Exploring the origins of the power-law properties of energy landscapes: An egg-box model

TL;DR

This paper introduces an egg-box model to investigate how Gaussian energy distributions in multidimensional energy landscapes can lead to power-law basin area distributions, revealing the importance of inhomogeneous basin arrangements.

Contribution

The study proposes a novel egg-box model to analyze the relationship between Gaussian energy distributions and power-law basin areas in energy landscapes.

Findings

Increasing Gaussian variance broadens basin area distribution.

Power-law basin areas do not naturally emerge from the model.

Inhomogeneous basin distribution is crucial for power-law behavior.

Abstract

Multidimensional potential energy landscapes (PELs) have a Gaussian distribution for the energies of the minima, but at the same time the distribution of the hyperareas for the basins of attraction surrounding the minima follows a power-law. To explore how both these features can simultaneously be true, we introduce an ``egg-box'' model. In these model landscapes, the Gaussian energy distribution is used as a starting point and we examine whether a power-law basin area distribution can arise as a natural consequence through the swallowing up of higher-energy minima by larger low-energy basins when the variance of this Gaussian is increased sufficiently. Although the basin area distribution is substantially broadened by this process,it is insufficient to generate power-laws, highlighting the role played by the inhomogeneous distribution of basins in configuration space for actual PELs.