Applications of Grassmannian flows to coagulation systems

TL;DR

This paper shows how various coagulation models, including Smoluchowski-type equations, can be represented as Grassmannian flows, revealing their integrable structure and broadening the understanding of their mathematical properties.

Contribution

It demonstrates that a wide class of coagulation models can be realized as Grassmannian flows, establishing their integrability and connecting different models through this geometric framework.

Findings

Smoluchowski-type equations are realizable as Grassmannian flows.

Several models like Gallay--Mielke and Derrida--Retaux are also realizable.

Additive and multiplicative kernels correspond to rank-one flows.

Abstract

We demonstrate how many classes of Smoluchowski-type coagulation models can be realised as multiplicative Grassmannian flows and are therefore linearisable, and thus integrable in this sense. First, we prove that a general Smoluchowski-type equation with a constant frequency kernel, that encompasses a large class of such models, is realisable as a multiplicative Grassmannian flow. Second, we establish that several other related constant kernel models can also be realised as such. These include: the Gallay--Mielke coarsening model; the Derrida--Retaux depinning transition model and a general mutliple merger coagulation model. Third, we show how the additive and multiplicative frequency kernel cases can be realised as rank-one analytic Grassmannian flows.