Mathematical Theory of Exchange-driven Growth

TL;DR

This paper establishes the fundamental mathematical properties of exchange-driven growth models, analyzing solution existence and uniqueness for different kernel classes, and conjecturing gelation phenomena in intermediate regimes.

Contribution

It provides the first rigorous analysis of the mean field kinetic equations for exchange-driven growth, distinguishing behaviors based on kernel symmetry and growth rates.

Findings

Global existence for non-symmetric kernels with $K(j,k) \\leq Cjk$

Existence loss for kernels with $K(j,k) \\geq Cj^{\beta}$, $\beta>1$

Symmetric kernels with $K(j,k) \\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ have global solutions under certain conditions

Abstract

Exchange-driven growth is a process in which pairs of clusters interact and exchange a single unit of mass. The rate of exchange is given by an interaction kernel $K(j,k)$ which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field kinetic equations of this process for the first time. We find two different classes of behaviour depending on whether $K(j,k)$ is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying $K(j,k)\leq Cjk$. This result is optimal in the sense that we show for a large class of initial conditions with kernels satisfying $K(j,k)\geq Cj^{\beta}$ ($\beta>1)$ the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for $K(j,k)\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ ($\mu,\nu\leq2,$ $\mu+\nu\leq3),$ while existence is lost for $K(j,k)\geq Cj^{\beta}$ ($\beta>2).$ In the intermediate regime $3<\mu+\nu\leq4,$ we can only show local existence. We conjecture that the intermediate regime exhibits finite-time gelation in accordance with the heuristic results obtained for particular kernels.