Emergence of Gamma distributions in granular materials and packing models

Abstract

We study the distribution of volume fluctuations in experiments and numerical simulations concerning equal-sized sphere packings prepared with different techniques. We show that the distribution of the local volumes (Voronoï cells) and also the distributions of the global volumes (whole samples) follow remarkably well a shifted and rescaled Gamma distribution that we name a k-Gamma distribution. Such agreement is robust over a broad range of packing fractions and it is observed for several distinct systems. This distribution is characterized by the average packing fraction and a shape parameter "k" which is very sensitive to changes in the structural organization. A statistical mechanics approach predicts such k-Gamma distribution at statistical equilibrium and it links the parameter k with the number of elementary cells which are exchanging volume during the system preparation. The thermodynamical equivalent of k and its relation with the "granular temperature" are also discussed.