next up previous
Next: Separation of timescales: general Up: Setting the stage II: Previous: Generating functional.

One-mode statistics

Of particular interest are one-mode densities which can be conveniently obtained using a one-amplitude generating function

$\displaystyle Z(t,\lambda)=\langle e^{\lambda \vert a_k\vert^2}\rangle, $

where $ \lambda$ is a real parameter. Then PDF of the wave intensities $ s= \vert a_k\vert^2$ at each $ {\bf k}$ can be written as a Laplace transform,

$\displaystyle P^{(a)}(s,t) = \langle \delta (\vert a_k\vert^2 -s) \rangle = {1 \over 2 \pi} \int_{0}^{\infty} Z(\lambda, t)e^{-s \lambda}d\lambda.$ (26)

For the one-point moments of the amplitude we have
$\displaystyle M_k^{(p)} \equiv \langle \vert a_k\vert^{2p}\rangle =\langle \vert a\vert^{2p}e^{\lambda \vert a\vert^2}\rangle\vert _{\lambda=0}=$      
$\displaystyle Z_{\lambda \cdots \lambda}\vert _{\lambda=0} = \int_{0}^{\infty} s^p P^{(a)}(s, t) \, ds,$     (27)

where $ p \in {\cal N}$ and subscript $ \lambda$ means differentiation with respect to $ \lambda$ $ p$ times.

The first of these moments, $ n_k = M_k^{(1)}$, is the waveaction spectrum. Higher moments $ M_k^{(p)}$ measure fluctuations of the waveaction $ k$-space distributions about their mean values [22]. In particular the r.m.s. value of these fluctuations is

$\displaystyle \sigma_k = \sqrt{ M_k^{(2)} - n_k^2}.$ (28)

next up previous
Next: Separation of timescales: general Up: Setting the stage II: Previous: Generating functional.
Dr Yuri V Lvov 2007-01-23