Mean-field theory of hard sphere glasses and jamming

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. There remain many weak points in our theory that should be better investigated. We hope that this paper can be useful to present the state of the art of the method and to stimulate new research in this field

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

“…Here, amorphous jammed packings are seen as infinite pressure glassy states [38,41]. Therefore, the properties of the jamming transition are intimately related to those of the glass transition [38].…”

“…Thus, by defining the upper bound at the frictionless isostatic limit we also exclude from the ensemble the partially crystalline packings. This is an important point, akin to mathematical tricks employed in replica approaches to glasses [38].…”

“…The jammed state is modelled as granular media composed of soft particles, typically Hertz-Mindlin [32,35,36] grains under external pressure and emulsions compressed under osmotic pressure [37], as they approach the jamming transition from the solid phase above the critical density: φ → φ + c . (c) Hard-sphere glasses: In parallel to studies in the field of jamming there exists a community attempting to understand the packing problem approaching jamming from the liquid phase [38,39,40], that is, φ → φ − c . Here, amorphous jammed packings are seen as infinite pressure glassy states [38,41].…”

“…(c) Hard-sphere glasses: In parallel to studies in the field of jamming there exists a community attempting to understand the packing problem approaching jamming from the liquid phase [38,39,40], that is, φ → φ − c . Here, amorphous jammed packings are seen as infinite pressure glassy states [38,41]. Therefore, the properties of the jamming transition are intimately related to those of the glass transition [38].…”

Structural Properties of Dense Hard Sphere Packings

“…Packings of HS have also been used in solving important problems of information and optimization theories [1,2]. In this Letter we focus on the structural properties of dense three dimensional (3D) HS systems at different packing fractions φ = π 6 ρσ 3 , where ρ = N/V is the density of N hard spheres in a system volume V and σ is the diameter of the spheres.…”