Structural analysis of high-dimensional basins of attraction

TL;DR

This paper introduces a Monte Carlo approach combined with the multistate Bennett acceptance-ratio method to efficiently compute volumes and structural properties of high-dimensional bodies, exemplified by analyzing basins of attraction in sphere packings.

Contribution

The paper presents a novel Monte Carlo method that accurately estimates high-dimensional volumes and structural features, improving upon existing techniques like thermodynamic integration.

Findings

Method yields results in excellent agreement with thermodynamic integration.

Provides direct estimates of statistical uncertainties.

Analyzes the impact of structural disorder on basins of attraction.

Abstract

We propose an efficient Monte Carlo method for the computation of the volumes of high-dimensional bodies with arbitrary shape. We start with a region of known volume within the interior of the manifold and then use the multistate Bennett acceptance-ratio method to compute the dimensionless free-energy difference between a series of equilibrium simulations performed within this object. The method produces results that are in excellent agreement with thermodynamic integration, as well as a direct estimate of the associated statistical uncertainties. The histogram method also allows us to directly obtain an estimate of the interior radial probability density profile, thus yielding useful insight into the structural properties of such a high-dimensional body. We illustrate the method by analyzing the effect of structural disorder on the basins of attraction of mechanically stable packings of soft repulsive spheres.