Random very loose packings

We measure the number Omega(phi) of mechanically stable states of volume fraction phi of a granular assembly under gravity. The granular entropy S(phi)=log Omega(phi) vanishes both at high density, at phi similar or equal to phi(rcp), and a low density, at phi similar or equal to phi(rvlp), where phi(rvlp) is a new lower bound we call random very loose pack. phi(rlp) is the volume fraction where the entropy is maximal. These findings allow for a clear explanation of compaction experiments and provide the first first-principle definition of the random loose volume fraction. In the context of the statistical mechanics approach to static granular materials, states with phi <phi(rlp) are characterized by a negative temperature.