Force network ensemble for the triangular lattice: A tale of tiles

In granular materials, contact forces between neighboring grains are organized in disordered force networks. At present, it is unclear whether or not theories based on statistical mechanics can correctly predict the statistics of these force networks. The force network ensemble is a convenient model system for studying force networks. This ensemble comprises all sets of noncohesive contact forces on a fixed underlying contact network and a fixed load, i.e., global stress tensor, for which all grains are in static force and torque balance. Each force network is assigned an equal a priori probability. The corresponding phase space can be sampled using the Monte Carlo method. Figure 1 a shows a typical force network for the frictionless triangular lattice. Each force network has a complementary representation known as a reciprocal tiling Fig. 1 b in which each grain maps to a particular tile and each contact force to a tile face. The face is oriented at a /2 rotation to the contact force f with a length proportional to f . Due to local force balance, the tile faces form closed loops and, because forces come in action-reaction pairs, all tiles fit together without any gaps. The local pressure of a single grain is equal to its tile perimeter. One can show that fixing the global stress tensor  leads to a conservation of the total tile area A. In Ref. 2, we derived an analytical expression for the distribution of the local pressure on individual grains, by maximizing the entropy while conserving  and A. Figure 1 c shows that this theoretical prediction is in excellent agreement with the numerical result from the force network ensemble. Due to these constraints, large local stresses obey Gaussian statistics, in sharp contrast to the common belief that exponential force statistics are characteristic for granular materials. This observation is robust to changes in contact network including disordered networks and finite friction coefficient.