Continuous time anomalous diffusion in a composite medium.
Continuous time anomalous diffusion in a composite medium.
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复合介质中的连续时间反常扩散。
DOI：
10.1103/physreve.84.021116
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
E. Schachinger
中科院分区：
文献类型：
--
作者：
B. Stickler;E. Schachinger
研究了有限层直接接触复合介质中一维连续时间反常扩散问题。扩散过程本身借助于两个概率密度函数（PDF）来描述，其中一个是任意跳跃长度的PDF，另一个是由等待时间指数β∈（0，1）表征的长尾等待时间PDF。前者被假定为空间坐标x和时间坐标t的函数，而后者是x和时间间隔的函数。对于这样的环境中的扩散方程的一个非常一般的形式，它描述了连续时间的异常扩散的复合介质。这个结果，然后专门为两个特定形式的跳跃长度PDF，即连续时间的Lévy飞行PDF和连续时间截断的Lévy飞行PDF。在这两种情况下，PDF的特征在于Lévy指数α∈（0，2），它被认为是x和t的函数。可以证明，对于指数α和β的特定选择，可以立即得出从文献中熟知的反常扩散的其他方程。这证明了非常普遍的适用性的推导和由此产生的分数阶微分方程在这里讨论。
The one-dimensional continuous time anomalous diffusion in composite media consisting of a finite number of layers in immediate contact is investigated. The diffusion process itself is described with the help of two probability density functions (PDFs), one of which is an arbitrary jump-length PDF, and the other is a long-tailed waiting-time PDF characterized by the waiting-time index β∈(0,1). The former is assumed to be a function of the space coordinate x and the time coordinate t while the latter is a function of x and the time interval. For such an environment a very general form of the diffusion equation is derived which describes the continuous time anomalous diffusion in a composite medium. This result is then specialized to two particular forms of the jump-length PDF, namely the continuous time Lévy flight PDF and the continuous time truncated Lévy flight PDF. In both cases the PDFs are characterized by the Lévy index α∈(0,2) which is regarded to be a function of x and t. It is possible to demonstrate that for particular choices of the indices α and β other equations for anomalous diffusion, well known from the literature, follow immediately. This demonstrates the very general applicability of the derivation and of the resulting fractional differential equation discussed here.