Cluster Coagulation

Abstract:

We introduce a general class of coagulation models, where clusters of given types may coagulate in more than one way and where the rate at which this happens may depend on the cluster types. In the continuum version of these models there is a generalization of Smoluchowski's coagulation equation. We introduce a notion of strong solution for this equation and prove the existence of a maximal strong solution, which while it persists is the only solution. When the total rate of coagulation for particles is bounded above and below by constant multiples of the product of their masses, we show that the maximal strong solution coincides with the maximal mass-conserving solution and does not persist for all time. Thus, for these models, loss of mass (to infinity) coincides with divergence of the second moment of the mass distribution and takes place in a finite time. When the total rate of coagulation of large particles is proportional to their masses, we establish the existence and uniqueness of solutions for all time. In a restricted class of “polymer” models, we allow coagulation of weighted shapes in a finite number of ways. For this class we establish a discrete approximation scheme for the continuum dynamics.

For each continuum coagulation model, there is a corresponding finite-particle-number stochastic model. We show that, in the polymer case, which includes the case of simple mass coalescence, as the number of particles becomes large, the stochastic model converges weakly to the deterministic continuum model, at an exponential rate.