Steady, one-dimensional Brownian motion with an absorbing boundary

We consider the solution of the Fokker–Planck equation for the case of steady, one-dimensional flow with prescribed flux at the outer boundary and complete absorption at the inner boundary. An approximate solution is obtained using a modified version of the bimodal Maxwellian moment method due to Lees, which has been used previously with success in treating boundary layer problems in the context of the Boltzmann equation. We obtain explicit results for the density in the physical space which is characterized by a boundary layer of order (inverse velocity relaxation time/thermal velocity)−1 and a Milne extrapolation length (distance beyond the boundary at which the extrapolated asymptotic value of the density is zero) of 1.44 in appropriately normalized units. This latter value compares surprisingly well with recent analytical–numerical results which find a value of 1.46 for this quantity.