FisMat2017 - Submission - View

Abstract's title: A fractional Fick's law for anomalous transport in turbulent plasmas
Submitting author: Gaetano Zimbardo
Affiliation: Universita della Calabria
Affiliation Address: Dipartimento di Fisica Universita della Calabria Ponte P. Bucci, Cubo 31C 87036 Arcavacata di Rende
Country: Italy
Oral presentation/Poster (Author's request): Oral presentation
Other authors and affiliations:
Non-diffusive, anomalous transport is often observed in both laboratory and astrophysical plasmas. In particular, superdiffusive transport implies that the mean square displacement of particles grows superlinearly with time. While probabilistic models based on a continuos time random walk are able to describe superdiffusion, the descriptions based on fractional diffusion equation are not yet fully satisfying, due to the divergence of the second order moment of Levy distributions. Here we propose to advance our understanding of superdiffusive transport by introducing a fractional Fick's law for turbulent plasmas. We quickly review the most common forms of fractional derivatives and of fractional Fick's law discussed so far, and then we propose to use the Caputo fractional derivative to express the particle flux in a bounded system. A heuristic motivation for the use of the Caputo derivative for transport in turbulent plasmas is given. Then we consider an advection-diffusion equation in steady-state, and we show that the steady-state solutions are given by the Mittag-Leffler functions. A comparison with spacecraft observations of energetic particles at interplanetary shocks is also done.