Stress fluctuations in granular force networks

The heterogeneous force networks in static granular media are distinguished from other network structures in that they must satisfy constraints of mechanical equilibrium on every vertex/grain. Here we study the statistics of ensembles of hyperstatic frictionless force networks, which are composed of more forces than can be determined uniquely from force balance. Hyperstatic force networks possess degrees of freedom that rearrange one balanced network into another. We calculate the equation of state and demonstrate that the number of rearrangements governs the macroscopic statistical properties of the ensemble, in particular the macroscopic fluctuations of stress, which scale with distance to the isostatic point. We then show that a maximum entropy postulate allows one to quantitatively capture many features of the microscopic statistics. All predictions are tested against highly accurate Monte Carlo numerical simulations employing umbrella sampling.