Statistical properties of nearest-neighbor distances at an imperfect trap

There have been a number of recent investigations of statistical properties of the nearest-neighbor distance of the closest diffusing particle in the presence of a trap. These have been shown to be useful characterizations of self-organizing properties of simple binary reactions of the form A+A A or A+A 0. In this paper we extend our results to diffusion in the presence of an imperfect trap in one and three dimensions. The imperfect trap is modeled in terms of a radiation boundary condition. Our exact solution permits one to follow the transition in the shape of the probability density for the nearest-neighbor distance from the exponential in one dimension corresponding to total reflection, to the skewed Gaussian form for perfect reaction. Similar results are given for the case of a reactive sphere in the presence of mobile particles in three dimensions.