Anomalous diffusion

Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), ${\displaystyle \langle r^{2}(\tau )\rangle }$, and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process described by Einstein and Smoluchowski, where the MSD is linear in time (namely, ${\displaystyle \langle r^{2}(\tau )\rangle =2dD\tau }$ with d being the number of dimensions and D the diffusion coefficient).^[1][2]

It has been found that equations describing normal diffusion are not capable of characterizing some complex diffusion processes, for instance, diffusion process in inhomogeneous or heterogeneous medium, e.g. porous media. Fractional diffusion equations were introduced in order to characterize anomalous diffusion phenomena.

Examples of anomalous diffusion in nature have been observed in ultra-cold atoms,^[3] harmonic spring-mass systems,^[4] scalar mixing in the interstellar medium, ^[5] telomeres in the nucleus of cells,^[6] ion channels in the plasma membrane,^[7] colloidal particle in the cytoplasm,^[8][9][10] moisture transport in cement-based materials,^[11] and worm-like micellar solutions.^[12]