Evolution of the Generating Functional and Multi-particle PDF

Let us first derive an evolution equation for the generating functional $Z\{\lambda, \mu\}$ exploiting the separation of the linear and nonlinear time scales. ³ To do this, we have to calculate $Z$ at the intermediate time $t=T$ based on its value at $t=0$. The derivation, although standard for WT, is quite lengthy and will have to be published in a longer paper. Here, we will only outline the main steps and give the result. First, we need to substitute the $\epsilon$-expansion of $a$ from (12) into the expressions $e^{\lambda_j \vert a_j\vert^2}$ and $\psi_j^{\mu_j} ={1 \over 2 } (\ln {a_j \over \bar a_j})^{\mu_j}$. Second, the phase averaging should be done. Note that, because, we assume that initial phase factors are independent at $t=0$ with required accuracy, we can do such phase averaging independently of the amplitude averaging (which we do not do yet). Thirdly, we take $N \to \infty$ limit followed by $T \sim 1/\epsilon \to \infty$ (this order of the limits is essential!). Taking into account that $\lim\limits_{T\to\infty}E(0,x)= T (\pi \delta(x)+iP(\frac{1}{x}))$, and $\lim\limits_{T\to\infty}\vert\Delta(x)\vert^2=2\pi T\delta(x)$ and, replacing $(Z(T) -Z(0))/T$ by $\dot Z$ (because the nonlinear time $\sim 1/\epsilon^2 \gg T)$ we have

$$\dot Z = 4 \pi {\epsilon}^2 \int \big\{ (\lambda_{j}+\lambda_{j}^2 {\delta... ...delta_{j+n}^{m} \right] {\delta^2 Z\over \delta \lambda_{m} \delta \lambda_{n}}$$

$$+ 2 \lambda_j \left[ - \vert V_{mn}^j\vert^2 \delta(\omega_{mn}^j... ...a \over \delta \lambda_{n}} \right) \right] {\delta Z \over \delta \lambda_{j}}$$

$$+ 2 \lambda_j\lambda_m \left[ -2 \vert V_{mn}^j\vert^2 \delta_{m+... ...lta \lambda_{j} \delta \lambda_{n} \delta \lambda_{m}} \big\}\, dk_j dk_m dk_n.$$ (14)

Here variational derivatives appeared instead of partial derivatives because of the $N \to \infty$ limit. This expression is valid up to the $[1+O({\epsilon}^2)]$ factor. Equation (15) does not contain $\mu$ dependence which means that that these variables separate from $\lambda$'s and the solution is a purely-amplitude $Z$ times an arbitrary function of $\mu$'s which is going to be stationary in time. The latter corresponds to preservation of the initial $\Pi \delta(\mu_l)$ dependence by equation (15) which means that no angular harmonics of the PDF higher than zeroth will be excited. In the other words, all the phases will remain statistically independent and uniformly distributed on $S^1$ with the accuracy of the equation (15) integrated over the nonlinear time $1/{\epsilon}^2$, i.e. with the $O({\epsilon}^2)$ accuracy. This proves the first of the ``essential RPA'' properties. In fact, this result was already obtained before in [15] for a narrower class of 3-wave systems (see footnote 2). Note that we still have not used any assumption about the statistics of $A$'s and, therefore, (15) could be used in future for studying systems with random phases but correlated amplitudes.

Taking the inverse Laplace transform of (15) we have the following equation for the PDF,

$$\dot {\cal P} = - \int {\delta F_j \over \delta s_j} \, dk_j,$$ (15)

where $F_j$ is a flux of probability in the space of the amplitude $s_j$,

$$-{F_j \over 4 \pi {\epsilon}^2 s_j} = \int \big\{ (\vert V_{mn}^{j}\vert^2 \delta(\omega_{mn}^{j}) \del... ...(\omega_{jm}^{n}) \delta_{j+m}^{n} ) s_n s_m {\delta {\cal P} \over \delta s_j}$$

$$+2 {\cal P} ( \vert V_{jm}^{n}\vert^2 \delta(\omega_{jm}^{n}) \de... ...+m}^{n} - \vert V_{mn}^{j}\vert^2 \delta(\omega_{mn}^{j}) \delta_{m+n}^{j} )s_m$$

$$+2 (\vert V_{jm}^{n}\vert^2 \delta(\omega_{jm}^{n}) \delta_{j+m}^... ...elta_{m+n}^{j} ) s_n s_m {\delta {\cal P} \over \delta s_m} \big\} \, dk_m dk_n$$ (16)

This equation is identical to the Zaslavski-Sagdeev (ZS) [13] equation (Brout-Prigogine in the physics of crystals context [15,16]). Note that ZS equation was originally derived in [13] for a much narrower class of systems, see footnote 2, whereas the result above indicates that it is also valid in the most general case of 3-wave systems. Here we should again emphasize the importance of the order of limits, $N \to \infty$ first and ${\epsilon} \to 0$ second. Physically this means that the frequency resonance is broad enough to cover a great many modes. Some authors, e.g. ZS and BP leave the sum notation in the PDF equation even after the ${\epsilon} \to 0$ limit taken giving $\delta(\omega_{jm}^{n})$. One has to be careful interpreting such a formula because formally the RHS is null in most of the cases because there may be no exact resonances between the discrete $k$ modes (as it is the case, e.g. for the capillary waves). Thus, our functional integral notation is a more accurate way to write the result.