Gaussian processes in complex media: new vistas on anomalous diffusion

Di Tullio F., Paradisi P., Spigler R., Pagnini G.

time-subordinated process  Mathematical Physics  Anomalous diffusion  Biophysics  heterogeneity  General Physics and Astronomy  Materials Science (miscellaneous)  Heterogeneity  complex medium  Gaussian process  fractional diffusion  Fractional diffusion  Continuous time random walk  Generalized gray Brownian motion  continuous time random walk  anomalous diffusion  generalized gray Brownian motion  Complex medium  Time-subordinated process  Physical and Theoretical Chemistry 

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 modeling approaches are: the continuous time random walk, the generalized gray 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.

Source: Frontiers in Physics 7 (2019): 123-1–123-11. doi:10.3389/fphy.2019.00123

Publisher: Frontiers, Lausanne

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