virial Expansion Bounds

TL;DR

This paper reviews and generalizes cluster expansion bounds used to analyze the virial expansion in statistical mechanics, highlighting improvements for positive potentials with hard cores.

Contribution

It extends classical bounds to more recent cluster expansion techniques and optimizes them using Lambert W-function expressions.

Findings

Improved bounds on virial expansion convergence for positive potentials.

Generalization of classical cluster expansion bounds.

Optimization of bounds using Lambert W-function.

Abstract

In the 1960s, the technique of using cluster expansion bounds in order to achieve bounds on the virial expansion was developed by Lebowitz and Penrose (1964) and Ruelle (1969). This technique is generalised to more recent cluster expansion bounds by Poghosyan and Ueltschi (2009), which are related to the work of Procacci (2007) and the tree-graph identity, detailed by Brydges (1986). The bounds achieved by Lebowitz and Penrose can also be sharpened by doing the actual optimisation and achieving expressions in terms of the Lambert W-function. The different bound from the cluster expansion shows some improvements for bounds on the convergence of the virial expansion in the case of positive potentials, which are allowed to have a hard core.