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

Source: [Duality Between Relaxation and First Passage in Reversible Markov Dynamics: Rugged Energy Landscapes Disentangled](<https://www.sota2.com/research/paper/68fa38ca1e209131ce8bd376>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T23:08:49.560Z

Authors: David Hartich, Aljaz Godec

Published: 2018-02-27T17:53:04.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1802.10046>)

[PDF](<https://arxiv.org/pdf/1802.10046.pdf>)