Fractional Fick's law: the direct way
Fractional Fick's law: the direct way
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DOI：
10.1088/1751-8113/40/29/007
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发表时间：
2007-07
期刊：
影响因子：
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通讯作者：
M. Néel;Ali Abdennadher;M. Joelson
中科院分区：
文献类型：
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作者：
M. Néel;Ali Abdennadher;M. Joelson
Lévy flights是一种马尔可夫连续时间随机游动，可能解释了极端事件，经常用作异质介质中物质传播的小尺度模型。其中，布朗运动是菲克定律成立的一个特殊情况：对于步行者云，通量与在某个地方找到粒子的概率密度的梯度成正比。李维飞行类似于布朗运动，除了跳跃长度是根据α稳定的李维定律分布的，可能表现出重尾和偏斜。当α介于1和2之间时，费克定律的分数形式在无限介质中成立：通量正比于分数阶导数的组合，或者步行者密度的α − 1阶是由分数阶色散方程得到的。我们提出了一个直接的和自然的证明这一结果的基础上，一个新的定义通常的分数阶导数，涉及卷积和限制过程。考虑到这样得到的分数阶Fick定律，得到光滑密度的分数阶色散方程。该方法适用于域，限制边界可能意味着非平凡的修改这个方程。
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.