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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