The kinetic boundary layer for the Fokker-Planck equation: Selectively absorbing boundaries

Abstract

We study the joint distribution function for position and velocity of a Brownian particle near a wall. The wall absorbs all particles that hit it with sufficiently high velocity and reflects all slower ones, either specularly or diffusely. We determine in particular stationary distributions in the absence of external forces. Appreciable deviations from local equilibrium occur in a kinetic boundary layer near the wall; its details depend strongly on the way in which the slow particles are reflected. The resulting effective absorption rate is calculated and compared with the result of approximations analogous to the transition state theory of chemical reactions. The method used is a generalization of the one used in an earlier paper for the case of a completely absorbing wall; a numerical algorithm based on an expansion of the distribution function in terms of a presumably complete set of boundary layer solutions.