Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach

TL;DR

This paper establishes a mathematical framework linking metastable continuous Markov processes with jump Markov models using quasi-stationary distributions, justifying transition state theory and Eyring-Kramers formula in kinetic modeling.

Contribution

It introduces a quasi-stationary distribution approach to connect continuous metastable processes with jump models, providing error quantification and theoretical justification for transition state methods.

Findings

Provides a framework to analyze exit events from metastable states.

Quantifies errors in jump model parametrization using Eyring-Kramers formula.

Justifies the use of transition state theory in kinetic Monte Carlo models.

Abstract

We are interested in the connection between a metastable continuous state space Markov process (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring-Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.