Fractional diffusion and medium heterogeneity: the case of the continuous time random walk

Sposini V., Vitali S., Paradisi P., Pagnini G.

Time-fractional diffusion  Continuous time random walk  Anomalous diffusion  Continuos time random walk  Medium heterogeneity 

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 relate 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 environmental heterogeneity. The results are supported by simulations.

Source: Nonlocal and Fractional Operators, edited by Beghin L., Mainardi F., Garrappa R., pp. 275–286, 2021

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