Incorporation of the intensive and extensive entropy contributions in the disk intersection theory of a hard disk system

TL;DR

This paper develops a unified theory for the entropy of hard disk systems by incorporating both extensive and intensive contributions, providing exact formulas based on circle intersections that facilitate numerical simulations.

Contribution

It introduces a novel approach that unifies and explicitly calculates both entropy contributions in hard disk systems using circle intersection geometry.

Findings

Derived exact formulas for entropy contributions from circle intersections.

Enabled direct computation of entropy from disk coordinates in simulations.

Applicable to hard sphere systems without additional geometric constructions.

Abstract

The one-body free volume, which determines the entropy of a hard disk system, has extensive (cavity) and intensive (cell) contributions. So far these contributions have not been unified and considered separately. The presented theory incorporates both contributions, and their sum is shown to determine the free volume and partition function. The approach is based on multiple intersections of the circles concentric with the disks but of twice larger radius. The result is exact formulae for the extensive and intensive entropy contributions in terms of the intersections of just two, three, four, and five circles. The method has an important advantage for applications in numerical simulations: the formulae enable one to convert the disk coordinates into the entropy contribution directly without any additional geometric construction. The theory can be straightforwardly applied to a system of hard spheres.