# Anomalous diffusion

Mean squared displacement $\langle r^2(\tau)\rangle$ for different types of anomalous diffusion

Anomalous diffusion is a diffusion process with a non-linear relationship to time, in contrast to a typical diffusion process, in which the mean squared displacement (MSD), σr2, of a particle is a linear function of time. Physically, the MSD can be considered the amount of space the particle has "explored" in the system.

Unlike typical diffusion, anomalous diffusion is described by a power law,[1][2] σr2 ~ Dtα, where D is the diffusion coefficient and t is the elapsed time. In a typical diffusion process, α = 1. If α > 1, the phenomenon is called super-diffusion. Super-diffusion can be the result of active cellular transport processes. If α < 1, the particle undergoes sub-diffusion.[3]

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. Sub-diffusion has been proposed as a measure of macromolecular crowding in the cytoplasm.

Recently, anomalous diffusion was found in several systems including ultra-cold atoms,[4] Telomeres in the nucleus of cells, [5] in single particle movements in cytoplasm,[6] and in in worm-like micellar solutions.[7] Anomalous diffusion was also found in other biological systems, including heartbeat intervals and in DNA sequences.[8]

## Types of Anomalous diffusion

Of interest within the scientific community, when one an anomalous-type diffusion process, is to then understand the underlying mechanism which has caused it. There are a number of frameworks which give rise to anomalous diffusion that are currently in vogue within the statistical physics community. These are continuous-time random walks (CTRW) and fractional Brownian motion (fBm), and diffusion on fractal topology.