The relaxation rate of a stochastic spreading process in a closed ring

TL;DR

This paper investigates how the relaxation rate of a stochastic spreading process in a closed ring depends on bias, ring length, and weak links, revealing a transition to under-damped behavior and spectral properties.

Contribution

It provides a detailed analysis of the relaxation dynamics in a diffusive ring, including effects of bias, system size, and weak links, and discusses continuum limit subtleties.

Findings

Relaxation rate depends on affinity and ring length.

Under-damped relaxation occurs beyond a critical bias.

Weak links influence spectral properties and relaxation dynamics.

Abstract

The relaxation process of a diffusive ring becomes under-damped if the bias (so called affinity) exceeds a critical threshold value, aka delocalization transition. This is related to the spectral properties of the pertinent stochastic kernel. We find the dependence of the relaxation rate on the affinity and on the length of the ring. Additionally we study the implications of introducing a weak-link into the circuit, and illuminate some subtleties that arise while taking the continuum limit of the discrete model.