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

Abstract

The target and trapping problems refer to the reaction A+B→B. In the target problem a single A particle is fixed in space and surrounded by B’s allowed 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 ballistic regime, in all dimensions. We show that the solution of the target problem provides a mean-field approximation to the solution of the trapping problem, not only in the diffusive regime, but also for arbitrary noise. This approximate solution works well in the diffusive regime, but not when motion 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 remain finite at t=0 for arbitrarily strong noise intensities. This behavior is in contrast to the diffusion theory prediction that the coefficient diverges at t=0. A recently developed model that discretizes the velocity, allowing only three values, ±v and 0, is used to study the reaction kinetics of both the trapping 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 problem this time decreases monotonically with increasing noise, while in the trapping problem this time exhibits a turnover behavior. We argue that a similar turnover occurs in the one-dimensional trapping problem when particle motion is governed by a Langevin equation.