**Yeontaek Choi, Yuri V. Lvov and Sergey Nazarenko**

*
Mathematics Institute, The University of Warwick, Coventry, CV4 7AL, UK*

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180

Random Phase Approximation (RPA) provides a very convenient
tool to study the ensembles of weakly interacting
waves, commonly called Wave Turbulence.
In its traditional
formulation, RPA assumes that phases of interacting waves are random
quantities but it usually ignores randomness of their amplitudes.
Recently, RPA was generalised in a way that takes into account the
amplitude randomness and it was applied to study of the higher momenta
and probability densities of wave amplitudes. However, to have a
meaningful description of wave turbulence the RPA properties assumed
for the initial fields must be proven to survive over the nonlinear
evolution time, and such a proof is the main goal of the present
paper. We derive an evolution equation for the full
probability density function which contains the complete information about
the joint statistics of all wave amplitudes and phases. We show that,
for any initial statistics of the amplitudes, the phase factors remain
statistically independent uniformly distributed variables.
If in addition the initial amplitudes are also independent variables
(but with arbitrary distributions) they will remain independent when
considered in small sets which are much less than the total number of modes.
However, if the size of a set is of order of the total number of modes
then the joint probability density for this set is not factorisable into
the product of one-mode probabilities. In the other words, the modes
in such a set are involved in a ``collective'' (correlated) motion.
We also study new type of correlators describing
the phase statistics.

- Introduction
- Fields with Random Phases and Amplitudes.

- Weak nonlinearity and separation of time scales
- Evolution of the multi-mode PDF
- Generating functional.
- Asymptotic expansion of the generating functional.
- Statistical averaging and graphs.
- Equation for
- Equation for the PDF

- Approximate independence of the amplitudes.
- One-mode statistics
- Phase statistics.
- Discussion
- Appendix 1
- Appendix 2

- Bibliography
- About this document ...