Convergence of cluster coagulation dynamics

We study hydrodynamic limits of the cluster coagulation model; a coagulation model introduced by Norris [Comm. Math. Phys., 209(2):407-435 (2000)]. In this process, pairs of particles x, y in a measure space E, merge to form a single new particle z according to a transition kernel K(x, y, dz), in such a manner that a quantity, one may regard as the total mass of the system, is conserved. This model is general enough to incorporate various inhomogeneities in the evolution of clusters, for example, their shape, or their location in space. Our focus is concentration of trajectories from the underlying microscopic stochastic model on solutions of a generalised form of a macroscopic deterministic equation introduced by Norris. To prove tightness and convergence, we exploit conserved quantities (such as mass conservation or, in spatial models, preservation of the centre of mass) that also appear in the limiting equation. This approach does more than just generalise – it also weakens the assumptions needed to establish tightness and convergence for the classical Marcus–Lushnikov model. We also apply criteria for gelation in this process to derive sufficient conditions for this equation to exhibit gelling solutions. When this occurs, this multi-type Flory equation encodes, via the associated conserved property, the interaction between the gel and the finite size sol particles.