A model for coagulation with mating

TL;DR

This paper introduces a new coagulation model with male and female arms, analyzes its solutions using PDEs and generating functions, and connects it to Galton-Watson trees, providing both theoretical and microscopic process insights.

Contribution

It develops a novel coagulation model with mating arms, solves the associated nonlinear PDEs, and links the model to Galton-Watson trees and microscopic Marcus-Lushnikov processes.

Findings

Unique solutions exist up to a critical time.

Explicit solutions can be obtained using Lagrange Inversion.

Convergence to arm-disappearance state and relation to Galton-Watson trees.

Abstract

We consider in this work a model for aggregation, where the coalescing particles initially have a certain number of potential links (called arms) which are used to perform coagulations. There are two types of arms, male and female, and two particles may coagulate only if one has an available male arm, and the other has an available female arm. After a coagulation, the used arms are no longer available. We are interested in the concentrations of the different types of particles, which are governed by a modification of Smoluchowski's coagulation equation -- that is, an infinite system of nonlinear differential equations. Using generating functions and solving a nonlinear PDE, we show that, up to some critical time, there is a unique solution to this equation. The Lagrange Inversion Formula allows in some cases to obtain explicit solutions, and to relate our model to two recent models for limited aggregation. We also show that, whenever the critical time is infinite, the concentrations converge to a state where all arms have disappeared, and the distribution of the masses is related to the law of the size of some two-type Galton-Watson tree. Finally, we consider a microscopic model for coagulation: we construct a sequence of Marcus-Lushnikov processes, and show that it converges, before the critical time, to the solution of our modified Smoluchowski's equation.