On the rate of absorption of Brownian particles by a black sphere: The connection between the Fokker–Planck equation and the diffusion equation

It is shown that, if one takes the first two moments of the Fokker–Planck equation, one obtains, in the lowest order, the telegraph equation irrespective of whether one uses a full-range or a half-range decomposition of the distribution function; it is pointed out that the exact boundary condition, according to which the distribution function for emerging particles vanishes at the surface of an absorbing (black) sphere of radius R, may be replaced, in either case, by Marshak’s boundary condition j+(R, t) =0, where j+ is the outward radial current. Drawing on Wilemski’s work, the second moment equation is finally replaced by Fick’s law, obtaining thereby the diffusion equation and the radiation boundary condition.