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## Probability Distribution Function.

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. 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 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,

 (14)

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

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 (17),

 (15)

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,

 (16)

which depends only on the amplitudes marked by labels .

Next: Definition of an ideal Up: Setting the stage II: Previous: Setting the stage II:
Dr Yuri V Lvov 2007-01-23