https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

/pdf/the-restaurant-at-the-end-of-the-random-walk-recent-1w7yr3snyb.pdf

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

Maximizing information exchange between complex networks

We investigate the physical background and implications of a space- and time-fractional diffusion equation which corresponds to a random walker which combines competing long waiting times and Levy flight properties. Explicit solutions are examined, and the corresponding fractional Fokker–Planck–Smoluchowski equation is presented. The framework of fractional kinetic equations which control the systems relaxation to either Boltzmann–Gibbs equilibrium, or a far from equilibrium Levy form is explored, putting the fractional approach in some perspective from the standard non-equilibrium dynamics point of view, and its generalisation.

Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation

We present a generalisation of the classical exponential relaxation based on the fractional Fokker–Planck equation framework. We show how fractional dynamics modifies the Brownian dynamics underlying standard relaxation processes, and gives rise to the Mittag–Leffler relaxation of modes and moments. The latter is characterised through a turnover from an initial stretched exponential to a final inverse power-law pattern and the associated complex susceptibility corresponds to the Cole–Cole pattern.

From stretched exponential to inverse power-law: fractional dynamics, Cole-Cole relaxation processes, and beyond

A non-Markovian generalization of the Chapman−Kolmogorov transition equation for continuous time random processes governed by a waiting time distribution is investigated. It is shown under which conditions a long-tailed waiting time distribution with a diverging characteristic waiting time leads to a fractional generalization of the Klein−Kramers equation. From the latter equation a fractional Rayleigh equation and a fractional Fokker−Planck equation are deduced. These equations are characterized by a slow, nonexponential relaxation of the modes toward the Gibbs−Boltzmann and the Maxwell thermal equilibrium distributions. The derivation sheds some light on the physical origin of the generalized diffusion and friction constants appearing in the fractional Fokker−Planck equation.

Ralf Metzler and

Papers

From a Generalized Chapman−Kolmogorov Equation to the Fractional Klein−Kramers Equation†