The kinetic boundary layer for the Fokker-Planck equation: A Brownian particle in a uniform field

We study stationary solutions of the Fokker-Planck equation for velocity and position of a Brownian particle in the presence of a constant external field and a completely or selectively absorbing plane wall. Adapting a procedure used earlier for the field-free case, we determine analytically a set of boundary layer solutions and combine them numerically with the Chapman-Enskog solutions to satisfy various boundary conditions at the wall, albeit approximately. Various aspects of the solutions so obtained are presented, in particular the profiles of the density and the kinetic energy density in the boundary layer. The most important quantity that can be extracted from them is the effective absorption rate at the wall corresponding to complete or partial absorption of particles hitting the wall. This quantity is noticeably increased by the presence of the external field; as in the field-free case it also depends on the reflection mechanism adopted for the non-absorbed particles. The trends in the absorption rate, and even their numerical values, are predicted remarkably well by a simple analytical approximation, a self-consistent modification of the usual transition state theory for chemical reactions.