The discrete generalized exchange-driven system

TL;DR

This paper introduces a discrete model for generalized exchange-driven growth, extending existing models to include larger particle exchanges, and proves existence, conservation properties, and uniqueness of solutions under certain conditions.

Contribution

It develops a unified framework for exchange-driven systems with larger exchange sizes, establishing existence, conservation laws, and uniqueness results.

Findings

Existence of admissible solutions for the model.

Conservation of total particles and mass in isolated models.

Uniqueness of solutions under restrictive growth conditions.

Abstract

We study a discrete model for generalized exchange-driven growth in which the particle exchanged between two clusters is not limited to be of size one. This set of models include as special cases the usual exchange-driven growth system and the coagulation-fragmentation system with binary fragmentation. Under reasonable general condition on the rate coefficients we establish the existence of admissible solutions, meaning solutions that are obtained as appropriate limit of solutions to a finite-dimensional truncation of the infinite-dimensional ODE. For these solutions we prove that, in the class of models we call isolated both the total number of particles and the total mass are conserved, whereas in those models we can non-isolated only the mass is conserved. Additionally, under more restrictive growth conditions for the rate equations we obtain uniqueness of solutions to the initial value problems.