Variables do not separate in the above equation for the PDF. Indeed, substituting
To answer to this question let us differentiate the discrete version of the equation (64) with respect to 's to get equations for the amplitude moments. We can easily see that
Similarly, one can show that the modes will remain
independent over the nonlinear time
in any subset of modes with accuracy
) if they were initially independent in
every subset of size . Namely
Mismatch arises from some terms in the ZS equation with coinciding indices . For there is only one such term in the -sum and, therefore, the corresponding error is which is much less than (due to the order of the limits in and ). However, the number of such terms grows as and the error accumulates to which can greatly exceed for sufficiently large .
We see that the accuracy with which the modes remain independent in a subset is worse for larger subsets and that the independence property is completely lost for subsets approaching in size the entire set, . One should not worry too much about this loss because is the biggest parameter in the problem (size of the box) and the modes will be independent in all -subsets no matter how large. Thus, the statistical objects involving any finite number of modes are factorisable as products of the one-mode objects and, therefore, the WT theory reduces to considering the one-mode objects. This results explains why we re-defined RPA in its relaxed ``essential RPA'' form. Indeed, in this form RPA is sufficient for the WT closure and, on the other hand, it remains valid over the nonlinear time. In particular, only property (77) is needed, as far as the amplitude statistics is concerned, for deriving the 3-wave kinetic equation, and this fact validates this equation and all of its solutions, including the KZ spectrum which plays an important role in WT.
The situation were modes can be considered as independent when taken in relatively small sets but should be treated as dependent in the context of much larger sets is not so unusual in physics. Consider for example a distribution of electrons and ions in plasma. The full -particle distribution function in this case satisfies the Liouville equation which is, in general, not a separable equation. In other words, the -particle distribution function cannot be written as a product of one-particle distribution functions. However, an -particle distribution can indeed be represented as a product of one-particle distributions if where is the number of particles in the Debye sphere. We see an interesting transition from a an individual to collective behaviour when the number of particles approaches . In the special case of the one-particle function we have here the famous mean-field Vlasov equation which is valid up to corrections (representing particle collisions).