Let us consider a wavefield
in a periodic cube of with
side and let the Fourier transform of this field be where
index
marks the mode with wavenumber
on the grid in the -dimensional Fourier space. For
simplicity let us assume that there is a maximum wavenumber
(fixed e.g. by dissipation) so that no modes with wavenumbers greater
than this maximum value can be excited. In this case, the total
number of modes is
. Correspondingly, index
will only take values in a finite box,
which is centred at 0 and all sides of which are equal to
. To consider homogeneous turbulence, the
large box limit
will have to be taken.
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Let us write the complex as
where is a
real positive amplitude and is a phase factor which takes
values on
, a unit circle centred at zero in the
complex plane. Let us define the -mode joint PDF
as the probability for the wave intensities to be in the
range
and for the phase factors to be on
the unit-circle segment between and
for all
. In terms of this PDF, taking the averages will
involve integration over all the real positive 's and along all
the complex unit circles of all 's,

where notation means that depends on all 's and all 's in the set (similarly, means , etc). The full PDF that contains the complete statistical information about the wavefield in the infinite -space can be understood as a large-box limit

The full PDF defined above involves all modes (for either finite or in the limit). By integrating out all the arguments except for chosen few, one can have reduced statistical distributions. For example, by integrating over all the angles and over all but amplitudes,we have an ``-mode'' amplitude PDF,

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which depends only on the amplitudes marked by labels .