Large Time Behavior of Exchange-driven Growth

TL;DR

This paper analyzes the long-term behavior of exchange-driven growth models, showing conditions under which systems reach equilibrium or exhibit convergence, depending on kernel properties and initial mass.

Contribution

It provides a rigorous mathematical analysis of EDG equations' large time behavior, including equilibrium existence and convergence rates for different kernel types.

Findings

Existence of equilibrium solutions up to a critical mass for type I kernels.

Weak convergence to critical equilibrium if initial mass exceeds critical mass.

Exponential convergence results for type II kernels under certain conditions.

Abstract

Exchange-driven growth (EDG) is a process in which pairs of clusters interact by exchanging single unit with a rate given by a kernel $K(j,k)$. Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel $(i)$ $K(j,k)=b_{j}a_{k}$ and $(ii)$ $K(j,k)=ja_{k}+b_{j} +\varepsilon\beta_{j}\alpha_{k}$. For type I kernels, under the detailed balance assumption, we show that the system admits equilibrium solutions up to a critical mass $\rho_{s}$ above which there is no equilibrium. We prove that if the system has an initial mass above $\rho_{s}$ then the solutions converge to critical equilibrium distribution in a weak sense while strong convergence can be shown when initial mass is below $\rho_{s}$. For type II kernels, we make no assumption of detailed balance and equilibrium is obtained via a contraction property. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we show exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.