Entropy maximization in the force network ensemble for granular solids.

A long-standing issue in the area of granular media is the tail of the force distribution, in particular, whether this is exponential, Gaussian, or even some other form. Here we resolve the issue for the case of the force network ensemble in two dimensions. We demonstrate that conservation of the total area of a reciprocal tiling, a direct consequence of local force balance, is crucial for predicting the local stress distribution. Maximizing entropy while conserving the tiling area and total pressure leads to a distribution of local pressures with a generically Gaussian tail that is in excellent agreement with numerics, both with and without friction and for two different contact networks.

Tail of the contact force distribution in static granular materials.

We numerically study the distribution P(f) of contact forces in frictionless bead packs, by averaging over the ensemble of all possible force network configurations. We resort to umbrella sampling to resolve the asymptotic decay of P(f) for large f , and determine P(f) down to values of order 10{-45} for ordered and disordered systems in two (2D) and three dimensions (3D). Our findings unambiguously show that, in the ensemble approach, the force distributions decay much faster than exponentially: P(f) approximately exp(-cf(alpha)}) , with alpha approximately 2.0 for 2D systems, and alpha approximately 1.7 for 3D systems.