Brownian motion in phase space

Abstract

In most treatments Brownian motion is considered either in position or in velocity space. While the description of Brownian motion as a Markov process via a master equation in phase space can, in principle, deal with the full problem, assumptions are usually introduced which lead to the formulation of the generalized Fokker-Planck equation. In the following we want to show how, in the field-free one-dimensional case, the transport equation for a time-dependent conditional average of an arbitrary physical quantity (depending on position and velocity coordinates) of a tagged particle in phase space can be derived and solved successively if the time- and position-independent moments of the transition probability can be expanded in terms of a parameter ${\ensuremath{\Omega}}^{\ensuremath{-}1}$. The transport equation can then be transformed into a set of $l$ linear partial differential equations, whose solutions provide the $l\mathrm{th}$ approximation in the expansion parameter ${\ensuremath{\Omega}}^{\ensuremath{-}1}$ to the full solution of the transport equation. The system of linear partial differential equations is closed in the sense that the solution of the $l\mathrm{th}$ partial differential equation is determined by the solutions of the $l\ensuremath{-}1$ partial differential equations. It turns out that the $l=0$ partial differential equation describes the macroscopic (nonfluctuating) motion of the tagged particle. It is discussed in which sense a bivariate Gaussian distribution describes the full Markov process and how the full phase-space treatment provides more insight into Brownian motion. It is shown how the special cases described by the Langevin equation and the Fokker-Planck equation are contained in the new method presented.