Fractional Diffusion and Medium Heterogeneity: The Case of the Continuous Time Random Walk

Abstract

In this contribution we show that fractional diffusion emerges from a simple Markovian Gaussian random walk when the medium displays a power-law heterogeneity. Within the framework of the continuous time random walk, the heterogeneity of the medium is represented by the selection, at any jump, of a different time-scale for an exponential survival probability. The resulting process is a non-Markovian non-Gaussian random walk. In particular, for a power-law distribution of the time-scales, the resulting random walk corresponds to a time-fractional diffusion process. We relates the power-law of the medium heterogeneity to the fractional order of the diffusion. This relation provides an interpretation and an estimation of the fractional order of derivation in terms of environment heterogeneity. The results are supported by simulations.

Anomalous diffusion originated by two Markovian hopping-trap mechanisms

“…This last was originally based on the over-damped fractional Brownian motion (fBm) but it could be extended to the under-damped Langevin equation as well [74]. In terms of physical interpretation, the approach based on the CTRW describes a diffusion process in an inhomogeneous medium [90,91] while the approach based on the gBm describes a diffusion process by an heterogeneous ensemble of walkers [74]. Another stochastic modelling, whose walker's PDF is governed by the time-fractional diffusion equation, is the subordination approach [68] and it is somehow related to the CTRW [92].…”

Anomalous diffusion originated by two Markovian hopping-trap mechanisms

“…This last was originally based on the over-damped fractional Brownian motion (fBm) but it could be extended to the underdamped Langevin equation as well [74]. In terms of physical interpretation, the approach based on the CTRW describes a diffusion process in an inhomogeneous medium [90,91] while the approach based on the gBm describes a diffusion process by an heterogeneous ensemble of walkers [74]. Another stochastic modelling, whose walker's PDF is governed by the time-fractional diffusion equation, is the subordination approach [68] and it is somehow related to the CTRW [92].…”