Statistical properties of nearest-neighbor distances in a diffusion-reaction model

We consider the problem of a mobile trap (B) diffusing in the presence of stationary particles (A’s) which may be caught by the trap. In particular, we calculate the probability density for the distance of the nearest free A from the trap, as a function of time. The mean value of this quantity is found to vary as $t^{1/2}$ in one dimension, in contrast to the case in which the A’s are mobile and B is stationary, where the mean distance is proportional to $t^{1/4}$ for the analogous quantity. In two dimensions we find by simulation that the mean distance increases as lnt, while in three dimensions it is asymptotically constant.