The Random Walk and Diffusion Theory

Abstract

The random walk, a classical example of Markov-chains, is used as an entry-point for a more involved discussion of diffusion theory. After a complete analysis of the random walk, the Wiener process and its relation to Brownian motion is presented. In fact, the classic diffusion equation can be derived from the unbiased random walk within certain limits. Langevin’s stochastic differential equation is introduced and as a direct consequence the Ornstein–Uhlenbeck process follows. A more generalized description of the diffusion process is possible due to the introduction of a jump pdf which in turn allows the definition of a jump length pdf and a waiting time pdf. This extension appears to be necessary because many diffusive processes (not only in physics) cannot be understood on the level of Brownian motion. A consequence of this extension is the introduction of Levy flights and of fractal time random walks on the stochastic level.