  Randomly packing spheres of equal size into a container consistently results
in a static configuration with a density of ~64%. The ubiquity of random close
packing (RCP) rather than the optimal crystalline array at 74% begs the
question of the physical law behind this empirically deduced state. Indeed,
there is no signature of any macroscopic quantity with a discontinuity
associated with the observed packing limit. Here we show that RCP can be
interpreted as a manifestation of a thermodynamic singularity, which defines it
as the "freezing point" in a first-order phase transition between ordered and
disordered packing phases. Despite the athermal nature of granular matter, we
show the thermodynamic character of the transition in that it is accompanied by
sharp discontinuities in volume and entropy. This occurs at a critical
compactivity, which is the intensive variable that plays the role of
temperature in granular matter. Our results predict the experimental conditions
necessary for the formation of a jammed crystal by calculating an analogue of
the "entropy of fusion". This approach is useful since it maps
out-of-equilibrium problems in complex systems onto simpler established
frameworks in statistical mechanics.
