Random walk on spheres method for solving drift-diffusion problems

摘要

The well-known random walk on spheres method (RWS) for the Laplace equation is here extended to drift-diffusion problems. First we derive a generalized spherical mean value relation which is an extension of the classical integral mean value relation for the Laplace equation. Next we give a probabilistic interpretation of the kernel. The distribution on the sphere generated by this kernel is then related to the von Mises-Fisher distribution on the sphere which can be efficiently simulated. The rigorous expressions are given for the case of constant velocity drift, but the algorithm is then extended to solve drift-diffusion problems with arbitrary varying drift velocity vector. Applications to cathodoluminescence and EBIC imaging of defects and dislocations in semiconductors are discussed.