Equation for $Z$

Now we can observe that all contributions to the evolution of $Z$ (namely $J_1 - J_5$, see the previous section and Appendix 2) contain factor $\prod_{l}\delta(\mu_l)$ which means that the phase factors $\{\psi\}$ remain a set of statistically independent (of each each other and of $A$'s) variables uniformly distributed on $S^1$. This is true with accuracy $O({\epsilon}^2)$ (assuming that the $N$-limit is taken first, i.e. $1/N \ll {\epsilon}^2$) and this proves persistence of the first of the ``essential RPA'' properties. Similar result for a special class of three-wave systems arising in the solid state physics was previously obtained by Brout and Prigogine [16]. This result is interesting because it has been obtained without any assumptions on the statistics of the amplitudes $\{ A \}$ and, therefore, it is valid beyond the RPA approach. It may appear useful in future for study of fields with random phases but correlated amplitudes.

Let us now derive an evolution equation for the generating functional. Using our results for $J_1 - J_5$ in (18) and (17) we have

$$Z(T) - Z(0) = {\epsilon}^2 \sum_{j,m,n}(\lambda_{j}+\lambda_{j}^2 {\partial \ov... ...n}^{m} \right] {\partial^2 Z(0)\over \partial \lambda_{m} \partial \lambda_{n}}$$

$$+ 4{\epsilon}^2 \sum_{j,m,n} \lambda_j \left[ - \vert V_{mn}^j\ve... ...partial \lambda_{n}} \right) \right] {\partial Z(0) \over \partial \lambda_{j}}$$

$$+ 2 {\epsilon}^2 \sum_{j\neq k,n}\lambda_j\lambda_k \left[ -2 \ve... ...0)\over \partial \lambda_{j} \partial \lambda_{n} \partial \lambda_{k}} \; +cc.$$ (33)

Here partial derivatives with respect to $\lambda_l$ appeared because of the $A_l$ factors. This expression is valid up to $O({\epsilon}^4)$ and $O(\epsilon^2/N)$ corrections. Note that we still have not used any assumption about the statistics of $A$'s. This is a linear equation: as usual in statistics we traded nonlinearity for higher dimensions. The last term here ``spoils'' the separation of variables and, therefore, puts a question mark on the independence of variables $\{ A \}$ from each other on the nonlinear time.

Let us now $N \to \infty$ limit followed by $T \sim 1/\epsilon \to \infty$ (we re-iterate that 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 $\dot Z$ we have

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

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

$$+ 2 \lambda_j\lambda_m \left[ -2 \vert V_{mn}^j\vert^2 \delta_{m+... ...lta \lambda_{j} \delta \lambda_{n} \delta \lambda_{m}} \big\}\, dk_j dk_m dk_n.$$ (34)

Here variational derivatives appeared instead of partial derivatives because of the $N \to \infty$ limit.