Normal and Anomalous Diffusion: A Tutorial

By Loukas Vlahos, Heinz Isliker, Yannis Kominis, Kyriakos Hizanidis. 2008

132 citations recorded by the source corpus.

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 formal­ izing 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 equa­ tion), 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 Fokker­Planck 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 prob­ ability 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.

Subfields: Modeling and Simulation, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dba5cdb6bdd41eb2c27fb8cbf18536f5414b37ed

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Citations in the corpus

• Hippocampal Reactivation of Random Trajectories Resembling Brownian Diffusion. (2019, background): “In physics, diffusion processes are described by a power law between mean distance and time, which represents an exponential relationship characterized by the value of the exponent (Metzler et al., 2012; Vlahos et al., 2008).”
• Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, background): “A main result is the property of infinite speed of propagation.”
• A critical fractional equation with concave-convex power nonlinearities (2013, background): “For more details and applications, see [8, 11, 25, 34, 59, 60] and the references therein.”
• BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL NONLOCAL FRACTIONAL LAPLACIAN PROBLEMS (2016, unknown): “, the papers [2, 7, 10, 16, 21, 34, 35, 36] and references therein).”