The distribution of forces in a granular system under external stress is a spinglass problem

Abstract

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 to A, ω. A microcanonical ensemble defines the entropy from volume V, where V equals 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 P(external) = ∑(forcemomentsingrains)/V via exp(−Π(∂S/∂P)). The canonical ensemble immediately gives the exponential distribution of intergranular forces, found experimentally. The distribution must depend on the configuration and any physical property will have a value averaged over configurations, i.e. will give rise to a spinglass problem.