EXACT SOLUTION OF THE ONE-DIMENSIONAL IMMOBILE TRAPPING PROBLEM WITH AND WITHOUT SOURCES

We consider the one-dimensional trapping problem A+SS when the traps S as well as the reactants A are immobile. The trapping rate k(r) depends on the distance r between the reactant and the trap, and is characterized by an effective reaction radius r0, which is a measure of the first moment of k(r). We find that the decay of an initial density of A particles is exponential in time at short times and an inverse power law at long times. In the presence of sources, we find that a steady state exists only if the effective reaction radius is larger than half of the average distance between sinks, and we find an inverse-power-law approach to the steady state. If this condition is not met, then there is an unbounded accumulation of A particles in regions that can not be effectively depleted by the reaction. The growth of the density in this case is of power-law form. © 1990 The American Physical Society.