Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems

Abstract

Abstract We suggest in this paper a Random Walk on Spheres (RWS) method for solving transient drift-diffusion-reaction problems which is an extension of our algorithm we developed recently [26] for solving steady-state drift-diffusion problems. Both two-dimensional and three-dimensional problems are solved. Survival probability, first passage time and the exit position for a sphere (disc) of the drift-diffusion-reaction process are explicitly derived from a generalized spherical integral relation we prove both for two-dimensional and three-dimensional problems. The distribution of the exit position on the sphere has the form of the von Mises–Fisher distribution which can be simulated efficiently. Rigorous expressions are derived in the case of constant velocity drift, but the algorithm is then extended to solve drift-diffusion-reaction problems with arbitrary varying drift velocity vector. The method can efficiently be applied to calculate the fluxes of the solution to any part of the boundary. This can be done by applying a reciprocity theorem which we prove here for the drift-diffusion-reaction problems with general boundary conditions. Applications of this approach to methods of cathodoluminescence (CL) and electron beam induced current (EBIC) imaging of defects and dislocations in semiconductors are presented.