Scaling Theory and Exactly Solved Models In the Kinetics of Irreversible Aggregation

TL;DR

This paper reviews the scaling theory of irreversible aggregation, extends it to complex scenarios, discusses gelation, and examines exactly solved models to evaluate the validity of the scaling hypothesis.

Contribution

It provides a comprehensive formulation of the scaling hypothesis, extends the theory to complex reactions, and analyzes exactly solved models to test the hypothesis's applicability.

Findings

Scaling hypothesis holds in most models analyzed.

Counterexamples exist for stronger forms of the hypothesis.

The theory explains crossover phenomena and gelation effects.

Abstract

The scaling theory of irreversible aggregation is discussed in some detail. First, we review the general theory in the simplest case of binary reactions. We then extend consideration to ternary reactions, multispecies aggregation, inhomogeneous situations with arbitrary size dependent diffusion constants as well as arbitrary production terms. A precise formulation of the scaling hypothesis is given as well as a general theory of crossover phenomena. The consequences of this definition are described at length. The specific issues arising in the case in which an infinite cluster forms at finite times (the so-called gelling case) are discussed, in order to address discrepancies between theory and recent numerical work. Finally, a large number of exactly solved models are reviewed extensively with a view to pointing out precisely in which sense the scaling hypothesis holds in these various models. It is shown that the specific definition given here will give good results for almost all cases. On the other hand, we show that it is usually possible to find counterexamples to stronger formulations of the scaling hypothesis.