Propagation of chaos and the higher order statistics in the wave kinetic theory
Abstract
This manuscript continues and extends in various directions the result in arXiv:2104.11204, which gave a full derivation of the wave kinetic equation (WKE) from the nonlinear Schrödinger (NLS) equation in dimensions $d\geq 3$. 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 asymptotically 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 shown to be given by solutions of the socalled wave kinetic heirarchy (WKH) with factorized initial data, hence giving a rigorous justification of the latter system. We treat both Gaussian and nonGaussian 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 nonGaussian 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.04565
 Bibcode:
 2021arXiv211004565D
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35Q55 (Primary) 35Q82 (Secondary)
 EPrint:
 43 pages, 6 figures. arXiv admin note: text overlap with arXiv:2104.11204