Rigorous results in the scaling theory of irreversible aggregation kinetics

TL;DR

This paper rigorously analyzes the scaling behavior of irreversible aggregation kinetics, providing proofs for the existence of scaling solutions in systems with mass conservation and gelation, supported by numerical evidence.

Contribution

It offers the first rigorous proofs of scaling solutions in aggregation systems, including those with gelation, advancing the theoretical understanding of these processes.

Findings

Rigorous proof of scaling solutions with conserved mass

Numerical evidence for scaling in gelation systems

Explicit examples of scaling behavior in gelation phenomena

Abstract

The kinetic equations describing irreversible aggregation and the scaling approach developed to describe them in the limit of large times and large sizes are tersely reviewed. Next, a system is considered in which aggregates can only react with aggregates of their own size. The existence of a scaling solution of the kinetic equations can then be shown rigorously in the case in which the total mass of the system is conserved. A large number of detailed properties of the solution, previously predicted by qualitative arguments, can be shown rigorously as well in this system. In the case in which gelation occurs, some sketchy rigorous results are shown, and numerical evidence for the existence of a scaling solution is presented. These are the first explicit examples of typical scaling behaviour for systems exhibiting gelation.