  Granular matter is comprised of a large number of particles whose collective
behavior determines macroscopic properties such as flow and mechanical
strength. A comprehensive theory of the properties of granular matter,
therefore, requires a statistical framework. In molecular matter, equilibrium
statistical mechanics, which is founded on the principle of conservation of
energy, provides this framework. Grains, however, are small but macroscopic
objects whose interactions are dissipative since energy can be lost through
excitations of the internal degrees of freedom. In this work, we construct a
statistical framework for static, mechanically stable packings of grains, which
parallels that of equilibrium statistical mechanics but with conservation of
energy replaced by the conservation of a function related to the mechanical
stress tensor. Our analysis demonstrates the existence of a state function that
has all the attributes of entropy. In particular, maximizing this state
function leads to a well-defined granular temperature for these systems.
Predictions of the ensemble are verified against simulated packings of
frictionless, deformable disks. Our demonstration that a statistical ensemble
can be constructed through the identification of conserved quantities other
than energy is a new approach that is expected to open up avenues for
statistical descriptions of other non-equilibrium systems.
