Accurate volume estimates for basins of attraction of jammed systems

The Edwards hypothesis contends that all jammed packings of a disordered granular system are equally probable to be observed. A direct test of this hypothesis requires calculating the probability of observing a specific jammed packing by some dynamical protocol (e.g., steepest descent). This probability is proportional to the volume of phase space that converges to the chosen packing, i.e., its basin of attraction. Basin volume calculations require an accurate map from an initial unstable configuration to its corresponding jammed state, obtained by solving the steepest descent ODE. This map was previously implemented using optimizers such as FIRE because of their speed but sacrificed the detailed underlying energy landscape picture. We show that broadly adopted optimizers fail to capture simple geometric features of the energy landscape. For instance, the radial probability of remaining in a basin around a random point follows a stretched exponential, yielding a length scale that is missed when using an optimizer. Finally, we explore how Monte Carlo estimates of basin volumes change based on the methods used and implications for previous computational tests of the Edwards hypothesis.