Basin-volume distributions in monodisperse particle packings -- the soul of memory

Mechanically stable packings of $N$ particles in $d$ dimensions lie at the minima of an $Nd$-dimensional potential energy landscape. Starting from random initial particle positions, the system can relax using gradient-based optimization until it arrives at one of the equilibrium states; all initial conditions that end at the same minimum belong to the same catchment basin. We measure the distribution of the catchment basin volumes for indistinguishable monodisperse soft spheres in both $d=2$ and $d=3$. Ordering the basins at each system size, $N$, according to their volume, $P_N(n)$, from the largest at $n=1$ to smaller at larger $n$, we find a very wide distribution of volumes which is similar in both dimensions: $P_N(n) \approx A_Nn^{-\alpha}$ with $\alpha \approx 1$ which, in our most favorable cases, extends over $7$ decades. We explore aspects of the connectivity of the basins, show that their structure is highly contorted, and demonstrate how these results may be used to understand the imprinting of memories in cyclic strain studies of solids.