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.
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