Abstract
Normal or Brownian diffusion is historically identified by the linear growth in time of the variance and by a Gaussian shape of the displacement distribution. Processes departing from the at least one of the above conditions defines anomalous diffusion, thus a nonlinear growth in time of the variance and/or a non-Gaussian displacement distribution. Motivated by the idea that anomalous diffusion emerges from standard diffusion when it occurs in a complex medium, we discuss a number of anomalous diffusion models for strongly heterogeneous systems. These models are based on Gaussian processes and characterized by a population of scales, population that takes into account the medium heterogeneity. In particular, we discuss diffusion processes whose probability density function solves space- and time-fractional diffusion equations through a proper population of time-scales or a proper population of length-scales. The considered modelling approaches are: the continuous time random walk, the generalized grey Brownian motion, and the time-subordinated process. The results show that the same fractional diffusion follows from different populations when different Gaussian processes are considered. The different populations have the common feature of a large spreading in the scale values, related to power-law decay in the distribution of population itself. This suggests the key role of medium properties, embodied in the population of scales, in the determination of the proper stochastic process underlying the given heterogeneous medium.
Highlights
Normal diffusion has been widely investigated by means of different modeling approaches, such as: conservation of mass, constitutive laws, random walks based on central limit theorem (CLT), stochastic models, i.e., Wiener process, Langevin equation, Fokker–Planck equation, and other Markovian Master equations [1,2,3]
In this paper we studied a framework for explaining the emergence of anomalous diffusion in media characterized by random structures
We considered three different modeling approaches based on Gaussian processes but displaying a population of scales
Summary
Normal diffusion has been widely investigated by means of different modeling approaches, such as: conservation of mass, constitutive laws, random walks based on central limit theorem (CLT), stochastic models, i.e., Wiener process, Langevin equation, Fokker–Planck equation, and other Markovian Master equations [1,2,3]. Both stochastic models, FBM and CTRW, do not describe the observed features of the SPT data As a consequence, this implies that the above two minimal models (FBM and CTRW) do not take into account some microscopic dynamics affecting the particle motion and determining the emergence of long-range correlations, anomalous diffusion, non-Gaussian power-law distributions, ergodicity breaking, and aging [38]. This implies that the above two minimal models (FBM and CTRW) do not take into account some microscopic dynamics affecting the particle motion and determining the emergence of long-range correlations, anomalous diffusion, non-Gaussian power-law distributions, ergodicity breaking, and aging [38] For this reason, the scientific community is focusing on the role of the system’s heterogeneity, which was at first neglected in the above mentioned modeling approaches.
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