Convergence of cluster coagulation dynamics

We study hydrodynamic limits of the cluster coagulation model; a coagulation model introduced by Norris [$\mathit{\text{Comm.}}\mathit{\text{Math.}}\mathit{\text{Phys.}}$, 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 $K(x,y,\mathrm{d}z)$, 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. We derive sufficient criteria for trajectories associated with this process to concentrate among solutions of a generalisation of the $\mathit{\text{Flory}}\mathit{\text{equation}}$, and, in some special cases, by means of a uniqueness result for solutions of this equation, prove a weak law of large numbers. This multi-type Flory equation is associated with $\mathit{\text{conserved}}\mathit{\text{quantities}}$ associated with the process, which may encode different information to conservation of mass (for example, conservation of centre of mass in spatial models). We also apply criteria for gelation in this process to derive sufficient criteria 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.