Geometric and Topological Entropies of Sphere Packing

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 and weakly under-constrained packings in 2D and 3D. The theory generalizes concepts of granular and glassy configurational entropies for the case of non-jammed systems.