Exponentially slow mixing and hitting times of rare events for a reaction--diffusion model

TL;DR

This paper studies the mixing and hitting times in a reaction-diffusion model, revealing exponential lower bounds for mixing times and exponential distribution convergence for hitting times of rare events, depending on the potential landscape.

Contribution

It provides new exponential bounds on mixing times and characterizes hitting times for rare events in a reaction-diffusion system with complex potential landscapes.

Findings

Mixing time has exponential lower bound when potential has multiple minima.

Hitting times of rare events converge to an exponential distribution with mean one.

Results depend on the shape of the potential in the hydrodynamic equation.

Abstract

We consider the superposition of symmetric simple exclusion dynamics speeded-up in time, with spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We show that the mixing time has an exponential lower bound in the system size if the potential of the hydrodynamic equation has more than two local minima. We also apply our estimates to show that the normalized hitting times of rare events converge to a mean one exponential random variable if the potential has a unique minimum.