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Probability Distribution Function.

Let us consider a wavefield $ a({\bf x}, t)$ in a periodic cube of with side $ L $ and let the Fourier transform of this field be $ a_l(t)$ where index $ l {\in } {\cal Z}^d$ marks the mode with wavenumber $ k_l = 2
\pi l /L$ on the grid in the $ d$-dimensional Fourier space. For simplicity let us assume that there is a maximum wavenumber $ k_{max}$ (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 $ N = (k_{max} / \pi L)^d$. Correspondingly, index $ l$ will only take values in a finite box, $ l \in {\cal B}_N \subset
{\cal Z}^d$ which is centred at 0 and all sides of which are equal to $ k_{max} / \pi L = N^{1/3}$. To consider homogeneous turbulence, the large box limit $ N \to \infty $ will have to be taken. 1

Let us write the complex $ a_l$ as $ a_l =A_l \psi_l $ where $ A_l$ is a real positive amplitude and $ \psi_l $ is a phase factor which takes values on $ {\cal S}^{1} $, a unit circle centred at zero in the complex plane. Let us define the $ N$-mode joint PDF $ {\cal P}^{(N)}$ as the probability for the wave intensities $ A_l^2 $ to be in the range $ (s_l, s_l +d s_l)$ and for the phase factors $ \psi_l $ to be on the unit-circle segment between $ \xi_l$ and $ \xi_l + d\xi_l$ for all $ l \in {\cal B}_N$. In terms of this PDF, taking the averages will involve integration over all the real positive $ s_l$'s and along all the complex unit circles of all $ \xi_l$'s,

$\displaystyle \langle f\{A^2, \psi \} \rangle
= \left(
\prod_{ l {\cal 2 B}_N }...
... S}^{1} }
\vert d \xi_l\vert \right) \; {\cal P}^{(N)} \{s, \xi \}
f\{s, \xi \}$     (14)

where notation $ f\{A^2,\psi\}$ means that $ f$ depends on all $ A_l^2 $'s and all $ \psi_l $'s in the set $ \{A_l^2, \psi_l; l {\cal 2 B}_N \}$ (similarly, $ \{s, \xi \}$ means $ \{s_l, \psi_l; l \in {\cal B}_N \}$, etc). The full PDF that contains the complete statistical information about the wavefield $ a({\bf x}, t)$ in the infinite $ x$-space can be understood as a large-box limit

$\displaystyle {\cal P} \{ s_k, \xi_k \} = \lim_{N \to \infty}
{\cal P}^{(N)} \{s, \xi \},

i.e. it is a functional acting on the continuous functions of the wavenumber, $ s_k$ and $ \xi_k$. In the the large box limit there is a path-integral version of (17),

$\displaystyle \langle f\{A^2, \psi \} \rangle = \int {\cal D}s \oint \vert{\cal D} \xi\vert \; {\cal P} \{s, \xi \} f\{s, \xi \}$ (15)

The full PDF defined above involves all $ N$ modes (for either finite $ N$ or in the $ N \to \infty $ 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 $ M$ amplitudes,we have an ``$ M$-mode'' amplitude PDF,

$\displaystyle {\cal P}^{(M)}_{j_1, j_2, \dots , j_M} = \left( \prod_{ l \ne j_1...
...\oint_{{\cal S}^{1} } \vert d \xi_m\vert \; \right) {\cal P}^{(N)} \{s, \xi \},$ (16)

which depends only on the $ M$ amplitudes marked by labels $ j_1, j_2, \dots , j_M {\cal 2 B}_N$.

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