Langevin equation with fluctuating diffusivity: a two-state model

TL;DR

This paper introduces a two-state Langevin model with fluctuating diffusivity to explain anomalous diffusion phenomena observed in experiments, revealing transient subdiffusion, aging, and non-Gaussian behaviors through theoretical analysis.

Contribution

It develops a two-state renewal theory to analytically describe fluctuating diffusivity effects on diffusion, including transient behaviors and slow convergence to Gaussian distributions.

Findings

Ensemble-averaged MSD shows transient subdiffusion in non-equilibrium.

Time-averaged MSD exhibits normal diffusion with aging effects.

RSD of the time-averaged MSD reveals slow relaxation and long-time correlations.

Abstract

Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a non-equilibrium ensemble, the ensemble-averaged mean square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time, and converges to a Gaussian distribution in a long time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion, and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterize the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized, and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.