From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion

TLDR

Einstein’s random‑walk explanation of Brownian motion underpins modern stochastic theory, describing Markovian diffusion via the diffusion equation, yet many natural processes are non‑Markovian and display anomalous diffusion. The study investigates subdiffusive, semi‑Markovian continuous‑time random walks with heavy‑tailed waiting times, deriving two equivalent kinetic equations that reduce to fractional diffusion or Fokker‑Planck forms for power‑law distributions. The authors model subdiffusion as a normal diffusion subordinated to an operational time governed by a heavy‑tailed waiting‑time distribution, and show that for non‑power‑law waiting times one kinetic equation is preferable over the other depending on whether the process slows down or accelerates.

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.