The Fokker - Planck Equation for Stochastic Motion

We set up now the Fokker - Planck equation for the special case that the noise is small 

We have used the chain rule for functions with more than one argument. Therefore, the equation results with the same form as the well-known equation of diffusion. 

The function P(x, t) is the probability to find a particle at time t at position x. The particle is moving with a constant velocity. This means, it has a kinetic energy, which is modified by a small stochastic term. The solution of Eq. (21.18) is 

If we are again moving with the velocity v, then a Gaufi distribution will arise that broadens in time. Often this behavior is addressed that the particle flows away or melts in time. Thus, we are finding in the stochastic mechanics a very similar situation to that in comparison to the quantum mechanics. The concepts of the position and the momentum must be revisited and replaced by a new concept. In stochastic mechanics, we can reasonably justify that a particle never can be observed in an isolated manner. Here an interaction with other particles in the neighborhood appears. This interaction cannot be described by a deterministic method. Therefore, we chose a statistic description of the motion. Quantum mechanics starts with certain postulates that cannot be justified in detail. The success of the method defends it, even when the vividness suffers. 

We introduce now several stochastic processes. These processes are generalizations of the classic random walk. A particle should be at a certain time t in one box of a collection of boxes. After a time step, it moves into another box, or it stays in this box. We can set up Fig. 21.1. The arrows show, where a particle can move within the next time step. The boxes have a width of Ax. 

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