On Smoluchowski Equations for Coagulation Processes with Multiple Absorbing States

Smoluchowski coagulation equations propose a model for the stochastic time evolution of a particles population in which particle clusters merge to form larger clusters, at some given rates. These equations represent the dynamics of the expected cluster size distribution. Since Smoluchowski equations were not derived as a rigorous description of the underlying stochastic process, their quality in this context is not obvious. Here, we consider the case of a finite particles population and raise the following question: to what limit do the solutions of Smoluchowski equations converge as t→ ∞? In particular, we are concerned with the case where the population size is N and the coagulation rates restrict the maximal group sizes to D. For D = N, the stochastic process has only one absorbing state, but if D< N it may have many absorbing states. We demonstrate here that when the D ~ N, the solutions of Smoluchowski equations do not converge, as t→ ∞, to the expected cluster size distribution, but when D « N the convergence is to a limit which is close to the exact solution.