异常扩散的时空结构
Time-space fabric underlying anomalous diffusion
摘要：
这项研究揭示了异常扩散过程背后的时空转换。在此基础上，我们提出了分形时空结构对物理行为影响的两个假设，并由此导出了能量与频率、动量与波数之间的分数阶量子关系，进而导出了分数阶薛定谔方程。作为标准分数阶导数的另一种建模方法，我们引入了度量时空的Hausdorff维数的Hausdorff导数的概念。在此基础上，利用Hausdorff导数导出了一个线性反常输运扩散方程。它的格林函数解是一种新型的拉伸高斯分布，并与Richardson扩散方程的格林函数解进行了比较。
英文摘要：
This study unveils the time-space transforms underlying anomalous diffusion process. Based on this finding, we present the two hypotheses concerning the effect of fractal time-space fabric on physical behaviors and accordingly derive fractional quantum relationships between energy and frequency, momentum and wavenumber which further give rise to fractional Schrodinger equation. As an alternative modeling approach to the standard fractional derivatives, we introduce the concept of the Hausdorff derivative underlying the Hausdorff dimensions of metric spacetime. And in terms of the proposed hypotheses, the Hausdorff derivative is used to derive a linear anomalous transport-diffusion equation underlying anomalous diffusion process. Its Green's function solution turns out to be a new type of stretched Gaussian distribution and is compared with that from the Richardson's diffusion equation.
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