论文标题
与产品内核的交换驱动增长模型的自相似行为
Self-similar behavior of the exchange-driven growth model with product kernel
论文作者
论文摘要
我们研究了交换驱动的增长模型的自相似行为，该模型描述了一个过程，其中成对由整数数量的单体组成，通过单个单体的交换相互作用。交换速率由交互内内核$ k（k，l）$给出，该$取决于两个交互簇的尺寸$ k $和$ l $，并被认为是乘积$ $（k \，l）^λ$的$（k \，l）^λ$，for $λ\ in [0,2）$ in [0,2）$。我们严格地建立了Ben-Naim和Krapivsky [7]发现的自我相似概况的粗糙率和融合。对于显式内核，进化与非线性时间变化在正整数上的离散加权热方程有关。对于这个方程式，我们建立了一种新的加权NASH不平等，可产生缩放不变的衰减和连续性估计。加上将离散操作员与其连续类似物联系起来的替代身份，我们得出了加权热方程的离散到核缩放尺度的极限。在使用额外力矩估计的情况下，恢复时间变化，对线性方程的分析产生了交换驱动驱动的增长模型的粗略速率和自相似收敛。
We study the self-similar behavior of the exchange-driven growth model, which describes a process in which pairs of clusters, consisting of an integer number of monomers, interact through the exchange of a single monomer. The rate of exchange is given by an interaction kernel $K(k,l)$ which depends on the sizes $k$ and $l$ of the two interacting clusters and is assumed to be of product form $(k\,l)^λ$ for $λ\in [0,2)$. We rigorously establish the coarsening rates and convergence to the self-similar profile found by Ben-Naim and Krapivsky [7]. For the explicit kernel, the evolution is linked to a discrete weighted heat equation on the positive integers by a nonlinear time-change. For this equation, we establish a new weighted Nash inequality that yields scaling-invariant decay and continuity estimates. Together with a replacement identity that links the discrete operator to its continuous analog, we derive a discrete-to-continuum scaling limit for the weighted heat equation. Reverting the time-change under the use of additional moment estimates, the analysis of the linear equation yields coarsening rates and self-similar convergence of the exchange-driven growth model.
