非定域动力学和扩散区的交叉
Nonlocal-in-time dynamics and crossover of diffusive regimes
摘要：
我们研究了一个简单的非局部的时间动力系统提出的复杂的扩散制度在非均匀介质中的有效建模。我们提出了它的解决方案和他们共同研究的统计，如均方距离。这个有趣的模型采用了一个非局部算子来代替传统的一阶时间导数。它引入了通过核函数编码的恒定长度的有限内存效应。非局部时间算子一方面与依赖于整个时间历程的分数阶时间导数有关，另一方面，如果记忆窗口的长度减小，则会减少到经典时间导数。这使我们能够证明非局部时间模型在捕获初始子扩散和长时间正态扩散之间的自然界中广泛观察到的交叉的有效性。
英文摘要：
We study a simple nonlocal-in-time dynamic system proposed for the effective modeling of complex diffusive regimes in heterogeneous media. We present its solutions and their commonly studied statistics such as the mean square distance. This interesting model employs a nonlocal operator to replace the conventional first-order time-derivative. It introduces a finite memory effect of a constant length encoded through a kernel function. The nonlocal-in-time operator is related to fractional time derivatives that rely on the entire time-history on one hand, while reduces to, on the other hand, the classical time derivative if the length of the memory window diminishes. This allows us to demonstrate the effectiveness of the nonlocal-in-time model in capturing the crossover widely observed in nature between the initial sub-diffusion and the long time normal diffusion.
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