  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.
