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.

A continuous variant for Grünwald–Letnikov fractional derivatives

“…It yields quite a number of approximations to fractional derivatives, since kernel F only has to match oscillation conditions and to behave asymptotically as Grünwald-Letnikov weights. It generalizes to all positive orders a result, previously obtained for α between 0 and 1 [9], and presented in a slightly different form. For those values of α, the integral on the right-hand side of (1) represents an essential step in computing fluxes of particles performing random walks [10][11][12][13], allowing for heavy tails connected with the value of α.…”

“…[9] for mass spreading. It shows that the flux of a cloud of particles performing a broad class of random walks, is a fractional derivative, on the macroscopic scale.…”

“…Then, fractional Fick's law (8) implies the space-fractional diffusion equation (9). In an infinite domain and with initial condition in the form of a Dirac impulse (applied anywhere), (9) can be obtained directly from Fourier analysis and function P is smooth.…”

“…Assuming that they are realizations of a random variable of mean τ as in Ref. [9] would yield a similar issue. The prefactor in front of the integrals comes up naturally when we count particles crossing a given location per unit of time, provided we assume a scaling law of the form of h α /τ = K [14].…”

Mass transport with sorption in porous media

“…Hence, Lemma 4 is a consequence of the Lemma 5, already presented in slightly different forms in [10,11,17,18].…”