Ideal mixture approximation of cluster size distributions at low density

TL;DR

This paper rigorously analyzes a low-density interacting particle system, demonstrating that it closely resembles an ideal mixture of clusters, with bounds on free energy and cluster distributions, including effects of mixing entropy.

Contribution

It extends previous work by incorporating mixing entropy into the ideal mixture approximation at low density, providing rigorous bounds for free energy and cluster distributions.

Findings

Bounds on constrained free energy for cluster size distributions

Bounds on free energy minimized over cluster distributions

Inclusion of mixing entropy improves approximation accuracy

Abstract

We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture. The present paper improves our earlier results by taking into account the mixing entropy.