Anomalous diffusion

Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), σ_r², and time, in contrast to a typical diffusion process, in which the MSD is a linear function of time. Physically, the MSD can be considered the amount of space the particle has "explored" in the system. An example of anomalous diffusion in nature is the subdiffusion that has been observed in the cell nucleus, plasma membrane and cytoplasm.[1]

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

Unlike typical diffusion, anomalous diffusion is described by a power law,[2][3] σ_r² ~ Dt^α, 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. [4]

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. 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,[5] scalar mixing in the interstellar medium, [6] telomeres in the nucleus of cells,[7] ion channels in the plasma membrane,[8] colloidal particle in the cytoplasm,[9] moisture transport in cement-based materials,[10] and worm-like micellar solutions.[11] Anomalous diffusion was also found in other biological systems, including heartbeat intervals and in DNA sequences.[12]

The daily fluctuations of climate variables such as temperature can be regarded as steps of a random walker or diffusion and have been found to be anomalous.[13]

In 1926, using weather balloons, Lewis Richardson demonstrated that the atmosphere exhibits super-diffusion.[14] In a bounded system, the mixing length (which determines the scale of dominant mixing motions) is given by the Von Kármán constant according to the equation ${\displaystyle l_{m}={\kappa }z}$, where ${\displaystyle l_{m}}$ is the mixing length, ${\displaystyle {\kappa }}$ is the Von Kármán constant, and ${\displaystyle z}$ is the distance to the nearest boundary.[15] Because the scale of motions in the atmosphere is not limited, as in rivers or the subsurface, a plume continues to experience larger mixing motions as it increases in size, which also increases its diffusivity, resulting in super-diffusion.[16]