论文标题
具有广义Lennard的一维离散系统的能量缩放和渐近性能 - JONES $（M，N）$相互作用
Energy Scaling and Asymptotic Properties of One-Dimensional Discrete System with Generalized Lennard--Jones $(m,n)$ Interaction
论文作者
论文摘要
众所周知，弹性效应会导致表面不稳定。在本文中，我们分析了一个一维离散系统，该系统可以揭示类似于“替代”现象的模式形成机制，用于替代表面上的外延生长。表面步骤以远程成对相互作用的形式受到一般的Lennard-Jones（LJ）型电势的形式。它的特点是两个指数$ m $和$ n $，描述了在小和大距离处的相互作用潜力的奇异和腐烂行为，此后称为广义lj $（m，n）$势。我们提供了对步骤配置的渐近性能和最小能量的值的系统分析，尤其是它们对$ M $和$ n $的依赖性以及指示交互范围的附加参数$α$。我们的结果表明，束和非束缚制度之间存在相变。此外，我们的某些陈述适用于能源的任何关键点，不一定是最小化的人。这项工作扩展了[Luo等，Siam MMS，2016]的技术和结果，该技术集中在LJ（0,2）电位的情况下（起源于步骤之间的弹性力单极和偶极相互作用）。作为副产品，我们的结果也导致了一个众所周知的事实，即经典的LJ（6,12）潜力并未显示出阶梯式现象。
It is well known that elastic effects can cause surface instability. In this paper, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the "step-bunching" phenomena for epitaxial growth on vicinal surfaces. The surface steps are subject to long-range pairwise interactions taking the form of a general Lennard--Jones (LJ) type potential. It is characterized by two exponents $m$ and $n$ describing the singular and decaying behaviors of the interacting potential at small and large distances, and henceforth are called generalized LJ $(m,n)$ potential. We provide a systematic analysis of the asymptotic properties of the step configurations and the value of the minimum energy, in particular, their dependence on $m$ and $n$ and an additional parameter $α$ indicating the interaction range. Our results show that there is a phase transition between the bunching and non-bunching regimes. Moreover, some of our statements are applicable for any critical points of the energy, not necessarily minimizers. This work extends the technique and results of [Luo et al, SIAM MMS, 2016] which concentrates on the case of LJ (0,2) potential (originated from the elastic force monopole and dipole interactions between the steps). As a by-product, our result also leads to the well-known fact that the classical LJ (6,12) potential does not demonstrate step-bunching type phenomena.
