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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 autocorrelation for the ultraslow diffusion models are also briefly discussed.

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This paper proposes a novel structural derivative approach to tackle the perplexing modeling problem of ultraslow diffusion. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space fabric of physical systems. Ultraslow diffusion has been observed in numerous lab experiments and field observations, whose behaviors deviate dramatically from the standard anomalous diffusion models characterizing power function of time. The logarithmic diffusion model has since been used to describe bizarre process of ultraslow diffusion but with very limited success. This study applies the inverse Mittag-Leffler function as the structural function in the structural derivative modeling ultraslow diffusion of a random system of two interacting particles. It is observed that the dynamics of two interacting particles are respectively the ballistic motion at the short time scale and the Sinai ultraslow diffusion at the long time scale. Compared with the logarithmic diffusion model, the inverse Mittag-Leffler diffusion model has higher accuracy and manifests clearer physical mechanism. Numerical experiments show that the structural derivative is a feasible mathematical tool to model the ultraslow diffusion using the inverse Mittag-Leffler function as its structural function.

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Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can be encoded in the attenuation of the observed MRI signal through the fractional order of the time- and space-derivatives. Specifically, in solving the Bloch-Torrey equation, fractional order imaging biomarkers are identified that connect the continuous time random walk model of Brownian motion to the structure and composition of cells, cell membranes, proteins, and lipids. In this way, the decay of the induced magnetization is influenced by the micro- and meso-structure of tissues, such as the white and gray matter of the brain or the cortex and medulla of the kidney. Fractional calculus provides new functions (Mittag-Leffler and Kilbas-Saigo) that characterize tissue in a concise way. In this paper, we describe the exponential, stretched exponential, and fractional order models that have been proposed and applied in MRI, examine the connection between the model parameters and the underlying tissue structure, and explore the potential for using diffusion-weighted MRI to extract biomarkers associated with normal growth, aging, and the onset of disease.