  A general Hamiltonian wave system with quartic resonances is considered, in
the standard kinetic limit of a continuum of weakly interacting dispersive
waves with random phases. The evolution equation for the multimode
characteristic function $Z$ is obtained within an "interaction representation"
and a perturbation expansion in the small nonlinearity parameter. A frequency
renormalization is performed to remove linear terms that do not appear in the
3-wave case. Feynman-Wyld diagrams are used to average over phases, leading to
a first order differential evolution equation for $Z$. A hierarchy of
equations, analogous to the Boltzmann hierarchy for low density gases is
derived, which preserves in time the property of random phases and amplitudes.
This amounts to a general formalism for both the $N$-mode and the 1-mode PDF
equations for 4-wave turbulent systems, suitable for numerical simulations and
for investigating intermittency.
