Convergence of cluster coagulation dynamics

TL;DR

This paper investigates the hydrodynamic limits of a general cluster coagulation model, deriving conditions for solutions to follow a generalized Flory equation, and explores gelation phenomena and conserved quantities in the process.

Contribution

It introduces criteria linking the coagulation process to a multi-type Flory equation and establishes conditions for convergence, uniqueness, and gelation in this generalized framework.

Findings

Derived criteria for trajectories to concentrate on solutions of the generalized Flory equation.

Proved a weak law of large numbers for the coagulation process under certain conditions.

Identified conditions under which gelation occurs and how it affects conserved quantities.

Abstract

We study hydrodynamic limits of the cluster coagulation model; a coagulation model introduced by Norris [$\textit{Comm. Math. Phys.}$, 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 $K(x, y, \mathrm{d} z)$, 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 $\textit{Flory equation}$, 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 $\textit{conserved quantities}$ 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.