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

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).