Fractional diffusion and Lévy stable processes

Abstract

Anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in ''time'' has been observed in all manner of physical and biological systems from macroscopic surface growth to DNA sequences. Herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model. This model is a simple two-state process with long-time correlations and is shown to produce a random walk described by an exact fractional diffusion equation. Fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of Fox functions, including Lévy ␣-stable processes in the superdiffusive domain (1/2ϽHϽ1).

Strange kinetics: conflict between density and trajectory description

“…This is the main reason why the later paper of Ref. [27] was fraught by internal contradictions, correctly pointed out by Metzler and Nonnemacher [15] in a subsequent paper. The authors of Ref.…”

“…We hope that the present paper might serve at least the good purpose of explaining the mathematical and physical reasons behind the contradictory conclusions of the papers of Refs. [5,15,27,12]. We think that this might bear also the significant consequences of establishing the borders between dynamics and thermodynamics (Section IV) and between quantum and classical mechanics (Section V).…”

Untitled

“…The formulae for the Green function derived in the second Section can be easily adapted to numerical simulation of the diffusive motion in any potential of the mentioned type. For example, the force can be assumed to be a semi-Markovian or a non-Markovian variant of the dichotomic noise [27]- [28], it can exhibit jumps of random magnitudes (kangaroo process [26]), etc. For any such process, our analysis is valid up to Subsection 3.3.…”

Bayesian estimation of self-similarity exponent

“…Now, I discuss the limitation of the proposed Bayesian method to a process which is self-similar, but has no properties resembling a Gaussian process, such the α-stable Lévy motion. I therefore perform the analysis with Lévy process following a stable distribution as discussed next [26,96,115,127].…”