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

The Edward's statistical mechanics of jammed sphere packings [Song et al., Nature (London) 453, 629 (2008)] is generalized to different systems: polydisperse sphere packings in three dimensions, and high-dimensional monodisperse sphere packings. The theory predicts the density of random close packing and random loose packing of polydisperse systems for a given distribution of particle size and describes packings for any interparticle friction coefficient. In the high-dimensional limit, an asymptotic solution of the self-consistent relation is obtained by saddle-point evaluation and checked numerically. The resulting random close packing density scaling is consistent with that of other approaches, such as replica theory and density-functional theory. The theory could serve as a starting point to solve more difficult problems: such as predicting the optimal density of non-spherical packings, and understanding the higher-order correlations present in amorphous jammed packings.