Basins of Attraction in the Jamming Energy Landscape Have Power-Law Length Distributions

The energy landscape of harmonic sphere packings is a high-dimensional space with an astronomical number of local minima. This landscape is tesselated into basins of attraction, with each basin comprising the set of all configurations that fall into the same local minima under overdamped dynamics. The distribution of basin volumes gives the probability distribution of stable structures and allows for the calculation of the configurational entropy of the system. The high dimensionality, however, makes the energy landscape too vast to exhaustively explore. Further, the basins are not typically compact objects, rather, they have long skinny tendrils that hold much of the basin volume far from their minima. These features make statistical sampling of the energy landscape difficult. In this work, we employ a, “transect method,” whereby we measure the lengths of basins along a random line drawn through the energy landscape. We find that the basin lengths along a given line follow a power law distribution with a power that approaches one as we sample this line at ever smaller length scales. This distribution does not significantly change with packing fraction as we approach jamming from above. The existence of a power law result is consistent with the claim that this landscape is ultrametric. Thus, our results are further evidence that the energy landscape of sphere packings is fractally rough and in a Gardner phase for packing fractions above jamming.