2004/11/01 by Igor M. Sokolov, I. M. Sokolov, J. Klafter · 466 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Anomalous diffusion #Brownian motion #Computer science #Continuous-time random walk #Differential equation #Diffusion #Diffusion equation #Diffusion process #Einstein #Fokker–Planck equation #Fractional Brownian motion #Fractional Differential Equations Solutions #Innovation diffusion #Markov process #Mathematical analysis #Mathematical physics #Mathematics #Physics #Power law #Quantum mechanics #Random walk #Statistical Mechanics and Entropy #Statistical physics #Statistics #Stochastic process #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1063/1.1860472
published in Chaos An Interdisciplinary Journal of Nonlinear Science 15(2), 26103 (American Institute of Physics)
arxiv created 2004/11/01 · openalex publication_date 2005/06/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.