Constructing and exploring wells of energy landscapes

TL;DR

This paper introduces a dynamic exploration method for analyzing energy landscape topography, defining wells of duration and altitude, and provides a constructive algorithm for their determination using set-valued analysis and viability theory.

Contribution

It offers a quantitative, dynamic approach to landscape analysis, introducing wells of duration and altitude, and presents a constructive algorithm for identifying these features.

Findings

Quantitative characterization of landscape topography via dynamic exploration.

An alternative to stochastic gradient dynamics for landscape exploration.

Constructive method for determining wells and inherent dynamic features.

Abstract

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.