Target and trapping problems: From the ballistic to the diffusive regime

The target and trapping problems refer to the reaction A+B-->B. In the targ et problem a single particle is fixed in space and surrounded by B's allowe d to move freely, while in the trapping problem the B's are fixed and the A is able to move. Exact solutions are found for both problems in the ballis tic regime, in all dimensions. We show that the solution of the target prob lem provides a mean-field approximation to the solution of the trapping pro blem, not only in the diffusive regime, but also for arbitrary noise. This approximate solution works well in the diffusive regime, but not when motio n is ballistic, since it breaks down at very early times. We show that the time-dependent rate coefficients in both the target and trapping problems r emain finite at t=0 for arbitrarily strong noise intensities. This behavior is in contrast to the diffusion theory prediction Chat the coefficient div erges at t=0. A recently developed model that discretizes the velocity, all owing only three values, +/-nu and 0, is used to study the reaction kinetic s of both the crapping and target problems in one dimension over the entire range of noise intensities. The solutions are used to study the effects of noise intensity on the mean survival time. We show that in the target, pro blem this time decreases monotonically with increasing noise, while in the trapping problem this time exhibits a turnover behavior. We argue that a si milar turnover occurs in the one-dimensional trapping problem when particle motion is governed by a Langevin equation. (C) 1999 American Institute of Physics. [S0021-9606(99)51902-8].