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## Why 's and not 's?

Importantly, RPA formulation involves independent phase factors and not phases themselves. Firstly, the phases would not be convenient because the mean value of the phases is evolving with the rate equal to the nonlinear frequency correction [24]. Thus one could not say that they are distributed uniformly from to ''. Moreover the mean fluctuation of the phase distribution is also growing and they quickly spread beyond their initial -wide interval [24]. But perhaps even more important, 's build mutual correlations on the nonlinear time whereas 's remain independent. Let us give a simple example illustrating how this property is possible due to the fact that correspondence between and is not a bijection. Let be a random integer and let and be two independent (of and of each other) random numbers with uniform distribution between and . Let

Then

and

Thus,

which means that variables and are correlated. On the other hand, if we introduce

then

and

which means that variables and are statistically independent. In this illustrative example it is clear that the difference in statistical properties between and arises from the fact that function does not have inverse and, consequently, the information about contained in is lost in .

Summarising, statistics of the phase factors is simpler and more convenient to use than because most of the statistical objects depend only on . This does not mean, however, that phases are not observable and not interesting. Phases can be tracked'' in numerical simulations continuously, i.e. without making jumps to when the phase value exceeds . Such continuous in function can achieve a large range of variation in values due to the dependence of the nonlinear rotation frequency with . This kind of function implies fastly fluctuating which is the mechanism behind de-correlation of the phase factors at different wavenumbers.

Next: Wavefields with long spatial Up: Setting the stage II: Previous: Definition of an essentially
Dr Yuri V Lvov 2007-01-23