The Scaling Attractor and Ultimate Dynamics for Smoluchowski’s Coagulation Equations

Abstract

We describe a basic framework for studying dynamic scaling that has roots in dynamical systems and probability theory. Within this framework, we study Smoluchowski’s coagulation equation for the three simplest rate kernels K(x,y)=2, x+y and xy. In another work, we classified all self-similar solutions and all universality classes (domains of attraction) for scaling limits under weak convergence (Menon and Pego in Commun. Pure Appl. Math. 57, 1197–1232, [2004]). Here we add to this a complete description of the set of all limit points of solutions modulo scaling (the scaling attractor) and the dynamics on this limit set (the ultimate dynamics). A key tool is Bertoin’s Lévy-Khintchine representation formula for eternal solutions of Smoluchowski’s equation (Bertoin in Ann. Appl. Probab. 12, 547–564, [2002a]). This representation linearizes the dynamics on the scaling attractor, revealing these dynamics to be conjugate to a continuous dilation, and chaotic in a classical sense. Furthermore, our study of scaling limits explains how Smoluchowski dynamics “compactifies” in a natural way that accounts for clusters of zero and infinite size (dust and gel).