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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 centered at 0 and all sides of which are equal to
. To consider homogeneous turbulence, the
large box limit will have to be taken.
1
Let us write the complex as
where is a
real positive amplitude and is a phase factor which takes
values on , a unit circle centered at zero in the
complex plane. Let us define the -particle 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,
|
|
|
(1) |
where the 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
i.e. it is a functional acting on the continuous functions of the
wavenumber, and . In the the large box limit there is a
path-integral version of (1),
|
(2) |
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 ``-particle'' amplitude PDF,
|
(3) |
which depends only on the amplitudes marked by labels
.
Statistical derivations are greatly facilitated by the introduction
of a generating functional
|
|
|
(4) |
where
is a set of parameters,
and
.
|
(5) |
where
is a set of indices enumerating the angular harmonics and
stands for the inverse Laplace transform with respect to all
.
Subsections
Next: Definition of an essentially
Up: Probability densities and preservation
Previous: Introduction
Dr Yuri V Lvov
2007-01-17