In [5] the following equation for for the -mode PDF was obtained for the four-wave systems,

where

Here limit has already been taken and means the variational derivative. Using this equation, one can prove that RPA property holds over the nonlinear time, i.e. the -mode PDF remains of the product factorized form with accuracy sufficient for the WT closures to work [5]. Using RPA, we get for the one-point marginals [5],

with is a probability flux in the s-space,

where

Here we introduced the wave-action spectrum,

(22) |

From (14) we get the following equation for the moments :

which, for gives the standard wave kinetic equation (WKE),