Amorphous packings of hard spheres in large space dimension

TL;DR

This paper derives asymptotic expressions for the glass transition and random close packing densities of hard spheres in very large dimensions, using a diagrammatic expansion of the replicated free energy.

Contribution

It shows that the full diagrammatic expansion can replace the HNC approximation to analyze hard sphere packings in high dimensions.

Findings

Exact entropy of hard sphere liquid in large dimensions

Asymptotic expressions for glass transition density

Asymptotic expressions for random close packing density

Abstract

In a recent paper (cond-mat/0506445) we derived an expression for the replicated free energy of a liquid of hard spheres based on the HNC free energy functional. An approximate equation of state for the glass and an estimate of the random close packing density were obtained in d=3. Here we show that the HNC approximation is not needed: the same expression can be obtained from the full diagrammatic expansion of the replicated free energy. Then, we consider the asymptotics of this expression when the space dimension d is very large. In this limit, the entropy of the hard sphere liquid has been computed exactly. Using this solution, we derive asymptotic expressions for the glass transition density and for the random close packing density for hard spheres in large space dimension.