From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion

TL;DR

This paper explores the extension of Einstein's Brownian motion to anomalous diffusion, specifically subdiffusive processes with non-Markovian properties, deriving fractional kinetic equations for complex waiting-time distributions.

Contribution

It introduces two equivalent forms of kinetic equations for subdiffusive processes, applicable to non-power-law waiting-time distributions, advancing understanding of anomalous diffusion.

Findings

Derived fractional diffusion and Fokker-Planck equations for power-law waiting times

Presented two forms of kinetic equations suitable for different temporal behaviors

Extended classical diffusion theory to non-Markovian, semi-Markovian processes

Abstract

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 to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and 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 know 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.