Cavity volume and free energy in many-body systems

TL;DR

This paper derives a universal expression for the free energy of many-body systems using cavity volume, connecting it to classical results and providing new insights and computational methods for hard sphere systems.

Contribution

It introduces a generalised cavity volume approach to derive an exact and universal equation of state applicable across densities and coordinate spaces.

Findings

Reclaims exact results for the classical Tonks gas.

Provides a novel derivation of Onsager's free energy.

Develops a local lattice ansatz for approximating cavity volume in hard sphere systems.

Abstract

Within this work we derive and analyse an expression for the free energy of a single-species system in the thermodynamic limit in terms of a generalised cavity volume, that is exact in general, and in principle applicable to systems across their entire range of density, as well as to particles within a general coordinate space. This provides a universal equation of state, and can thus relate the cavity volume to classical results, such as Mayer's cluster expansions. Through this we are able to provide some insight into the connections between cavity volume and free energy density, as well as their consequences. We use examples which permit explicit computations to further probe these results, reclaiming the exact results for a classical Tonks gas and providing a novel derivation of Onsager's free energy for a single species, isotropic system. Given the complexity of the problem we also provide a local lattice ansatz, exact in one dimension, with which we may approximate the cavity volume for hard sphere systems to provide an accurate equation of state in the cases of hard disks and spheres in both dilute regimes as well as beyond the freezing transition.