Gelation and hydrodynamic limits in cluster coagulation processes

Magnanini Elena, WIAS Berlin

A coagulation process describes the behavior of a particle system where families of particles (or clusters) merge as time passes. Specifically, we consider the so-called cluster coagulation model introduced by Norris in 2000. One of the main differences w.r.t. the classical Marcus-Lushnikov process is that this model is able to incorporate various features (not only the mass) in the evolution of clusters, such as their locations. The coagulation mechanism is regulated by a kernel, that is a positive symmetric function of the two clusters. Depending on the choice of this function, clusters whose masses are on a significantly larger scale than most others in the system, could emerge in finite time. This kind of phase transition is known as gelation. We provide sufficient criteria for gelation extending (and sometimes improving) results related to the classical Marcus-Lushnikov process. In some special cases the cluster coagulation process can be coupled with an inhomogeneous random graph; in this context we are able to identify an upper bound for the gelation time in terms of the critical parameter that determines the appearance of a giant component in the inhomogeneous random graph. We complete our analysis by showing that the trajectories associated with the process concentrate on solutions of an infinite system of measure valued differential equations that generalize the Flory equation. Joint work with L. Andreis and T. Iyer.

Area: CS27 - Particle systems with spatial and self-interactions (Alice Callegaro)

Keywords: Cluster coagulation, Marcus-Luschnikov process, Flory equation, gelation, inhomogeneous random graphs

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