论文标题
波动力学理论的严格理由
Rigorous justification of the wave kinetic theory
论文作者
论文摘要
该说明性注释的主要目的是简要说明数学波动力学理论的最新发展。在审查了物理理论之后，我们解释了缩放定律概念的重要性，该概念规定了动力学极限时渐近参数之间的关系。据我们所知，这对动力学近似设定了一些自然局限性。然后，我们描述了我们最近的即将到来的作品，这些作品赋予了自然动力学时间尺度上波动力学理论的第一个完整，数学上严格的衍生。关键的新成分是对图解扩展的微妙分析，该分析允许a）在任意的大图中发现高度详细的取消，b）克服了从扩张和问题的批判性中遇到的困难。本说明中提到的结果出现在我们最近的作品[16，17]以及[18]中的即将发生的结果中。
The main purpose of this expository note is to give a short account of the recent developments in mathematical wave kinetic theory. After reviewing the physical theory, we explain the importance of the notion of a scaling law, which dictates the relation between the asymptotic parameters as the kinetic limit is taken. This sets some natural limitations on the kinetic approximation that were not precisely understood in the literature as far as we know. We then describe our recent and upcoming works that give the first full, mathematically rigorous, derivation of the wave kinetic theory at the natural kinetic timescale. The key new ingredient is a delicate analysis of the diagrammatic expansion that allows to a) uncover highly elaborate cancellations at arbitrary large order of diagrams, and b) overcome difficulties coming from factorial divergences in the expansion and the criticality of the problem. The results mentioned in this note appear in our recent works [16, 17] as well as an upcoming one in [18].
