Granular matter can be considered a non-equilibrium system, such that equilibrium statistics is insufficient to describe the dynamics. A phase transition occurs when granular materials are compressed such that a nonzero stress develops in response to a strain deformation. This transition, referred to as the jamming transition, occurs at a critical volume fraction, &phis;_c depending on friction and preparation protocol. Analysis of the jamming transition produces a phase diagram of jammed granular matter for identical spheres, characterized by the critical volume fraction, &phis;_c and the average coordination number, Z. The boundaries of the phase diagram are related to well-defined upper and lower limits in the density of disordered packings; random close packing (RCP) and random loose packing (RLP).

Frictional systems, such as granular matter, exhibit an inherent path dependency resulting in the loss of energy conservation, an important facet of equilibrium statistics. It has been suggested Edwards that the volume-force (V-F) ensemble, wherein volume replaces energy as the conservative quantity, may provide a sufficient framework to create a statistical ensemble for jammed granular matter. Treating a jammed system via the V-F ensemble introduces an analogue to temperature in equilibrium systems. This analogue, "compactivity", measures how compact a system could be and governs fluctuation in the volume statistics.

Randomness in statistical systems is typically characterized by entropy, the equation of state derived from the number of microstates available to the system. In equilibrium statistical mechanics, entropy provides the link between these microstates and the macroscopic thermodynamic properties of the system. Therefore, calculating the entropy within the V-F ensemble can relate the available microscopic volume for each grain to the macroscopic system properties.

The entropy is shown to be minimal at RCP and maximal at the minimum RLP limit, via several methods utilizing simulations and theoretical models. Within this framework RCP is achieved in the limit of minimal compactivity and RLP is achieved in the limit of maximal compactivity. The boundaries of a phase diagram for jammed matter could thereby be defined by the limits of zero and infinite compactivities, characterizing the RCP and RLP limits of granular matter.