Kinetic wave turbulence

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. 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, but is instead a reduced equation containing only a subset of the terms in that equation. Our equations are consistent with the Peierls equation in that the additional terms in the latter vanish as inverse powers of volume in the large-box limit. 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 (PDF’s) previously derived by other authors. We introduce the concepts of the “empirical spectrum” and the “empirical 1-mode PDF.” We show that the factorization of the hierarchy equations implies that these quantities are self-averaging and satisfy the wave-kinetic closure equations for almost any selection of phases and amplitudes from the initial ensemble. 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.