The full canonical ensemble of a granular system

There is now ample experimental and computational evidence that a well-defined and reproducible state can be achieved in a granular system under a repeated disturbance; e.g., if subjected to disturbance of amplitude A and frequency ω, a volume V(A,ω) is found which will be returned to if the system is subjected to A′, ω′ and then returned to A, ω. A microcanonical ensemble defines the entropy from volume V, equally the volume function W, just as E equals H in conventional statistical physics. A canonical version exists via a compactivity ∂V/∂S. Granular systems also have a distribution of intergranular forces generated by external forces or gravity. This paper shows that the idea that the configurations are determined by the Gibbsian formula exp(-W(∂S/∂V)) can be extended to the distribution of forces with a microcanonical condition . The canonical ensemble immediately gives the exponential distribution of intergranular forces, found experimentally.