The statistics of frictional families

We develop a theoretical description for mechanically stable frictional packings in terms of the difference between the total number of contacts required for isostatic packings of frictionless disks and the number of contacts in frictional packings, $m=N_c^0-N_c$. The saddle order $m$ represents the number of unconstrained degrees of freedom that a static packing would possess if friction were removed. Using a novel numerical method that allows us to enumerate disk packings for each $m$, we show that the probability to obtain a packing with saddle order $m$ at a given static friction coefficient $\mu$, $P_m(\mu)$, can be expressed as a power-series in $\mu$. Using this form for $P_m(\mu)$, we quantitatively describe the dependence of the average contact number on friction coefficient for static disk packings obtained from direct simulations of the Cundall-Strack model for all $\mu$ and $N$.