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## Equation for the multi-mode PDF

Taking the inverse Laplace transform of (64) we have the following equation for the PDF,

 (59)

where is a flux of probability in the space of the amplitude ,
 (60)

This equation is identical to the one originally obtained by Peierls [19] and later rediscovered by Brout and Prigogine [20] in the context of the physics of anharmonic crystals,

 (61)

where

 (62)

Zaslavski and Sagdeev [21] were the first to study this equation in the WT context. However, the analysis of [19,20,21] was restricted to the interaction Hamiltonians of the potential energy'' type, i.e. the ones that involve only the coordinates but not the momenta. This restriction leaves aside a great many important WT systems, e.g. the capillary, Rossby, internal and MHD waves. Our result above indicates that the Peierls equation is also valid in the most general case of 3-wave systems. Note that Peierls form (67) - (68) looks somewhat more elegant and symmetric than (65) - (66). However, form (65) - (66) has advantage because it is in a continuity equation form. Particularly for steady state solutions, one can immediately integrate it once and obtain,

 (63)

where is an arbitrary functional of and is the antisymmetric tensor,

In the other words, probability flux can be an arbitrary solenoidal field in the functional space of . One can see that (69) is a first order equation with respect to the -derivative. Special cases of the steady solutions are the zero-flux and the constant-flux solutions which, as we will see later correspond to a Gaussian and intermittent wave turbulence respectively.

Here we should again emphasise importance of the taken order of limits, first and second. Physically this means that the frequency resonance is broad enough to cover great many modes. Some authors, e.g. [19,20,21], leave the sum notation in the PDF equation even after the limit taken giving . One has to be careful interpreting such formulae because formally the RHS is nill in most of the cases because there may be no exact resonances between the discrete modes (as it is the case, e.g. for the capillary waves). In real finite-size physical systems, this condition means that the wave amplitudes, although small, should not be too small so that the frequency broadening is sufficient to allow the resonant interactions. Our functional integral notation is meant to indicate that the limit has already been taken.

Next: Evolution of statistics of Up: Evolution of statistics of Previous: Equation for the generating
Dr Yuri V Lvov 2007-01-23