Mean field theory of hard sphere glasses and jamming

TL;DR

This paper develops a mean field theory for amorphous packings and glassy states of hard spheres using the replica method, providing predictions on structure and thermodynamics that align with simulations in dimensions two to six and exploring the large dimension limit.

Contribution

It introduces an improved replica-based theoretical framework for understanding amorphous hard sphere packings, including new results on correlation functions and contact force distributions in three dimensions.

Findings

Predictions match numerical simulations in 2-6 dimensions.

Exact solutions are obtained in the large dimension limit.

New insights into correlation functions and contact forces in 3D.

Abstract

Hard spheres are ubiquitous in condensed matter: they have been used as models for liquids, crystals, colloidal systems, granular systems, and powders. Packings of hard spheres are of even wider interest, as they are related to important problems in information theory, such as digitalization of signals, error correcting codes, and optimization problems. In three dimensions the densest packing of identical hard spheres has been proven to be the FCC lattice, and it is conjectured that the closest packing is ordered (a regular lattice, e.g, a crystal) in low enough dimension. Still, amorphous packings have attracted a lot of interest, because for polydisperse colloids and granular materials the crystalline state is not obtained in experiments for kinetic reasons. We review here a theory of amorphous packings, and more generally glassy states, of hard spheres that is based on the replica method: this theory gives predictions on the structure and thermodynamics of these states. In dimensions between two and six these predictions can be successfully compared with numerical simulations. We will also discuss the limit of large dimension where an exact solution is possible. Some of the results we present here have been already published, but others are original: in particular we improved the discussion of the large dimension limit and we obtained new results on the correlation function and the contact force distribution in three dimensions. We also try here to clarify the main assumptions that are beyond our theory and in particular the relation between our static computation and the dynamical procedures used to construct amorphous packings.