Turning intractable counting into sampling: computing the configurational entropy of three-dimensional jammed packings

TL;DR

This paper introduces an efficient numerical method to compute the configurational entropy of three-dimensional jammed sphere packings, revealing a strong correlation between pressure and basin volume in the energy landscape.

Contribution

The authors extend and improve existing methods to calculate the total number of jammed configurations and their entropy as a function of pressure in 3D sphere packings.

Findings

Strong correlation between pressure and basin volume

Power-law relation describes the basin volume-pressure relation

Method applicable to various high-dimensional enumeration problems

Abstract

We report a numerical calculation of the total number of disordered jammed configurations $\Omega$ of $N$ repulsive, three-dimensional spheres in a fixed volume $V$. To make these calculations tractable, we increase the computational efficiency of the approach of Xu et al. (Phys. Rev. Lett. 106, 245502 (2011)) and Asenjo et al. (Phys. Rev. Lett. 112, 098002 (2014)) and we extend the method to allow computation of the configurational entropy as a function of pressure. The approach that we use computes the configurational entropy by sampling the absolute volume of basins of attraction of the stable packings in the potential energy landscape. We find a surprisingly strong correlation between the pressure of a configuration and the volume of its basin of attraction in the potential energy landscape. This relation is well described by a power law. Our methodology to compute the number of minima in the potential energy landscape should be applicable to a wide range of other enumeration problems in statistical physics, string theory, cosmology and machine learning, that aim to find the distribution of the extrema of a scalar cost function that depends on many degrees of freedom.