Coagulation equations with particle emission

TL;DR

This paper develops a kinetic model for particle coagulation with emission, establishing existence, uniqueness, and explicit formulas for cluster sizes, and explores gelation phenomena through numerical experiments.

Contribution

It introduces a novel coagulation model with particle emission, providing mathematical analysis and explicit formulas, extending classical equations to include emission effects.

Findings

Existence and uniqueness for systems with bounded or unbounded cluster sizes.

Explicit formulas for cluster sizes and moments in certain regimes.

Numerical experiments suggest formulas hold until gelation time.

Abstract

We present a model for sticky particles in which cluster sizes after a reaction have $\ell$ fewer total particles than the sum of their reactants. The finite particle system is modeled as a Markov process under a mean-field assumption for selecting reactants. The limiting kinetic equations form an infinite system of nonlinear differential equations similar to the Smoluchowski coagulation equations with multiplicative kernel. We show existence and uniqueness for systems whose cluster sizes are either bounded above or below by the emission size $\ell$. When clusters have at most $\ell$ particles, well-posedness can be extended until an exhaustion time in which certain cluster fractions vanish. For clusters with more than $\ell$ particles, we prove short-time well-posedness, along with explicit formulas for cluster sizes and moments. We also conduct numerical experiments which suggest these formulas hold until a gelation time, at which an infinite-sized cluster forms.