We consider general models of wave systems in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. We show in this limit that the leading-order, asymptotically valid dynamical equation for multimode amplitude distributions is not the well-known equation of Peierls, which contradicts the self-averaging property of the "empirical spectrum". We show that the asymptotic multimode equations possess factorized solutions for factorized initial data, which correspond to preservation in time of the property of “random phases & amplitudes”. The factors satisfy the equations for the 1-mode probability density functions previously derived by Choi et al. and Jakobsen & Newell. We show that these closure equations satisfy an H-theorem for an entropy defined by Boltzmann’s prescription. We also characterize the general solutions of our multimode distribution equations, for initial conditions with random phases but with no statistical assumptions on the amplitudes. Analogous to a result of Spohn for the Boltzmann hierarchy, these are “super-statistical solutions” that correspond to ensembles of solutions of the wave-kinetic closure equations with random initial conditions or random forces. This gives a possible kinetic explanation of intermittency and non-Gaussian statistics in wave turbulence.
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