Ergodic and Nonergodic Anomalous Diffusion in Coupled Stochastic Processes

TL;DR

This paper investigates how coupling with unobserved processes affects the diffusion behavior in stochastic systems, revealing conditions for anomalous, ergodic, or non-ergodic diffusion, with theoretical and numerical validation.

Contribution

It introduces a model for coupled Langevin processes with state-dependent friction, analyzing their long-term diffusion properties and ergodicity, which was not previously understood.

Findings

Anomalous diffusion occurs due to coupling effects.

Diffusion exponent cannot be predicted by simple scaling.

Coupling can induce ergodic or non-ergodic behavior.

Abstract

Inspired by problems in biochemical kinetics, we study statistical properties of an overdamped Langevin process whose friction coefficient depends on the state of a similar, unobserved process. Integrating out the latter, we derive the long time behaviour of the mean square displacement. Anomalous diffusion is found. Since the diffusion exponent can not be predicted using a simple scaling argument, anomalous scaling appears as well. We also find that the coupling can lead to ergodic or non-ergodic behaviour of the studied process. We compare our theoretical predictions with numerical simulations and find an excellent agreement. The findings caution against treating biochemical systems coupled with unobserved dynamical degrees of freedom by means of standard, diffusive Langevin descriptions.