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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.
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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⟩



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