Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities

Abstract

A growing number of biological, soft, and active matter systems are observed to exhibit normal diffusive dynamics with a linear growth of the mean-squared displacement, yet with a non-Gaussian distribution of increments. Based on the Chubinsky-Slater idea of a diffusing diffusivity, we here establish and analyze a minimal model framework of diffusion processes with fluctuating diffusivity. In particular, we demonstrate the equivalence of the diffusing diffusivity process with a superstatistical approach with a distribution of diffusivities, at times shorter than the diffusivity correlation time. At longer times, a crossover to a Gaussian distribution with an effective diffusivity emerges. Specifically, we establish a subordination picture of Brownian but non-Gaussian diffusion processes, which can be used for a wide class of diffusivity fluctuation statistics. Our results are shown to be in excellent agreement with simulations and numerical evaluations

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

“…Similarly, in simulations of crowded membranes it was shown that δ is between 1.3 and 1.6, and α below 1 [33] 4 . This behaviour can be explained by the fact that the considered medium is spatially or temporally heterogeneous [23,24,28,[35][36][37]. In such a system the thermal fluctuations are still Gaussian but the observed displacements are mixtures of these Gaussian contributions with random weights, effecting the non-Gaussian outcome.…”

“…X see [23][24][25][26][27] as well as the extensive list of references in [28]. These 'Brownian yet non-Gaussian' processes along with a more general class of non-Gaussian PDFs, discussed in more detail below, are in the focus of this study.…”

Escape rate and diffusion of a Stochastically Driven particle

“…An interesting feature of this model, which is also observed in a variety of soft-matter and biological systems92425, is the long coexistence of a van Hove distribution with non gaussian tails, and of a mean square displacement linear in time. An important open question ahead concerns the temporal evolution of the van Hove distribution in the underdamped limit, to rationalize how normal diffusion is recovered.…”

Codifference can detect ergodicity breaking and non-Gaussianity

“…In 2012 Chubynsky and Slater independently proposed a special case of the CIR process as a model of non-Gaussian diffusion [19,77]. This led the way to a wider range of models based on fluctuating diffusivity coefficient with a short time memory [78][79][80][81][82]. The evolution of the diffusion coefficient in the CIR model is defined by the stochastic equation , consequently D d 0 t > which causes the motion to stay positive.…”

“…The CIR process for ab  Î , can be proved to be a sum of squared independent Ornstein-Uhlenbeck processes, which follows directly from writing the stochastic differential equation of such a sum [83]. Thus, a natural generalisation is to consider D t being a square of a Gaussian process [80,81]. We will assume that the velocity can be decomposed as…”