Bounded diffusing diffusivities: Brownian yet non-Gaussian diffusion

Abstract

Brownian yet non-Gaussian diffusion has been recently reported in a huge number of biological and soft matter systems. Meanwhile, an archetypal theoretical model called “diffusing diffusivities” is proposed to interpret it. Based on this spirit of diffusing diffusivities, we extend the original diffusing diffusivities (with the unbounded exponential distribution) to the case that the diffusivity is constructed by a bounded stochastic process, i.e., corresponding diffusivities (with certain upper and lower bounds) obeying bounded power-law distribution. We demonstrate that Brownian yet non-Gaussian diffusion can be reproduced by this bounded diffusing diffusivities, via numerical simulations and analytic derivations. Specifically, the short-time distribution of displacement, as the indicator of the Brownian yet non-Gaussian diffusion, is derived analytically by means of superstatistical approach. This short-time distribution is distinct from the Laplace distribution that appears in the original model. The long-time Gaussian displacement distribution is obtained by utilizing the subordination concept. The bounded diffusing diffusivity here may be beneficial to further understanding the diffusive process of particles in complex and inhomogeneous environments. Our work enriches the diffusing diffusivity family and sheds new light on the concept of the Brownian yet non-Gaussian diffusion under stochastic process.