Why $\psi$'s and not $\phi$'s?

Importantly, RPA formulation involves independent phase factors $\psi = e^\phi$ and not phases $\phi$ themselves. Firstly, the phases would not be convenient because the mean value of the phases is evolving with the rate equal to the nonlinear frequency correction [24]. Thus one could not say that they are ``distributed uniformly from $-\pi$ to $\pi$''. Moreover the mean fluctuation of the phase distribution is also growing and they quickly spread beyond their initial $2 \pi$-wide interval [24]. But perhaps even more important, $\phi$'s build mutual correlations on the nonlinear time whereas $\psi$'s remain independent. Let us give a simple example illustrating how this property is possible due to the fact that correspondence between $\phi$ and $\psi$ is not a bijection. Let $N$ be a random integer and let $r_1$ and $r_2$ be two independent (of $N$ and of each other) random numbers with uniform distribution between $-\pi$ and $\pi$. Let

$$\phi_{1,2} = 2 \pi N + r_{1,2}.$$

Then

$$\langle \phi_{1,2}\rangle= 2 \pi \langle N \rangle,$$

and

$$\langle \phi_{1}\phi_{2}\rangle=4\pi^2 \langle N^2 \rangle.$$

Thus,

$$\langle \phi_{1}\phi_{2}\rangle - \langle \phi_{1} \rangle \langle \phi_{2}\rangle = 4 \pi^2 ( \langle N^2 \rangle - \langle N \rangle^2) > 0,$$

which means that variables $\phi_1$ and $\phi_2$ are correlated. On the other hand, if we introduce

$$\psi_{1,2} = e^{i \phi_{1,2}},$$

then

$$\langle \psi_{1,2}\rangle=0,$$

and

$$\langle \psi_{1}\psi_{2}\rangle=0.$$

$$\langle \psi_{1}\psi_{2}\rangle- \langle \psi_{1}\rangle \langle \psi_{2}\rangle =0,$$

which means that variables $\psi_1$ and $\psi_2$ are statistically independent. In this illustrative example it is clear that the difference in statistical properties between $\phi$ and $\psi$ arises from the fact that function $\psi(\phi)$ does not have inverse and, consequently, the information about $N$ contained in $\phi$ is lost in $\psi$.

Summarising, statistics of the phase factors $\psi$ is simpler and more convenient to use than $\phi$ because most of the statistical objects depend only on $\psi$. This does not mean, however, that phases $\phi$ are not observable and not interesting. Phases $\phi$ can be ``tracked'' in numerical simulations continuously, i.e. without making jumps to $-\pi$ when the phase value exceeds $\pi$. Such continuous in $k$ function $\phi(k)$ can achieve a large range of variation in values due to the dependence of the nonlinear rotation frequency with $k$. This kind of function implies fastly fluctuating $\psi(k)$ which is the mechanism behind de-correlation of the phase factors at different wavenumbers.