Normal and Anomalous Diffusion: A Tutorial

TL;DR

This tutorial comprehensively introduces normal and anomalous diffusion, covering mathematical models, key equations, and recent applications in physics, providing a clear understanding of diffusion processes from basic experiments to advanced theories.

Contribution

It systematically explains the mathematical modeling of diffusion, including CTRW and fractional equations, and connects classical and anomalous diffusion with recent physical applications.

Findings

Derivation of classical diffusion equation from random walk and Langevin models.

Explanation of CTRW and Levy distributions in anomalous diffusion.

Application of diffusion theories to laboratory and astrophysical plasmas.

Abstract

The purpose of this tutorial is to introduce the main concepts behind normal and anomalous diffusion. Starting from simple, but well known experiments, a series of mathematical modeling tools are introduced, and the relation between them is made clear. First, we show how Brownian motion can be understood in terms of a simple random walk model. Normal diffusion is then treated (i) through formalizing the random walk model and deriving a classical diffusion equation, (ii) by using Fick's law that leads again to the same diffusion equation, and (iii) by using a stochastic differential equation for the particle dynamics (the Langevin equation), which allows to determine the mean square displacement of particles. (iv) We discuss normal diffusion from the point of view of probability theory, applying the Central Limit Theorem to the random walk problem, and (v) we introduce the more general Fokker-Planck equation for diffusion that includes also advection. We turn then to anomalous diffusion, discussing first its formal characteristics, and proceeding to Continuous Time Random Walk (CTRW) as a model for anomalous diffusion. It is shown how CTRW can be treated formally, the importance of probability distributions of the Levy type is explained, and we discuss the relation of CTRW to fractional diffusion equations and show how the latter can be derived from the CTRW equations. Last, we demonstrate how a general diffusion equation can be derived for Hamiltonian systems, and we conclude this tutorial with a few recent applications of the above theories in laboratory and astrophysical plasmas.