Reaction kinetics of cluster impurities

We study the kinetics of clustered immobile reactants in diffusion-controlled single-species annihilation. We consider the initial conditions where the immobile reactants occupy a subspace of dimension ${\mathit{d}}_{\mathit{i}}$, while the rest of the d-dimensional space is occupied by identical mobile particles. The Smoluchowski rate theory suggests that the immobile reactant concentration s(t) exhibits interesting behavior as a function of the codimension, d¯≡d-${\mathit{d}}_{\mathit{i}}$. This survival probability undergoes a survival-to-extinction transition at d${\mathrm{¯}}_{\mathit{c}}$=2. For d¯d${\mathrm{¯}}_{\mathit{c}}$, a finite fraction of the immobile reactants survives, while for d¯≥d${\mathrm{¯}}_{\mathit{c}}$, s(t) decays indefinitely. The corresponding asymptotic properties of the concentration are discussed. The theoretical predictions are verified by numerical simulations in two and three dimensions. © 1996 The American Physical Society.