Equation of state for hard sphere fluids with and without Kac tails

TL;DR

This paper presents a straightforward derivation of the equation of state for hard sphere fluids, including cases with Kac tails, and extends the approach to higher dimensions, emphasizing simplicity and avoiding complex mathematics.

Contribution

It introduces a simple, mathematically accessible derivation method for the hard sphere equation of state, applicable with and without Kac tails, and extends to higher dimensions.

Findings

Derivation of the 1D hard rod equation of state

Extension to higher-dimensional hard sphere fluids

Recovery of the virial form of the equation of state

Abstract

In this note, we propose a simple derivation of the one dimensional hard rod equation of state, with and without a Kac tail (appended long range and weak potential). The case of hard spheres in higher dimension is also addressed and it is shown there that our arguments --which avoid any mathematical complication-- allow to recover the virial form of the equation of state in a direct way.