Nearest-neighbor distances at an imperfect trap in two dimensions

The distance between the nearest diffusing particle to a single trap has been shown to be a useful characterization of self-segregation in low-dimensional reaction kinetics. Recent studies in two dimensions show that the average of this distance increases asymptotically as (ln t${)}^{1/2}$. In this paper we study a two-dimensional system in which the trap is an imperfect one, modeled in terms of the radiation boundary condition. Our exact solution shows that there exists a spatial and temporal dependence on the trap absorptivity, but the asymptotic time dependence remains unchanged. These results enable us to follow the crossover between the two limiting cases, namely, perfect trapping and total reflection. Analytical expressions are also given for the concentration profile and the reaction rate at the trap.