Cluster renormalization in the Becker-Doring equations

TL;DR

This paper introduces a renormalization approach to the Becker-Doring equations, revealing a dynamical structure that classifies systems into three groups and aligns with previous asymptotic solutions, advancing understanding of cluster formation.

Contribution

It applies a novel coarse-graining and renormalization framework to the Becker-Doring model, uncovering a dynamical structure and classifying system behaviors.

Findings

Identification of nine archetypal systems

Classification into three behavior groups

Agreement with previous asymptotic solutions

Abstract

We apply ideas from renormalization theory to models of cluster formation in nucleation and growth processes. We study a simple case of the Becker-Doring system of equations and show how a novel coarse-graining procedure applied to the cluster aggregation space affects the coagulation and fragmentation rate coefficients. A dynamical renormalization structure is found to underlie the Becker-Doring equations, nine archetypal systems are identified, and their behaviour is analysed in detail. These architypal systems divide into three distinct groups: coagulation-dominated systems, fragmentation-dominated systems and those systems where the two processes are balanced. The dynamical behaviour obtained for these is found to be in agreement with certain fine-grained solutions previously obtained by asymptotic methods. This work opens the way for the application of renormalization ideas to a wide range of non-equilibrium physicochemical processes, some of which we have previously modelled on the basis of the Becker-Doring equations.