Ergodic property of random diffusivity system with trapping events

TL;DR

This paper investigates a complex Langevin system with random diffusivity and trapping events, deriving expressions for mean-squared displacements and demonstrating nonergodicity in such models, relevant to biological and active matter systems.

Contribution

It introduces a model combining random diffusivity with an $ ext{alpha}$-stable subordinator, providing analytical expressions for ergodicity breaking and displacement distributions.

Findings

The model exhibits nonergodicity regardless of diffusivity type.

Analytic formulas for ergodicity breaking parameter are derived.

The model captures the Brownian yet non-Gaussian phenomenon.

Abstract

Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous environment. This paper considers a Langevin system containing a random diffusivity and an $\alpha$-stable subordinator with $\alpha<1$. This model describes the particle's motion in complex media where both the long trapping events and random diffusivity exist. We derive the general expressions of ensemble- and time-averaged mean-squared displacements which only contain the values of the inverse subordinator and diffusivity. Further taking specific time-dependent diffusivity, we obtain the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement. The results imply the nonergodicity of the random diffusivity model for any kind of diffusivity, including the critical case where the model presenting normal diffusion.