两组分Becker-Döring系统的长时间行为
Long Time Behaviour of the Two-Component Becker-Döring System
AI总结：
本文严格证明两组分Becker-Döring系统在细致平衡下所有初始条件存在最小化相对熵的解，通过二维离散对数Sobolev不等式确定弱*极限由初始质量及混合比决定，并改进单组分系统的长时间行为假设。
AI中文摘要：
经典的Becker-Döring方程描述了通过单体的聚集和碎片化形成团簇的过程。如果总质量是超临界的，则会形成越来越大的团簇，导致在长时间极限中质量的渐近损失。当团簇由两种类型的单体构成时，就产生了两组分Becker-Döring系统。在此，我们研究单组分系统的自然扩展，其中没有能量或熵离开或进入系统——即所谓的细致平衡假设。我们严格证明了所有初始条件在时间趋近无穷时都允许一个最小化相对熵的解。这表明，在弱*拓扑中的长时间极限是通过初始的I型和II型质量来选择的。此外，损失的I型质量的比例由极限点通过最大化结合能的I型和II型单体的混合比来确定。两组分系统的主要困难在于没有相对熵是弱*连续的，而这对于经典论证[2]至关重要。相反，我们的证明基于一个二维离散对数Sobolev不等式，该不等式在正确的时间尺度上界定了熵耗散。我们的方法还改进了当前用于确定单组分系统长时间行为的假设。
英文摘要：
The classical Becker-Döring equations describe the formation of clusters by aggregation and fragmentation of monomers. If the total amount of mass is supercritical, larger and larger clusters are formed, leading to an asymptotic loss of mass in the long time limit. The two-component Becker-Döring system arises when clusters are built from two types of monomers. Here, we study the natural extension of the one-component system, where no energy or entropy is leaving or entering the system - the so called detailed balance assumption. We rigorously prove that all initial conditions admit a solution minimising the relative entropy as time approaches infinity. This shows, that the long time limit in the weak* topology is selected through the initial Type I and Type II masses. Furthermore, the proportion of lost Type I mass is determined by the limit point via the mixing ratio of Type I and Type II monomers that maximises the binding energy. The main difficulty of the two-component system is that no relative entropy is weak* continuous, which is crucial for the classical argument [2]. Instead, our proof is based on a discrete two-dimensional logarithmic Sobolev inequality, that bounds the entropy dissipation on the correct timescale. Our approach also improves current assumptions to determine the long time behaviour for the one-component system.
