Nonlinear Brownian Motion. Unified Description of Kinetic, Hydrodynamic and Diffusion Processes

Abstract

In both the kinetic theory and in the theory of Brownian motion we dealt with the transition from the kinetic description in six-dimensional phase space to a more coarse description in the space of three dimensions. In Ch. 11, for example, it was the transition from the Boltzmann equation to the comparatively simple equations of gas dynamics. Then, in the theory of Brownian motion, we went over from Fokker — Planck equation for the distribution function f(r,v,t) to the simpler Einstein — Smoluchowski equation in the spatial distribution f(r,t). As indicated in Sect. 15.10, this transition corresponds to the simplified hydrodynamic description of Brownian motion