Construction of a pathway map on a complicated energy landscape

TL;DR

This paper introduces a numerical method to construct comprehensive pathway maps on complex energy landscapes, aiding the understanding of transitions between stationary states in physical systems.

Contribution

The method enables systematic mapping of all stationary points and transition states from a single parent state on complex energy landscapes.

Findings

Successfully applied to liquid crystal model in a square box

Identified transition states and energy barriers

Mapped multiple stationary solutions and pathways

Abstract

How do we search for the entire family tree without unwanted random guesses, starting from a high-index and high-energy stationary state on the energy landscape? Here we introduce a general numerical method that constructs the pathway map clearly connecting all stationary points branched from a single parent state. The map guides our understanding of how a physical system moves on the energy landscape. In particular, the method allows us to identify the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau-de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.