On the Virial Series for a Gas of Particles with Uniformly Repulsive Pairwise Interaction

TL;DR

This paper derives and analyzes a formal power series solution for the pressure of a gas with uniformly repulsive interactions, providing bounds on the virial series convergence and comparing coefficient behaviors.

Contribution

It introduces a Hamilton-Jacobi framework for the virial series and establishes a lower bound on its convergence radius beyond classical methods.

Findings

Lower bound on virial series convergence radius.

Comparison of Mayer and virial coefficients across regimes.

Application of the Cauchy-Majorant method to the H-J equation.

Abstract

The pressure of a gas of particles with a uniformly repulsive pair interaction in a finite container is shown to satisfy (exactly as a formal object) a "viscous" Hamilton-Jacobi (H-J) equation whose solution in power series is recursively given by the variation of constants formula. We investigate the solution of the H-J and of its Legendre transform equation by the Cauchy-Majorant method and provide a lower bound to the radius of convergence on the virial series of the gas which goes beyond the threshold established by Lagrange's inversion formula. A comparison between the behavior of the Mayer and virial coefficients for different regimes of repulsion intensity is also provided.