Superfast diffusion exists in various complex anisotropic systems. Its mean square displacement is an exponential function of time proved by several theoretical and experimental investigations. Previous studies have studied the superfast diffusion based on the time-space scaling local structural derivatives without considering the memory of dynamic behavior. This paper proposes a nonlocal time structural derivative model based on the Caputo fractional derivative to describe superfast diffusion in which the structural function is a power law function of time. The obtained concentration of the diffusive particles, i.e. the solution of the structural derivative model is a double-sided exponential distribution. The derived mean square displacement is a Mittag–Leffler function of time, which generalizes the exponential case. To verify the feasibility of the model, the charge and energy transfer at nanoscale interfaces in solar cells and the dynamics of the dripplons between two graphene sheets are employed. Compared with the existing models, the fitting results indicate that the proposed model is more accurate with higher credibility. The properties of the nonlocal structural derivative model with different structural functions are also discussed.

中文翻译：

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基于 Caputo 分数阶导数的非局部结构导数模型，用于异质介质中的超快扩散

超快扩散存在于各种复杂的各向异性系统中。它的均方位移是时间的指数函数，已通过多项理论和实验研究证明。以前的研究已经研究了基于时空尺度局部结构导数的超快扩散，而没有考虑动态行为的记忆。本文提出了一种基于Caputo分数导数的非局部时间结构导数模型来描述超快扩散，其中结构函数是时间的幂律函数。得到的扩散粒子浓度，即结构导数模型的解为双边指数分布。导出的均方位移是时间的 Mittag-Leffler 函数，它概括了指数情况。为了验证模型的可行性，采用了太阳能电池纳米级界面的电荷和能量转移以及两个石墨烯片之间的液滴动力学。与现有模型相比，拟合结果表明，该模型更准确，可信度更高。还讨论了具有不同结构函数的非局部结构导数模型的性质。