Linear and Nonlinear Fokker-Planck Equations

Abstract

Glossary Linear Here, linear with respect to a probability density. Nonlinear Here, nonlinear with respect to a probability density. Markov process Process for which it is sufficient to have information about the presence in order to make best predictions about the future. Additional information about the past will not improve the predictions. Martingale process ^ Z Process for which the future mean value hZ(t + Dt)i of a set of realizations Z (i) that is passing at presence t through a certain common state z is the state z: z : hZ(t + Dt)i Z(t) = z = z. Additional information about states z 0 visited at times t 0 prior to t is irrelevant. Definition of the Subject Let ^ X denote a stochastic process defined on the space O and the time interval [t 0 , 1], where t 0 denotes the initial time of the process. We assume that the process ^ X can be described in terms of a random variable X O. More precisely, let X(t) denote the time-dependent evolution of the random variable X for t ! t 0. Then, we assume that the process ^ X can be described in terms of the infinitely large set of realizations X (i) (t) of X(t) with i = 1, 2,. . .. The realizations i = 1, 2,. .. constitute a statistical ensemble. At every time t the probability density P of the process ^ X can be computed from the realizations X (i) (t), that is, from the ensemble by means of P x, t ð Þ ¼ d x ¼ X t ð Þ ð Þ h i , (1) where hÁi denotes ensemble averaging and d(Á) is the delta function. We assume that at time t 0 the process is distributed like u. That is, the function u(x) describes the initial probability density of the random variable X and we have P(x,t 0) = u(x). In general, the evolution of P depends on how the process is distributed at initial time t 0. In order to emphasize this point, we will use in what follows the notation P(x,t;u). That is, we interpret Eq. (1) as a conditional probability density with the constraint given by the initial distribution u: P x, t; u ð Þ¼ d x ¼ X t ð Þ ð Þ h i d xÀX t 0