Propagation of chaos and the higher order statistics in the wave kinetic theory
Propagation of chaos and the higher order statistics in the wave kinetic theory
复制标题
波动理论中混沌的传播和高阶统计
DOI：
10.4171/jems/1488
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发表时间：
2021
影响因子：
2.6
通讯作者：
Z. Hani
中科院分区：
文献类型：
--
作者：
Yu Deng;Z. Hani
本文在多个方向上延续和扩展了[12]的结果，[12]给出了d≥3维的非线性Schrödinger （NLS）方程的波动动力学方程（WKE）的完整推导。波动动力学方程描述了NLS解的傅里叶模态的二阶矩在动力学时间尺度上的有效动力学，以及在系统的大小发散到无穷大和非线性强度消失到零的动力学极限下，根据特定的标度定律。在这里，我们研究了这些傅里叶模式的联合分布的行为，并推导了它们在动力学时间尺度上的有效极限动力学。特别地，我们证明了混沌在波设置中的传播：初始独立的傅立叶模在动力学极限中保持这种独立性。这些陈述是所有动力学理论形式推导的核心，可以追溯到玻尔兹曼（Stosszahlansatz）的工作。我们通过推导高傅立叶矩的渐近性来得到这一点，高傅立叶矩是由波动力学层次（WKH）的解给出的，具有因式初始数据。作为副产品，我们还为一般（不一定是分解的）初始数据提供了这种层次结构的严格证明。我们同时处理高斯和非高斯初始分布。在高斯条件下，我们证明了高斯性的传播性，证明了渐近分布在极限处保持了初始数据的高斯性。在非高斯情况下，我们推导了高阶矩的极限方程，以及解的密度函数（PDF）。我们证明的一些结果是在物理文献中推测出来的，其他的似乎是新的。给出了动力学极限解的统计量的完整描述。
This manuscript continues and extends in various directions the result in [12], which gave a full derivation of the wave kinetic equation (WKE) from the nonlinear Schrödinger (NLS) equation in dimensions d ≥ 3. The wave kinetic equation describes the effective dynamics of the second moments of the Fourier modes of the NLS solution at the kinetic timescale, and in the kinetic limit in which the size of the system diverges to infinity and the strength of the nonlinearity vanishes to zero, according to a specified scaling law. Here, we investigate the behavior of the joint distribution of these Fourier modes and derive their effective limit dynamics at the kinetic timescale. In particular, we prove propagation of chaos in the wave setting: initially independent Fourier modes retain this independence in the kinetic limit. Such statements are central to the formal derivations of all kinetic theories, dating back to the work of Boltzmann (Stosszahlansatz). We obtain this by deriving the asymptotics of the higher Fourier moments, which are given by solutions of the wave kinetic heirarchy (WKH) with factorized initial data. As a byproduct, we also provide a rigorous justification of this hierarchy for general (not necessarily factorized) initial data. We treat both Gaussian and non-Gaussian initial distributions. In the Gaussian setting, we prove propagation of Gaussianity as we show that the asymptotic distribution retains the Gaussianity of the initial data in the limit. In the non-Gaussian setting, we derive the limiting equations for the higher order moments, as well as for the density function (PDF) of the solution. Some of the results we prove were conjectured in the physics literature, others appear to be new. This gives a complete description of the statistics of the solutions in the kinetic limit.
DOI：
10.1017/fmp.2021.6
发表时间：
2019-12
期刊：
Forum of Mathematics, Pi
影响因子：
--
作者：
Yu Deng;Z. Hani
通讯作者：
Yu Deng;Z. Hani
波动动力学方程的全推导
影响因子：
3.1
作者：
Deng, Yu;Hani, Zaher
通讯作者：
Hani, Zaher
非线性薛定谔方程长期行为的波湍流描述
影响因子：
3.1
作者：
Buckmaster, T.;Germain, P.;Hani, Z.;Shatah, J.
通讯作者：
Shatah, J.