nLab anomalous diffusion

The usual diffusion comes from Brownian motion – the random walk with equal steps. Anomalous diffusion is about more general case which comes from more complicated random processes, including nonlocal in time and those having jumps.

• J-P Bouchaud, A Georges, Anomalous diffusion in disordered media: statistical mechanisms, Phys. Rep. 195 (1990) pdf

• wikipedia anomalous diffusion

• Ralf Metzler, Joseph Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports 339:1 (2000) 1–77 doi

• Michael F. Shlesinger, Joseph Klafter, Gert Zumofen, Above, below and beyond Brownian motion, Amer Jour Phys 67(12) (1999) 1253 doi

• J. Klafter, M. F. Shlesinger, G. Zumofen, Beyond Brownian motion, Physics Today 49(2) (1996) 33 doi

• Ralf Metzler, Eli Barkai, Joseph Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Physical Review Letters 82:18, 3563-3567 (1999)

• R. Metzler, E. Barkai, J. Klafter, Deriving fractional Fokker-Planck equations from a generalised master equation, Europhys. Lett. 46 431 (1991) doi

• V. Zaburdaev, S. Denisov, J. Klafter, Lévy walks, arxiv/1410.5100v1

• Eric Lutz, Fractional Langevin equation, Phys. Rev. E 64, 051106 doi cond-math/0103128v1

We investigate fractional Brownian motion with a microscopic random-matrix model and introduce a fractional Langevin equation. We use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath. We further compare fractional Brownian motion with the fractal time process. The respective mean-square displacements of these two forms of anomalous diffusion exhibit the same power-law behavior. Here we show that their lowest moments are actually all identical, except the second moment of the velocity. This provides a simple criterion that enable us to distinguish these two non-Markovian processes.

A well–known example of a process which is non–local in space is Lévy stable motion, for which the mean–square displacement is actually infinite due to the occurrence of very long jumps. In this Letter we focus on processes which are nonlocal in time and whence show memory effects. Specifically, we shall discuss and compare fractional Brownian motion (fBm) and the fractal time process (ftp). These two forms of anomalous diffusion are fundamentally different…