Abstract
Superstatistical approaches have played a crucial role in the investigations of mixtures of Gaussian processes. Such approaches look to describe non-Gaussian diffusion emergence in single-particle tracking experiments realized in soft and biological matter. Currently, relevant progress in superstatistics of Gaussian diffusion processes has been investigated by applying χ2-gamma and χ2-gamma inverse superstatistics to systems of particles in a heterogeneous environment whose diffusivities are randomly distributed; such situations imply Brownian yet non-Gaussian diffusion. In this paper, we present how the log-normal superstatistics of diffusivities modify the density distribution function for two types of mixture of Brownian processes. Firstly, we investigate the time evolution of the ensemble of Brownian particles with random diffusivity through the analytical and simulated points of view. Furthermore, we analyzed approximations of the overall probability distribution for log-normal superstatistics of Brownian motion. Secondly, we propose two models for a mixture of scaled Brownian motion and to analyze the log-normal superstatistics associated with them, which admits an anomalous diffusion process. The results found in this work contribute to advances of non-Gaussian diffusion processes and superstatistical theory.
Highlights
According to the history of diffusion processes, Brownian motion was reported for the first time in 1828, during experiments with tiny particles contained in pollen grains, immersed in water [1].This investigation was realized by Robert Brown, and for that reason, it became known as Brownian motion (BM)
Motivated by large amount of research in theoretical and experiment systems with fluctuating diffusivity [20,28,38,39,40,41,42,43], we investigated the log-normal superstatistics of BM to describe Brownian yet non-Gaussian diffusion, and in the following we consider the log-normal superstatistics associated with scaled Brownian motions to include anomalous diffusion
Thereby, the results suggest that the log-normal superstatistics approach of a mixture of Gaussian process implies heavy-tailed distributions, which maintain one of the Brownian features, the linear time relation for mean square displacement (MSD)
Summary
According to the history of diffusion processes, Brownian motion was reported for the first time in 1828, during experiments with tiny particles contained in pollen grains, immersed in water [1]. Recent experimental insights on single-particle tracking, realized in a variety of animate and inanimate systems, have reported non-Gaussian shapes for distribution, which maintain the MSD growing linearly in time [15] This new diffusion process is called Brownian yet non-Gaussian diffusion [16,17]. Motivated by large amount of research in theoretical and experiment systems with fluctuating diffusivity [20,28,38,39,40,41,42,43], we investigated the log-normal superstatistics of BM to describe Brownian yet non-Gaussian diffusion, and in the following we consider the log-normal superstatistics associated with scaled Brownian motions to include anomalous diffusion. These models admit diffusion processes that are non-Gaussian and anomalous. We present the conclusions together with new directions (open problems) associated with the obtained results
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