Anomalous diffusion

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, ${\displaystyle ⟨r^2(\tau )⟩=K_{\alpha }{\tau }^{\alpha }}$where ${\displaystyle K_{\alpha }}$ is the so-called generalized diffusion coefficient and ${\displaystyle \tau }$ is the elapsed time. The classes of anomalous diffusions are classified as follows:

• α < 1: subdiffusion. This can happen due to crowding or walls. For example, a random walker in a crowded room, or in a maze, is able to move as usual for small random steps, but cannot take large random steps, creating subdiffusion. This appears for example in protein diffusion within cells, or diffusion through porous media. Subdiffusion has been proposed as a measure of macromolecular crowding in the cytoplasm.
• α = 1: Brownian motion.
• ${\displaystyle 1<\alpha <2}$: superdiffusion. Superdiffusion can be the result of active cellular transport processes or due to jumps with a heavy-tail distribution.
• α = 2: ballistic motion. The prototypical example is a particle moving at constant velocity: ${\displaystyle r=v\tau }$.
• ${\displaystyle \alpha >2}$: hyperballistic. It has been observed in optical systems.

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 ${\displaystyle l_m=\kappa z}$, 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. 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.