Effective dynamics for a kinetic Monte-Carlo model with slow and fast time scales

TL;DR

This paper analyzes multiscale kinetic Monte Carlo models with fast and slow variables, deriving effective dynamics in the limit of large barriers or small time scale ratios, supported by numerical simulations.

Contribution

It introduces a rigorous derivation of effective dynamics for multiscale kinetic Monte Carlo models with slow and fast processes, including systems with energy exchange.

Findings

Effective dynamics between macro-states in high barrier limit

Convergence of processes to kinetic Monte Carlo models

Numerical validation of theoretical results

Abstract

We consider several multiscale-in-time kinetic Monte Carlo models, in which some variables evolve on a fast time scale, while the others evolve on a slow time scale. In the first two models we consider, a particle evolves in a one-dimensional potential energy landscape which has some small and some large barriers, the latter dividing the state space into metastable regions. In the limit of infinitely large barriers, we identify the effective dynamics between these macro-states, and prove the convergence of the process towards a kinetic Monte Carlo model. We next consider a third model, which consists of a system of two particles. The state of each particle evolves on a fast time-scale while conserving their respective energy. In addition, the particles can exchange energy on a slow time scale. Considering the energy of the first particle, we identify its effective dynamics in the limit of asymptotically small ratio between the characteristic times of the fast and the slow dynamics. For all models, our results are illustrated by representative numerical simulations.