Asymptotic behavior of diffusing atoms in a region with absorbing boundaries

The present paper calculates asymptotic behavior of the Green function of the Fokker–Plank equation, describing atom diffusion in a two-dimensional restricted isotropic medium with absorbing boundaries. The Green function is expressed in terms of the path integral over all possibble random walks of diffusion atoms inside the medium. To find the Green function asymptotic behavior we reduce the two-dimensional problem to atom uniform migration along an optimal path and one-dimensional random walks around it. Owing to this the Green function is presented as a product of two cofactors. The first factor corresponding to the migration along the optimal path is directly calculated by integrating. The second factor associated with the latter random walks may be obtained by solving the one-dimensional Fokker–Plank equation with a certain boundary condition, the particular form of which is determined by properties of both the medium boundary and the optimal path. Thus finding the asymptotic solution of this equation we carry the considered problem to completion.