HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Journal articles

Fractional Fick's law: the direct way

Abstract : Lévy flights, which are Markovian continuous time random walks possibly accounting for extreme events, serve frequently as small-scale models for the spreading of matter in heterogeneous media. Among them, Brownian motion is a particular case where Fick's law holds: for a cloud of walkers, the flux is proportional to the gradient of the probability density of finding a particle at some place. Lévy flights resemble Brownian motion, except that jump lengths are distributed according to an α-stable Lévy law, possibly showing heavy tails and skewness. For α between 1 and 2, a fractional form of Fick's law is known to hold in infinite media: that the flux is proportional to a combination of fractional derivatives or the order of α - 1 of the density of walkers was obtained as a consequence of a fractional dispersion equation. We present a direct and natural proof of this result, based upon a novel definition of usual fractional derivatives, involving a convolution and a limiting process. Taking account of the thus obtained fractional Fick's law yields fractional dispersion equation for smooth densities. The method adapts to domains, limited by boundaries possibly implying non-trivial modifications to this equation.
Document type :
Journal articles
Complete list of metadata

https://hal.inrae.fr/hal-02661818
Contributor : Migration Prodinra Connect in order to contact the contributor
Submitted on : Saturday, May 30, 2020 - 10:59:16 PM
Last modification on : Wednesday, March 17, 2021 - 3:37:45 AM

Links full text

Identifiers

Collections

Citation

Marie-Christine Neel, A. Abdennadher, Maminirina Joelson. Fractional Fick's law: the direct way. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2007, 40 (29), pp.8299-8314. ⟨10.1088/1751-8113/40/29/007⟩. ⟨hal-02661818⟩

Share

Metrics

Record views

7