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Original Research ARTICLE Provisionally accepted The full-text will be published soon. Notify me

Front. Phys. | doi: 10.3389/fphy.2019.00123

Gaussian processes in complex media: new vistas on anomalous diffusion

  • 1ISTI - CNR, Istituto di scienza e tecnologie dell'informazione 'Alessandro Faedo' (ISTI), Italy
  • 2BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque Country – Spain, Basque Center for Applied Mathematics, Spain
  • 3Department of Mathematics and Physics, Roma Tre University, Italy
  • 4Ikerbasque – Basque Foundation for Science, Calle de Marı́a Dı́az de Haro 3, 48013 Bilbao, Basque Country – Spain, IKERBASQUE Basque Foundation for Science, Spain

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.

Keywords: anomalous diffusion, Continuous time random walk, Fractional diffusion, Complex medium, Gaussian process, population of scales, heterogeneity, generalized grey Brownian motion, time-subordinated process

Received: 01 Jun 2019; Accepted: 14 Aug 2019.

Copyright: © 2019 Paradisi, Di Tullio, Spigler and Pagnini. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: PhD. Gianni Pagnini, Basque Center for Applied Mathematics, BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque Country – Spain, Bilbao, Basque Country, Spain, gpagnini@bcamath.org