Discussion

In the present paper, we considered the evolution of the full N-particle objects such as the generating functional and the probability density function for all the wave amplitudes and their phase factors. We proved that the phase factors, being statistically independent and uniform on $S^1$ initially, remain so over the nonlinear evolution time. This result does not rely on any assumptions about the amplitude statistics and, therefore, can be used in future for studying systems with correlated amplitudes (but random phases). If in addition the initial amplitudes are independent too, then they remain so over the nonlinear time in a coarse-grained sense. Namely, all joint PDF's for the number of modes $M \ll N$ split into products of the one-particle densities with $O(M/N)$ accuracy. Thus, the full $N$-particle PDF does not get factorized as a product of $N$ one-particle densities and the Fourier modes in the set considered as a whole are not independent. However, the wave turbulence closure only deals with the joint objects of the finite size $M$ of variables while taking the $N \to \infty$ limit. These objects do get factorized into products and, for the WT purposes, the Fourier modes can be interpreted as statistically independent. These results reduce the WT problem to the study of the one-particle amplitude PDF's and they validate the generalized RPA technique introduced in [19,21]. Such a study of the one-particle PDF and the high-order momenta of the wave amplitudes was done in [19,21] and the reader is referred to these papers for the discussion of WT intermittency.

Finally, we would like to mention the role of quasi-resonant interactions which, as we saw, do not produce any long-term effect at the $\epsilon^2$ order considered in this paper. However, these interactions do modify statistics at order as was shown in [22]. The correction can be important for the real space correlators which have Gaussian values at the $\epsilon^2$ order for any (not necessarily Rayleigh) amplitude distributions.