One-mode statistics

We have established above that the one-point statistics is at the heart of the WT theory. All one-point statistical objects can be derived from the one-point amplitude generating function,

$$Z_a (\lambda_j) = \left< e^{\lambda_j A_j^2} \right>$$

which can be obtained from the $N$-point $Z$ by taking all $\mu$'s and all $\lambda$'s, except for $\lambda_j$, equal to zero. Substituting such values to (35) we get the following equation for $Z_a$,

$$\frac{\partial Z_a}{\partial t} = \lambda_j \eta_j Z_a +(\la... ...- \lambda_j \gamma_j) \frac{\partial Z_a}{\partial \lambda_j},$$ (42)

where,

$$\eta_j = 4 \pi \epsilon^2 \int \left(\vert V^j_{lm}\vert^2 \delta... ...\delta^m_{jl} \delta(\omega^m_{jl} ) \right) n_{l} n_{m} \, d { k_l} d { k_m} ,$$ (43)

$$\gamma_j = 8 \pi \epsilon^2 \int \left( \vert V^j_{lm}\vert^2 \de... ...elta^m_{jl} \delta(\omega^m_{jl}) (n_{l}- n_{m}) \right) \, d { k_l} d { k_m} .$$ (44)

Correspondingly, for the one mode PDF $P_a (s_j)$ we have

$${\partial P_a \over \partial t}+ {\partial F \over \partial s_j} =0,$$ (45)

with $F$ is a probability flux in the s-space,

$$F=-s_j (\gamma P_a +\eta_j {\delta P_a \over \delta s_j}).$$ (46)

Equations (43) and (46) where previously obtained and studied in [20] in for the four-wave systems. The only difference for the four-wave case was different expressions for $\eta$ and $\gamma$. For the three-wave case, equation for the PDF was not considered before, but equations for its moments were derived and solved in [19]. In particular, equation for the first moment is nothing but the familiar kinetic equation $\dot n = - \gamma n + \eta$ which gives $\eta = \gamma n$ for any steady state. This, in turn means that in the steady state with $F=0$ we have $P^{(a)}_{j} = (1/n_j) \exp(-s_j /n_j)$ where $n_j$ can be any steady state solution of th kinetic equation including the KZ spectrum which plays the central role in WT [5,1]. However, it was shown in [20] that there also exist solutions with $F\ne 0$ which describe WT intermittency.