  The problem of finding the most efficient way to pack spheres has an
illustrious history, dating back to the crystalline arrays conjectured by
Kepler and the random geometries explored by Bernal in the 60's. This problem
finds applications spanning from the mathematician's pencil, the processing of
granular materials, the jamming and glass transitions, all the way to fruit
packing in every grocery. There are presently numerous experiments showing that
the loosest way to pack spheres gives a density of ~55% (RLP) while filling all
the loose voids results in a maximum density of ~63-64% (RCP). While those
values seem robustly true, to this date there is no physical explanation or
theoretical prediction for them. Here we show that random packings of
monodisperse hard spheres in 3d can pack between the densities 4/(4 + 2 \sqrt
3) or 53.6% and 6/(6 + 2 \sqrt 3) or 63.4%, defining RLP and RCP, respectively.
The reason for these limits arises from a statistical picture of jammed states
in which the RCP can be interpreted as the ground state of the ensemble of
jammed matter with zero compactivity, while the RLP arises in the infinite
compactivity limit. We combine an extended statistical mechanics approach 'a la
Edwards' (where the role traditionally played by the energy and temperature in
thermal systems is substituted by the volume and compactivity) with a
constraint on mechanical stability imposed by the isostatic condition.
Ultimately, our results lead to a phase diagram that provides a unifying view
of the disordered hard sphere packing problem.
