Reaction Kinetics of Clustered Impurities

Abstract

We study the density of clustered immobile reactants in the diffusion-controlled single species annihilation. An initial state in which these impurities occupy a subspace of codimension d&#39; leads to a substantial enhancement of their survival probability. The Smoluchowski rate theory suggests that the codimensionality plays a crucial role in determining the long time behavior. The system undergoes a transition at d&#39;=2. For d&#39;<2, a finite fraction of the impurities survive: ni(t) ~ ni(infinity)+const x log(t)/t^{1/2} for d=2 and ni(t) ~ ni(infinity)+const/t^{1/2} for d>2. Above this critical codimension, d&#39;>=2, the subspace decays indefinitely. At the critical codimension, inverse logarithmic decay occurs, ni(t) ~ log(t)^{-a(d,d&#39;)}. Above the critical codimension, the decay is algebraic ni(t) ~ t^{-a(d,d&#39;)}. In general, the exponents governing the long time behavior depend on the dimension as well as the codimension.