Extended Brownian dynamics. II. Reactive, nonlinear diffusion

An improved version of the "extended Brownian dynamics" algorithm recently proposed by the authors [J. Chem. Phys. 75, 365 (1981)] is given. This Monte Carlo procedure for solving the one-dimensional Smoluchowski diffusion equation is statistically exact near a boundary for a constant force and approximately correct for a linear force. The improved algorithm is both more accurate and simpler than the earlier version. In addition, the algorithm is extended to include diffusion near a reactive boundary or in a reactive optical potential. The treatment of diffusion for nonlinear forces is conveniently handled by choosing the time for a single diffusive jump locally. The algorithm converges as this jump time approaches zero. The appropriate modifications necessary to treat diffusion between two (possibly reactive) boundaries or diffusion with a spatially varying diffusion coefficient are also given. Finally, it is shown how multidimensional diffusion in a spherically symmetric force field may be treated by the one-dimensional algorithm described here. As in the earlier paper, numerical results are presented and compared with analytical and numerical descriptions of the diffusion process to demonstrate the validity of the algorithm.