Density of nearest-neighbor distances in diffusion-controlled reactions at a single trap

We derive the probability density function for the nearest-neighbor distance of the closest Brownian particle to an isolated trap in one and three dimensions. The asymptotic (in time) density of the nearest-neighbor distance in one dimension is a time-dependent skewed Gaussian function and the mean value of this distance increases asymptotically as t14. The large-distance form of this function is a simple exponential. In three dimensions the analogous result in the presence of an absorbing sphere is shown to resemble the Hertz density closely at large distances. The asymptotic reaction rate goes as c (density) in three dimensions but as ct-12 in one dimension. The results are related to exciton fusion and reaction kinetics.