Anomalous diffusion

Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), , 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, with d being the number of dimensions and D the diffusion coefficient). Examples of anomalous diffusion in nature have been observed in biology in the cell nucleus, plasma membrane and cytoplasm. Unlike typical diffusion, anomalous diffusion is described by a power law, where is the so-called generalized diffusion coefficient and is the elapsed time. In Brownian motion, α = 1. If α > 1, the process is superdiffusive. Superdiffusion can be the result of active cellular transport processes or due to jumps with a heavy-tail distribution. If α < 1, the particle undergoes subdiffusion. The role of anomalous diffusion has received attention within the literature to describe many physical scenarios, most prominently within crowded systems, for example protein diffusion within cells, or diffusion through porous media. Subdiffusion has been proposed as a measure of macromolecular crowding in the cytoplasm. 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. Recently, anomalous diffusion was found in several systems including ultra-cold atoms, harmonic spring-mass systems, scalar mixing in the interstellar medium, telomeres in the nucleus of cells, ion channels in the plasma membrane, colloidal particle in the cytoplasm, moisture transport in cement-based materials, and worm-like micellar solutions. In 1926, using weather balloons, Lewis Fry Richardson demonstrated that the atmosphere exhibits super-diffusion.