# Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation

## TL;DR

This paper develops a theoretical framework for Brownian motion with fluctuating diffusivity, revealing different relaxation behaviors such as stretched-exponential and power-law decay depending on the timescale and initial conditions.

## Contribution

It introduces a novel theory of alternating renewal processes to analyze relaxation functions in Brownian motion with fluctuating diffusivity, covering exponential and power-law sojourn-time distributions.

## Key findings

- Short-time relaxation shows exponential or stretched-exponential behavior.
- Long-time relaxation exhibits power-law decay with an exponential cutoff.
- Results are consistent with numerical simulations.

## Abstract

We investigate Brownian motion with diffusivity alternately fluctuating between fast and slow states. We assume that sojourn-time distributions of these two states are given by exponential or power-law distributions. We develop a theory of alternating renewal processes to study a relaxation function which is expressed with an integral of the diffusivity over time. This relaxation function can be related to a position correlation function if the particle is in a harmonic potential, and to the self-intermediate scattering function if the potential force is absent. It is theoretically shown that, at short times, the exponential relaxation or the stretched-exponential relaxation are observed depending on the power law index of the sojourn-time distributions. In contrast, at long times, a power law decay with an exponential cutoff is observed. The dependencies on the initial ensembles (i.e., equilibrium or non-equilibrium initial ensembles) are also elucidated. These theoretical results are consistent with numerical simulations.