Basin volumes have a Zipf distribution in a high-dimensional energy landscape

We study the energy landscape of disordered three-dimensional jammed packings of N repulsive soft spheres. Each mechanically stable configuration corresponds to a local minimum of an energy basin. We measure a basin’s volume as the probability of reaching that basin by sampling from random initial positions. In contrast to many previous studies, we use monodisperse particles, which avoids the need to distinguish between the N! permutations of the particles. This allows us to visit the same configurations multiple times providing a determination of their volumes even for large N. When ranking the basins by decreasing volume, we find V(n) ∝ 1/n, where n is the rank and V(n) is the volume of the n^th largest basin. This is an example of a Zipf distribution. For N=67, this scaling covers more than 97% of the total volume of the energy landscape. Even for N=151, the power-law regime accounts for at least 12% of the total volume, limited only by computational sampling. Individual basins have highly irregular shapes. This distribution shows that the probability of finding a basin between V and V+dV is uniform on a log V axis and influences how jammed systems store memory and respond to perturbations.