Energy landscapes and rare events

TL;DR

The paper addresses how to identify transition pathways and rates for systems with complex, high-dimensional energy landscapes and multiscale dynamics. It develops and analyzes the string method (zero- and finite-temperature variants) to compute minimum energy paths (MEPs) between metastable states, enabling a self-consistent, coordinate-free reduction of dynamics to a Markov chain on metastable regions. A key theoretical component is the MEP framework and Theorem 1.1, which links the stationary path ensemble to a Boltzmann-weighted cross-section via $\varphi^0(\alpha)=\frac{1}{Z(\alpha)}\int_{S^0(\alpha)} x e^{-{V(x)}/{(k_B T)}} dx$, providing a practical route to transition rates and free-energy landscapes. The approach offers a robust computational tool for activated processes across physics, chemistry, and biology, improving efficiency and accuracy in sampling rare events in high-dimensional systems.

Abstract

Many problems in physics, material sciences, chemistry and biology can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. Well-known examples include phase transitions of condensed matter, conformational changes of biopolymers, and chemical reactions. The energy landscape typically exhibits multiscale features, giving rise to the multiscale nature of the dynamics. This is one of the main challenges that we face in computational science. In this report, we will review the recent work done by scientists from several disciplines on probing such energy landscapes. Of particular interest is the analysis and computation of transition pathways and transition rates between metastable states. We will then present the string method that has proven to be very effective for some truly complex systems in material science and chemistry.