Large time behavior of exchange-driven growth

Abstract

Exchange-driven growth (EDG) is a model 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 + εβ j α k. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass ρs above which there is no equilibrium. We prove that if the system has an initial mass below ρs then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above ρs then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.