Fractional master equations and fractal time random walks

Abstract

Fractional master equations containing fractional time derivatives of order 0 < ω ≤ 1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density ψ(t) is obtained exactly as ψ(t) = (t ω−1 /C)E ω,ω (−t ω /C) where E ω,ω (x) is the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit. Contents 3 Appendix A. Definition of H-functions 4 References 5

The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

“…A waiting time PDF of the long-tailed inverse power-law form ψ(t) ∼ τ α /t 1+α (0 < α < 1), which enters the CTRW propagator (A.1) via the Laplace expansion ψ(u) ∼ 1 − (uτ ) α (u τ ), can be completed, for instance, to a one-sided Lévy stable density L [Hilfer and Anton (1995)]. In this case, the limit α = 1 becomes ψ(t) = e −t/τ .…”

“…Apart from other standard tools to describe anomalous dynamics such as continuous time random walks [Blumen et al (1986), Bouchaud and Georges (1990), Hughes (1995), ‡ Douglas Adams, The Restaurant at the End of the Universe, Tor Books, 1988 Klafter et al (1996), Shlesinger et al (1993)], fractional dynamical equations have become increasingly popular to model anomalous transport [Barkai (2001), Hilfer (2000), Metzler and Klafter (2000), Metzler and Klafter (2001), Sokolov et al (2002)]. In the presence of an external force field, in particular, the fractional Fokker-Planck equation provides a direct extension of the classical Fokker-Planck equation, being amenable to well-known methods of solution.…”

Short note on the emergence of fractional kinetics

“…In a pioneering paper by Hilfer & Anton in 1995 [6], it was shown that if the survival probability Ψ(τ) is…”

“…Hence Ψ(t) is the probability that, after a jump, the diffusing quantity does not change during the temporal interval of duration τ and it is the survival probability at the initial position [6]. When jumps and waiting times are statistically independent, the master equation of the CTRW model is [8] …”

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations

“…It was also recognized [HA95], [Com96], [MRGS00], [BMK00] that the fractional kinetic equations may be viewed as "hydrodynamic" (that is, long-time and long-space) limits of the CTRW (Continuous Time Random Walk) theory [MW65] which was succesfully applied to the description of anomalous diffusion phenomena in many areas, e.g., turbulence [KBS87], disordered medium [BG90], intermittent chaotic systems [ZK93], etc. However, the kinetic equations have two advantages over a random walk approach: firstly, they allow one to explore various boundary conditions (e.g., reflecting and/or absorbing) and, secondly, to study diffusion and/or relaxation phenomena in external fields.…”