Viscoelastic subdiffusion in a random Gaussian environment

TL;DR

This study investigates viscoelastic subdiffusion in a Gaussian random environment, revealing how disorder and correlation types influence diffusion behavior, ergodicity, and residence time distributions in a one-dimensional model relevant to cellular environments.

Contribution

It provides new insights into how viscoelastic subdiffusion interacts with spatial disorder, highlighting the effects of correlation decay and disorder strength on diffusion dynamics and ergodicity.

Findings

Weak ergodicity breaking occurs on long transient times.

Strong disorder leads to Sinai-type logarithmic diffusion.

Residence times follow a generalized log-normal distribution.

Abstract

Viscoelastic subdiffusion governed by a fractional Langevin equation is studied numerically in a random Gaussian environment modeled by stationary Gaussian potentials with decaying spatial correlations. This anomalous diffusion is archetypal for living cells, where cytoplasm is known to be viscoelastic and a spatial disorder also naturally emerges. We obtain some first important insights into it within a model one-dimensional study. Two basic types of potential correlations are studied: short-range exponentially decaying and algebraically slow decaying with an infinite correlation length, both for a moderate (several $k_BT$, in the units of thermal energy), and strong (5-10 $k_BT$) disorder. For a moderate disorder, it is shown that on the ensemble level viscoelastic subdiffusion can easily overcome the medium's disorder. Asymptotically, it is not distinguishable from the disorder-free subdiffusion. However, a strong scatter in single-trajectory averages is nevertheless seen even for a moderate disorder. It features a weak ergodicity breaking, which occurs on a very long yet transient time scale. Furthermore, for a strong disorder, a very long transient regime of logarithmic, Sinai-type diffusion emerges. It can last longer and be faster in the absolute terms for weakly decaying correlations as compare with the short-range correlations. Residence time distributions in a finite spatial domain are of a generalized log-normal type and are reminiscent also of a stretched exponential distribution. They can be easily confused for power-law distributions in view of the observed weak ergodicity breaking. This suggests a revision of some experimental data and their interpretation.