Power-law distributions for the areas of the basins of attraction on a potential energy landscape

TL;DR

This paper introduces a method to analyze the distribution of basin areas in potential energy landscapes, revealing a power-law pattern that suggests a fractal-like, scale-free structure similar to Apollonian packings.

Contribution

The study demonstrates that basin areas follow a power-law distribution and draws an analogy between energy landscape division and hypersphere packings, highlighting a fractal-like structure.

Findings

Basin area distribution follows a power law over many decades.

Lower-energy minima tend to have larger basins of attraction.

The power-law exponent is similar to that of Apollonian packings.

Abstract

Energy landscape approaches have become increasingly popular for analysing a wide variety of chemical physics phenomena. Basic to many of these applications has been the inherent structure mapping, which divides up the potential energy landscape into basins of attraction surrounding the minima. Here, we probe the nature of this division by introducing a method to compute the basin area distribution and applying it to some archetypal supercooled liquids. We find that this probability distribution is a power law over a large number of decades with the lower-energy minima having larger basins of attraction. Interestingly, the exponent for this power law is approximately the same as that for a high-dimensional Apollonian packing, providing further support for the suggestion that there is a strong analogy between the way the energy landscape is divided into basins, and the way that space is packed in self-similar, space-filling hypersphere packings, such as the Apollonian packing. These results suggest that the basins of attraction provide a fractal-like tiling of the energy landscape, and that a scale-free pattern of connections between the minima is a general property of energy landscapes.