Duality Between Relaxation and First Passage in Reversible Markov Dynamics: Rugged Energy Landscapes Disentangled

TL;DR

This paper establishes a duality between relaxation and first passage times in reversible Markov processes, providing insights into kinetics in complex energy landscapes relevant to biological and condensed matter systems.

Contribution

It proves a spectral duality linking relaxation and first passage processes and applies this to rugged energy landscapes, enhancing understanding of reaction trajectories.

Findings

Spectral duality between relaxation and first passage times.

Application to rugged energy landscapes and protein misfolding.

Enhanced understanding of reactive trajectory statistics.

Abstract

Relaxation and first passage processes are the pillars of kinetics in condensed matter, polymeric and single-molecule systems. Yet, an explicit connection between relaxation and first passage time-scales so far remained elusive. Here we prove a duality between them in the form of an interlacing of spectra. In the basic form the duality holds for reversible Markov processes to effectively one-dimensional targets. The exploration of a triple-well potential is analyzed to demonstrate how the duality allows for an intuitive understanding of first passage trajectories in terms of relaxational eigenmodes. More generally, we provide a comprehensive explanation of the full statistics of reactive trajectories in rugged potentials, incl. the so-called `few-encounter limit'. Our results are required for explaining quantitatively the occurrence of diseases triggered by protein misfolding.