Geometric and Topological Entropies of Sphere Packing

TL;DR

This paper develops a statistical mechanical framework for understanding the entropy of randomly packed spherical particles, incorporating geometric and topological contributions, applicable to colloids, granular, and glassy systems.

Contribution

It introduces a novel thermodynamic description of sphere packings that includes both geometric and topological entropy components, extending to non-jammed systems.

Findings

Calculated entropy contributions for 2D and 3D isostatic packings.

Extended the theory to floppy particle clusters.

Applicable to sticky colloids and non-jammed granular systems.

Abstract

We present a statistical mechanical description of randomly packed spherical particles, where the average coordination number is treated as a macroscopic thermodynamic variable. The overall packing entropy is shown to have two contributions: geometric, reflecting statistical weights of individual configurations, and topological, which corresponds to the number of topologically distinct states. Both of them are computed in the thermodynamic limit for isostatic packings in 2D and 3D, and the result is further expanded to the case of "floppy" particle clusters. The theory is directly applicable to sticky colloids, and in addition, generalizes concepts of granular and glassy configurational entropies for the case of non-jammed systems.