Role of Topology in Relaxation of One-Dimensional Stochastic Processes

TL;DR

This paper explores how topological properties, specifically a winding number, influence the relaxation dynamics of one-dimensional classical stochastic processes, revealing a link between topology and relaxation times.

Contribution

It introduces a topological invariant for stochastic processes and demonstrates its role in predicting spectral gaps and relaxation behavior, supported by numerical and experimental considerations.

Findings

Winding number predicts spectral gap and relaxation time.

Topological features influence transient behavior.

Numerical confirmation of system-size dependence.

Abstract

Stochastic processes are commonly used models to describe dynamics of a wide variety of nonequilibrium phenomena ranging from electrical transport to biological motion. The transition matrix describing a stochastic process can be regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems, the conservation of probability imposes additional constraints on the transition matrix, which can induce unique topological phenomena. Here, we reveal the role of topology in relaxation phenomena of classical stochastic processes. Specifically, we define a winding number that is related to topology of stochastic processes and show that it predicts the existence of a spectral gap that characterizes the relaxation time. Then, we numerically confirm that the winding number corresponds to the system-size dependence of the relaxation time and the characteristic transient behavior. One can experimentally realize such topological phenomena in magnetotactic bacteria and cell adhesions.