Appendix 1

Let us obtain $Z(T)$ in terms of the series in small nonlinearity up to the second order in ${\epsilon}$. As an intermediate step, we first consider separately the amplitude and the phase ingredients of $Z$ and substitute the $\epsilon$-expansion of $a$ from (8) into their expressions,

$$e^{\lambda_j \vert a_j\vert^2}= e^{\lambda_j \vert a_j^{(0)}+{\ep... ...ert a_j^{(1)}\vert^2+(a_j^{(2)}\bar a_j^{(0)}+ \bar a_j^{(2)}a_j^{(0)})\right]}$$

$$= e^{\lambda_j\vert a_j^{(0)}\vert^2}\left\{ 1+{\epsilon}\lambda_j(... ...}^2\lambda_j^2}{2}(a_j^{(1)}\bar a_j^{(0)}+ \bar a_j^{(1)}a_j^{(0)})^2 \right\}$$

$$= e^{\lambda_j A_j^{(0) 2}} ( 1+{\epsilon}{\alpha_{1j}} + {\epsilon}^2 {\alpha_{2j}}),$$ (63)

and

$$\psi_j^{\mu_j}=\left(\frac{a_j^{(0)}+{\epsilon}a_j^{(1)}+{\epsilo... ...}} +{\epsilon}^2\frac{\bar a_j^{(2)}}{\bar a_j^{(0)}}}\right)^{\frac{\mu_j}{2}}$$

$$= \psi_j^{(0)\mu_j} \left[1+{\epsilon}\frac{\mu_j}{2}\frac{a_j^{(1)... ...\left(\frac{\mu_j}{2}+1\right)\left(\frac{a_j^{(1)}}{a_j^{(0)}}\right)^2\right]$$

$$= \psi_j^{(0)\mu_j}\left\{ 1+{\epsilon} \frac{\mu_j}{2}\left(\frac{... ...ght)^2\right]-\frac{\mu_j^2\vert a_j^{(1)}\vert^2}{4A_j^{(0)2}}\right] \right\}$$

$$= \psi_j^{(0)\mu_j} (1+{\epsilon}{\beta_{1j}} +{\epsilon}^2 {\beta_{2j}} ),$$ (64)

where ${\alpha_{1j}},{\alpha_{2j}},{\beta_{1j}}$ and ${\beta_{2j}}$ denote the linear and quadratic contributions into the amplitude and phase parts of $Z$ respectively,

$${\alpha_{1j}} = \lambda_j(a_j^{(1)}\bar a_j^{(0)}+ \bar a_j^{(1)} a_j^{(0)}),$$ (65)

$${\alpha_{2j}} = {(\lambda_j +\lambda_j^2 A_j^{(0)2}\vert a_j^{(1)}\vert^2 +\lambd... ...frac{\lambda_j^2}{2}(a_j^{(1)}\bar a_j^{(0)})^2 + (\bar a_j^{(1)}a_j^{(0)})^2},$$ (66)

$${\beta_{1j}} = \frac{\mu_j}{2A_j^{(0)2}}(a_j^{(1)}\bar a_j^{(0)}-\bar a_j^{(1)} a_j^{(0)}),$$ (67)

$${\beta_{2j}} = \frac{\mu_j}{2A_j^{(0)2}}(a_j^{(2)}\bar a_j^{(0)}-\bar a_j^{(2)}a... ... a_j^{(0)}}\right)^2 \right]-\frac{\mu_j^2\vert a_j^{(1)}\vert^2}{4A_j^{(0)2}}.$$ (68)

Substituting expansions (64) and (65) into the expression for $Z$, we have

$$Z(T) = {1 \over (2 \pi)^{N}} \left<\prod_{l} e^{\lambda_l\vert a_l\vert^... ...a_{2l}]\psi_l^{(0)\mu_l}[1+{\epsilon} \beta_{1l}+{\epsilon}^2\beta_{2l}]\right>$$

$$= {1 \over (2 \pi)^{N}} \left<\prod_l e^{\lambda_l A_l^{(0)2}}\psi_... ...alpha_{1k}+\beta_{1j}\beta_{1k}) +\sum_{j,k}\alpha_{1j}\beta_{1k}\right]\right>$$

$$= {1 \over (2 \pi)^{N}} \left<\prod_l e^{\lambda_l A_l^{(0)2}}\psi_... ...j}\alpha_{1k}+\beta_{1j}\beta_{1k}+2\alpha_{1j}\beta_{1k})}_{I_3} \big]\right>,$$

For parts $I_1, I_2$ and $I_3$ in the above expression we have,

$$I_1 = \sum_j (\lambda_j +\frac{\mu_j}{2A_j^2})a_j^{(1)}\bar a_j^{(0)}+(\lambda_j -\frac{\mu_j}{2A_j^2})\bar a_j^{(1)}a_j^{(0)},$$

$$I_2 = \sum_j(\lambda_j +\lambda_j^2A_j^2-\frac{\mu_j^2}{2A_j^2})\vert a... ...j^{(2)}\bar a_j^{(0)}+ (\lambda_j -\frac{\mu_j}{2A_j^2})\bar a_j^{(2)}a_j^{(0)}$$

$$+ \left[\frac{\lambda_j^2}{2}+\frac{\mu_j}{4A_j^4}\left(\frac{\mu_j... ...{2}+1\right)-\frac{\lambda_j \mu_j}{2A_j^2} \right](\bar a_j^{(1)}a_j^{(0)})^2,$$

$$I_3 = {1 \over 2} \sum_{j \ne k} \big[ \lambda_j\lambda_k(a_j^{(1)}\bar... ...(0)}+\bar a_j^{(1)}a_j^{(0)})(a_k^{(1)}\bar a_k^{(0)}-\bar a_k^{(1)} a_k^{(0)})$$

$$+ \frac{\mu_j \mu_k}{4A_j^2 A_k^2}(a_j^{(1)}\bar a_j^{(0)}- \bar a_j^{(1)}a_j^{(0)})(a_k^{(1)}\bar a_k^{(0)}-\bar a_k^{(1)} a_k^{(0)}) \big],$$

Exploiting the property $\bar Z \{\lambda, -\mu\} = Z\{\lambda, \mu\}$ we can write

$$Z\{\lambda, \mu\} = X\{\lambda, \mu\} + \bar X \{\lambda, - \mu\}.$$ (69)

At $t=T$ we have for $X\{\lambda, \mu\}$

$$X(T) = X(0) + (2 \pi)^{2N} \left<\prod_{\Vert l\Vert<N} e^{... ...^2}[{\epsilon}J_1 +{\epsilon}^2(J_2 +J_3+J_4+J_5)] \right>_A,$$ (70)

where

$$J_1 = \left<\prod_l \psi_l^{(0)\mu_l} \sum_j (\lambda_j +\frac{\mu_j}{2\vert a_j^{(0)}\vert^2})a_j^{(1)}\bar a_j^{(0)}\right>_\psi,$$ (71)

$$J_2 = {1 \over 2} \left<\prod_l \psi_l^{(0)\mu_l} \sum_j (\lambda_j+ \l... ...^2-\frac{\mu_j^2}{2\vert a_j^{(0)}\vert^2})\vert a_j^{(1)}\vert^2 \right>_\psi,$$ (72)

$$J_3 = \left<\prod_l \psi_l^{(0)\mu_l} \sum_j (\lambda_j + \frac{\mu_j}{2\vert a_j^{(0)}\vert^2})a_j^{(2)}\bar a_j^{(0)} \right>_\psi,$$ (73)

$$J_4 = \left<\prod_l \psi_l^{(0)\mu_l} \sum_j \left[\frac{\lambda_j^2}{2... ...mu_j}{2\vert a_j^{(0)}\vert^2} \right](a_j^{(1)}\bar a_j^{(0)})^2 \right>_\psi,$$ (74)

$$J_5 = {1 \over 2} \left<\prod_l \psi_l^{(0)\mu_l} \sum_{j \ne k}\lambda... ...)}\bar a_k^{(0)} -\bar a_k^{(1)}a_k^{(0)})a_j^{(1)}\bar a_j^{(0)} \right>_\psi,$$ (75)

where $\left< \cdot \right>_A$ and $\left< \cdot \right>_\psi$ denote the averaging over the initial amplitudes and initial phases respectively. We remind that such individual averages are possible because the amplitudes and the phases are statistically independent from each other at $t=0$.