Taming of free volume in statistical mechanics of the hard disks model

TL;DR

This paper develops an exact analytical framework for the free volume in the hard disk model, linking it to intersection areas of exclusion circles, and derives the equation of state across a broad density range.

Contribution

It introduces a novel exact analytical expression for free volume in the hard disk model using intersection areas, enabling precise calculation of thermodynamic properties.

Findings

Exact formulas for free volume in terms of intersection areas

Partition function factorizes into free volumes with two limiting forms

Accurate equation of state up to close packing

Abstract

We turn the long time puzzle of the free volume, known for its highly irregular form, into exact analytical formulae and develop statistical mechanics of the hard disk model. The free volume is exactly expressed in terms of the intersection areas of up to five exclusion circles, which can be computed analytically as functions of disk coordinates. In turn, the free volume determines the partition function and entropy. The partition function is shown to factorize into a product of free volumes and admits two exact limiting forms corresponding to gaslike and liquidlike regimes. From this construction, using Monte Carlo-generated disk coordinates, the entropy and pressure are obtained analytically and recover the known equation of state of hard disks in almost entire density range up to the close packing. At intermediate densities, the theory reveals a mixed liquid regime associated with defect formation preceding the hexagonal ordering. The intersection area of five disks emerges as a scalar measure of the local hexagonal order. The theory can be directly adopted for the hard sphere model.