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 Lévy flights and of fractal time random walks on the stochastic level.
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Notes
- 1.
This assumption is known as the approximation of molecular chaos. In fact it represents the Markov approximation to the dynamics of a many particle system.
- 2.
For instance, one can employ Fermi’s golden rule [6] to obtain this function on a quantum mechanical level. We already came across an expression of the form (17.3) on the right hand side of the master equation, see Sect. 16.3, Eq. (16.42). However, the collision integral of the Boltzmann equation is non-linear.
- 3.
The function ρ(r, t) is referred to as a physical distribution function due to the normalization condition (17.6). This is in contrast to distribution functions we encountered so far within this book, which are normalized to unity.
- 4.
- 5.
In fact, it can be shown that W t is non-differentiable with probability one. This is the reason why it is defined as the formal derivative of W t . Let \(\varphi (t)\) be a test function and f(t) an arbitrary function which does not need to be differentiable with respect to t. Then the formal derivative \(\dot{f}(t)\) is defined by
$$\displaystyle{ \int _{0}^{\infty }\mathrm{d}t\,\dot{f}(t)\varphi (t) = -\int _{ 0}^{\infty }\mathrm{d}t\,f(t)\dot{\varphi }(t)\;. }$$ - 6.
A short introduction to fractional derivatives and integrals can be found in Appendix G.
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Stickler, B.A., Schachinger, E. (2016). The Random Walk and Diffusion Theory. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27265-8_17
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