Effect of Excluded Volume and Anisotropy on Granular Statistics: 'Fermi Statistics' and Condensation

We explore the consequences of the excluded volume interaction of hard spheres at high densities and present a theory for excited granular materials. We first demonstrate that, in the presence of gravity, the granular density crosses over from Boltzmann to Fermi statistics, when temperature is decreased in the weak excitation limit. Comparisons of numerical simulations with our predictions concerning the scaling behavior of temperature with agitation frequency, gravity and particle-diameter show satisfying agreement. Next, within the framework of the Enskog theory of hard spheres, we interpret this crossover as a 'condensation' of hard spheres from the dilute gas-state to a high density solid-like state. In the high density, low temperature limit Enskog theory fails because it predicts densities larger than the closed packed density below a certain temperature. We show how to extend the range of applicability of the Enskoq theory to arbitrarily low temperatures by constructing a physical solution: all particles that are situated in regions with densities larger than a certain maximum density are assumed to be 'condensed.'