In this study, the local time and space structural derivative diffusion model is introduced to describe non-Fickian diffusion in heterogeneous media from ultraslow to superfast. Its equivalent heterogeneous diffusion process (HDP) model with time and space-dependent diffusion coefficient is also given. The connection and difference between the proposed model and the existing HDP models are compared. Different types of diffusion processes can be captured by selecting proper structural functions in the structural derivative model, which is determined based on the concentration of tracers, mean squared displacement (MSD) and diffusion coefficient of the heterogeneous media. The dynamics of the free cooling granular gases with constant restitution coefficient and velocity-dependent restitution coefficient is investigated to test the proposed model. The results show that the diffusion law of the granular particles is not uniform for the short and long-time scales, but one uniform diffusion law derived from the proposed model can well fit the data in different time scales. The patterns of the time-dependent diffusion coefficient are also simulated to deeply understand the complicated non-Fickian diffusion in complex system.

中文翻译：

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时空波动的非菲克扩散扩散景观：从超快到超慢

本研究引入局部时空结构导数扩散模型来描述异质介质中从超慢到超快的非Fickian扩散。还给出了具有时间和空间相关扩散系数的等效非均匀扩散过程（HDP）模型。比较了提出的模型与现有 HDP 模型之间的联系和差异。通过在结构导数模型中选择适当的结构函数，可以捕获不同类型的扩散过程，该模型根据示踪剂的浓度、均方位移 (MSD) 和非均质介质的扩散系数确定。研究了具有恒定恢复系数和速度相关恢复系数的自然冷却颗粒气体的动力学，以测试所提出的模型。结果表明，颗粒颗粒的扩散规律在短时间尺度和长时间尺度上并不均匀，但从所提模型导出的一种均匀扩散规律可以很好地拟合不同时间尺度的数据。还模拟了时间相关扩散系数的模式，以深入理解复杂系统中复杂的非 Fickian 扩散。