Nearest-neighbor distances in diffusion-controlled reactions modelled by a single mobile trap

We consider a system consisting of an infinite number of identical particles on a lattice initially uniformly distributed, which diffuese in the presence of a singke mobile trap and ask for the time-dependent behavior of the distance of the trap from the nearest particle. This quantity is a measure of the tendency of the system to self-segregate. We show, by a simulation incorporating the exact enumeration method, that in one dimension the expected distance 〈L(t)〉 scales as 〈L(t)〉≈t^α as t→∞, where the exponent α depends only on the ratio of the diffusion constant. A heuristic expression for α is suggested, analogous to a rigorous exponent found by ben-Avraham for a similar but not identical problem. The flux into the trap is found to vary as t^{- 1 2} independent of the diffusion constants.