The Redner - Ben-Avraham - Kahng cluster system

TL;DR

This paper analyzes a unique coagulation model where cluster reactions produce smaller clusters, proving existence, uniqueness, and properties of solutions, and exploring their long-term and scaling behaviors.

Contribution

It establishes mathematical foundations for the Redner-Ben-Avraham-Kahng cluster system, including existence, uniqueness, and long-term behavior analysis.

Findings

Proved existence and uniqueness of solutions.

Analyzed long-time and scaling behavior of solutions.

Identified invariance properties of the system.

Abstract

We consider a coagulation model first introduced by Redner, Ben-Avraham and Krapivsky in [Redner, Ben-Avraham, Kahng: Kinetics of 'cluster eating', J. Phys. A: Math. Gen., 20 (1987), 1231-1238], the main feature of which is that the reaction between a j-cluster and a k-cluster results in the creation of a |j-k|-cluster, and not, as in Smoluchowski's model, of a (j+k)-cluster. In this paper we prove existence and uniqueness of solutions under reasonably general conditions on the coagulation coefficients, and we also establish differenciability properties and continuous dependence of solutions. Some interesting invariance properties are also proved. Finally, we study the long-time behaviour of solutions, and also present a preliminary analysis of their scaling behaviour.