Catalytic Coagulation

We introduce an autocatalytic aggregation model in which the rate at which two clusters merge to form a cluster is controlled by the presence of a third "catalytic" cluster whose mass must equal to the mass of one of the reaction partners. The catalyst is unaffected by the joining event and is available to either participate in or catalyze subsequent reactions. This model is meant to mimic the self-replicating reactions that occur in models for the origin of life. We solve the kinetics of this catalytic coagulation model for the case of mass-independent rates and show that the total cluster density decays as $t^{-1/3}$, while the density of clusters of any fixed mass decays as $t^{-2/3}$. These behaviors contrast with the corresponding $t^{-1}$ and $t^{-2}$ scalings for classic aggregation. We extend our model to mass-dependent reaction rates, to situations where only "magic" mass clusters can catalyze reactions, and to include steady monomer input.