A Survey of Models of Ultraslow Diffusion in Heterogeneous Materials

Ultraslow diffusion is characterized by a logarithmic growth of the mean squared displacement (MSD) as a function of time. It occurs in complex arrangements of molecules, microbes, and many-body systems. This paper reviews mechanical models for ultraslow diffusion in heterogeneous media from both macroscopic and microscopic perspectives. Macroscopic models are typically formulated in terms of a diffusion equation that employs noninteger order derivatives (distributed order, structural, and comb models (CM)) or employs a diffusion coefficient that is a function of space or time. Microscopic models are usually based on the continuous time random walk (CTRW) theory, but use a weighted logarithmic function as the limiting formula of the waiting time density. The similarities and differences between these models are analyzed and compared with each other. The corresponding MSD in each case is tabulated and discussed from the perspectives of the underlying assumptions and of real-world applications in heterogeneous materials. It is noted that the CMs can be considered as a type of two-dimensional distributed order fractional derivative model (DFDM), and that the structural derivative models (SDMs) generalize the DFDMs. The heterogeneous diffusion process model (HDPM) with time-dependent diffusivity can be rewritten to a local structural derivative diffusion model mathematically. The ergodic properties, aging effect, and velocity auto-correlation for the ultraslow diffusion models are also briefly discussed.