Gaussian Processes in Complex Media: New Vistas on Anomalous Diffusion

Abstract

Normal or Brownian diffusion is historically identified by the linear growth in time of the variance and by a Gaussian shape of the displacement distribution. Processes departing from the at least one of the above conditions defines anomalous diffusion, thus a nonlinear growth in time of the variance and/or a non-Gaussian displacement distribution. Motivated by the idea that anomalous diffusion emerges from standard diffusion when it occurs in a complex medium, we discuss a number of anomalous diffusion models for strongly heterogeneous systems. These models are based on Gaussian processes and characterized by a population of scales, population that takes into account the medium heterogeneity. In particular, we discuss diffusion processes whose probability density function solves space-and time-fractional diffusion equations through a proper population of timescales or a proper population of length-scales. The considered modeling approaches are: the continuous time random walk, the generalized gray Brownian motion, and the time-subordinated process. The results show that the same fractional diffusion follows from different populations when different Gaussian processes are considered. The different populations have the common feature of a large spreading in the scale values, related to power-law decay in the distribution of population itself. This suggests the key role of medium properties, embodied in the population of scales, in the determination of the proper stochastic process underlying the given heterogeneous medium.

Transient Anomalous Diffusion MRI Measurement Discriminates Porous Polymeric Matrices Characterized by Different Sub-Microstructures and Fractal Dimension

“…In soft condensed matter, the length scale (l D ) by which we can probe media structure is approximately equal to the square root of the mean-square displacement (MSD) of the diffusing particles: l D ≈ MSD 1/2 . In the normal Brownian diffusion approximation, the MSD of diffusing water molecules is linearly proportional to the diffusion coefficient D and the time t during which the diffusion process is observed (MSD ∝ D · t) [ 13 ]. Taking typical values for free water diffusion and diffusion time achievable on NMR scanners (e.g., D ≈ 3 μm 2 /ms and t ≈ 10–100 ms), molecular displacements occur over linear distances of about 10–40 micrometers.…”

“…Nowadays, it is recognized that anomalous subdiffusion [ 13 ] exists in complex and highly heterogeneous media, such as biological tissues [ 31 ] and biomaterials [ 32 , 33 ]. Therefore, anomalous subdiffusion is a probe of submicroscopic organization, as shown in theoretical [ 34 ] and experimental works performed by optical microscopy [ 35 , 36 , 37 , 38 ].…”

“…Therefore, anomalous subdiffusion is a probe of submicroscopic organization, as shown in theoretical [ 34 ] and experimental works performed by optical microscopy [ 35 , 36 , 37 , 38 ]. Specifically, anomalous subdiffusion is characterized by an MSD of diffusing particles growing nonlinearly in time, i.e., MSD ∝ t α (with α < 1) [ 13 ]. The α parameter, which quantifies subdiffusion processes due to entrapped diffusion in multicomponent or hierarchical structures and boundary micro-roughness, is sensitive to sub-microstructures occurring at different length scales.…”

Transient Anomalous Diffusion MRI in Excised Mouse Spinal Cord: Comparison Among Different Diffusion Metrics and Validation With Histology

“…The normal diffusion of bulk water in tissues is not capable of detecting any of the characteristics above, providing nonlocal, diffusion measurement averaged on the l D length scale. As well summarized by Di Tullio et al (2019) and introduced above, normal or Brownian diffusion is identified by the linear growth in time of the MSD and by the Gaussian shape of the molecular motion propagator. Processes departing from at least one of the above conditions define anomalous diffusion, thus a nonlinear growth in time of the MSD and/or a non-Gaussian displacement distribution.…”

Fractional Diffusion and Medium Heterogeneity: The Case of the Continuous Time Random Walk

“…Fractional diffusion is characterized by non-Gaussian statistics and nonlinear scaling in time of the mean-squared displacement [20][21][22]28]. Many different approaches have been implemented and extensively analyzed to reproduce this type of diffusion, see, e.g., [2,9,10]. In particular, we recall the continuous time random walk (CTRW) [11], where a power-law tailed distribution of the waiting times can be introduced to generate fractional diffusion processes.…”