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

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 nonequilibrium initial ensembles) are also elucidated. These theoretical results are consistent with numerical simulations.