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). 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, 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.

Classes of anomalous diffusion

Unlike typical diffusion, anomalous diffusion is described by a power law, where K_\alpha is the so-called generalized diffusion coefficient and \tau is the elapsed time. The classes of anomalous diffusions are classified as follows: In 1926, using weather balloons, Lewis Fry Richardson demonstrated that the atmosphere exhibits super-diffusion. 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, where l_m is the mixing length, {\kappa} is the Von Kármán constant, and z 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.

Models of anomalous diffusion

The types of anomalous diffusion given above allows one to measure the type, but how does anomalous diffusion arise? There are many possible ways to mathematically define a stochastic process which then has the right kind of power law. Some models are given here. These are long range correlations between the signals continuous-time random walks (CTRW) and fractional Brownian motion (fBm), and diffusion in disordered media. Currently the most studied types of anomalous diffusion processes are those involving the following These processes have growing interest in cell biophysics where the mechanism behind anomalous diffusion has direct physiological importance. Of particular interest, works by the groups of Eli Barkai, Maria Garcia Parajo, Joseph Klafter, Diego Krapf, and Ralf Metzler have shown that the motion of molecules in live cells often show a type of anomalous diffusion that breaks the ergodic hypothesis. This type of motion require novel formalisms for the underlying statistical physics because approaches using microcanonical ensemble and Wiener–Khinchin theorem break down.