  We consider a general model of Hamiltonian wave systems with triple
resonances, in the standard kinetic limit of a continuum of weakly interacting
dispersive waves with random phases. In this asymptotic limit we show that the
correct dynamical equation for multimode amplitude distributions is not the
well-known equation of Peierls but is instead a reduced equation with only a
subset of the terms in that equation. The equations that we derive are the
direct analogue of the Boltzmann hierarchy obtained from the BBGKY hierarchy in
the low-density limit for gases. 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 Jakobsen & Newell and Choi et al. We
show that the factorization of the hierarchy equations implies that these
quantities are self-averaging: they satisfy the wave-kinetic closure equations
of the spectrum and 1-mode PDF for almost any selection of phases and
amplitudes from the initial ensemble. We show that both of 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
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. On the basis of our results, we discuss possible
kinetic explanations of intermittency and non-Gaussian statistics in wave
turbulence.
