Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement, σr2, 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. Unlike typical diffusion, anomalous diffusion is described by a power law, 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. 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, scalar mixing in the interstellar medium, telomeres in the nucleus of cells, ion channels in the plasma membrane, colloidal particle in the cytoplasm, and worm-like micellar solutions. Anomalous diffusion was also found in other biological systems, including heartbeat intervals and in DNA sequences. 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. In 1926, using weather balloons, Lewis Richardson demonstrated that the atmosphere exhibits super-diffusion. In a bounded system, the mixing length is given by the Von Kármán constant according to the equation, where is the mixing length, is the Von Kármán constant, and is the distance to the nearest boundary. 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.
Types of Anomalous diffusion
Of interest within the scientific community, when an anomalous-type diffusion process is discovered, the challenge is to understand the underlying mechanism which causes 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 long range correlations between the signals continuous-time random walks and fractional Brownian motion, diffusion of colloidal particles in bacterial suspensions, and diffusion in disordered media. Anomalous subdiffusion in cellular cytosol can be an artifact resulting from using of polydisperse probes for measurements.
Hyper-ballistic diffusion
One important class of anomalous diffusion refers to the case when the scaling exponent of the MSD increases with value greater than 2. Such case is called hyper-ballistic diffusion and it has been observed in optical systems.
