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 [9], 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 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 so-called wave kinetic heirarchy (WKH) with factorized initial data, hence giving a rigorous justification of the latter system. 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.

Path large deviations for the kinetic theory of weak turbulence

“…Inspired by the Random Phase and Amplitude (RPA) approximation, it might be natural to consider the large deviations Hamiltonian for a system of modes that evolve via a mean-field Langevin model with interactions only through the empirical spectrum. This mean-field approach was shown previously in [5] (section 3.2.2) to yield the exact results for probability distributions of single-mode amplitudes [2,8,14]. However, we explain that for large deviations the actual wave-turbulence Hamiltonian and this mean-field Hamiltonian are different.…”

“…A.2 Computation of the moment generating function Z L, (21) From the definition of the moment generating function Z L, (21), using (8), one obtains…”

“…As we explain in more detail, a rigorous derivation would require the control of higher order terms. Given recent breakthroughs in the mathematical derivation of the wave kinetic equations [6,7], including for higher-order statistics [8], we believe that rigorous proof of our large-deviations theory may be within the realm of possibility.…”

Long time NLS approximation for the quasilinear Klein-Gordon equation on large domains under periodic boundary conditions

“…In other words we deal with wave numbers k 0 ∈ Z L belonging to the interval [1, C], which is independent of L. With this choice we fall in the small data regime of the NLS equation, which, on higher dimensional tori, is fundamental for the study of wave turbulence theory [6], [10], [11], [12]. Certainly other choices are possible, but covering the general case would require an extra huge technical effort.…”

“…However not all scaling laws are admissible for the kinetic theory, and the admissibility range can depend on the shape of the torus. For more details we refer to [12].…”