Anomalous diffusion

File:Msd anomalous diffusion.svg

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), σ_r², 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, 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.

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, Telomeres in the nucleus of cells, in single particle movements in cytoplasm, and in 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.

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 (CTRW ) and fractional Brownian motion (fBm), and diffusion in disordered media.

See also

• Lévy flight
• Random walks
• Percolation
• Long term correlations
• long range dependencies
• Hurst exponent
• Detrended fluctuation analysis [DFA]
• Fractals

References

• Matthias Weiss, Markus Elsner, Fredrik Kartberg, and Tommy Nilsson. "Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells." Biophysical Journal 87 : 3518-3524 (2004)
• Jean-Philippe Bouchaud, Antoine Georges, "Anomalous diffusion in disordered media." Physics Reports 195 (4-5) : 127-293 (1990).
• A. von Kameke et al., "Propagation of a chemical wave front in a quasi-two-dimensional superdiffusive flow", Phys. Rev. E 81 6 (2010).
• Wen Chen, HongGuang Sun, Xiaodi Zhang, Dean Korosak "Anomalous diffusion modeling by fractal and fractional derivatives". Computers and Mathematics with Applications, 59 (5): 1754-1758 (2010).
• HongGuang Sun, Mark M. Meerschaert, Yong Zhang, Jianting Zhu, Wen Chen. "A fractal Richards' equation to capture the non-Boltzmann scaling of water transport in unsaturated media". Advances in Water Resources, 52: 292-295 (2013).

External links

• Boltzmann's transformation, Parabolic law (animation)
• Anomalous interface shift kinetics (Computer simulations and Experiments)