Application of Edwards' statistical mechanics to high dimensional jammed sphere packings

TL;DR

This paper extends Edwards' statistical mechanics framework to high-dimensional jammed sphere packings, deriving a scaling law for packing density and validating it through simulations across dimensions.

Contribution

It generalizes Edwards' approach to arbitrary dimensions and provides a high-dimensional scaling law for packing density, supported by numerical validation.

Findings

The packing density scales as ^{-d} in high dimensions.

Numerical simulations from 3D to 6D support the theoretical predictions.

The approach improves in accuracy with increasing dimension.

Abstract

The isostatic jamming limit of frictionless spherical particles from Edwards' statistical mechanics [Song \emph{et al.}, Nature (London) {\bf 453}, 629 (2008)] is generalized to arbitrary dimension $d$ using a liquid-state description. The asymptotic high-dimensional behavior of the self-consistent relation is obtained by saddle-point evaluation and checked numerically. The resulting random close packing density scaling $\phi\sim d\,2^{-d}$ is consistent with that of other approaches, such as replica theory and density functional theory. The validity of various structural approximations is assessed by comparing with three- to six-dimensional isostatic packings obtained from simulations. These numerical results support a growing accuracy of the theoretical approach with dimension. The approach could thus serve as a starting point to obtain a geometrical understanding of the higher-order correlations present in jammed packings.