Generating functional.

Introduction of generating functionals simplifies statistical derivations. It can be defined in several different ways to suit a particular technique. For our problem, the most useful form of the generating functional is

$$Z^{(N)} \{\lambda, \mu \} = { 1 \over (2 \pi)^{N}} \langle \prod_{l \in {\cal B}_N } \; e^{\lambda_l A_l^2} \psi_l^{\mu_l} \rangle ,$$ (20)

where $\{\lambda, \mu \} \equiv \{\lambda_l, \mu_l ; l \in {\cal B}_N\}$ is a set of parameters, $\lambda_l {\cal 2 R}$ and $\mu_l {\cal 2 Z}$.

$${\cal P}^{(N)} \{s, \xi \} = {1 \over (2 \pi)^{N}} \sum_{\{\mu \}... ... \in {\cal B}_N } \delta (s_l - A_l^2) \, \psi_l^{\mu_l} \xi_l^{-\mu_l} \rangle$$ (21)

where $\{\mu \} \equiv \{ \mu_l \in {\cal Z}; {l \in {\cal B}_N } \}$. This expression can be verified by considering mean of a function $f\{A^2,\psi\}$ using the averaging rule (17) and expanding $f$ in the angular harmonics $\psi_l^m; \; m \in {\cal Z}$ (basis functions on the unit circle),

$$f\{A^2,\psi \} = \sum_{\{ m \} } g\{m, A\} \, \prod_{l \in {\cal B}_N } \psi_l^{m_l},$$ (22)

where $\{m\} \equiv \{ m_l \in {\cal Z}; {l \in {\cal B}_N } \}$ are indices enumerating the angular harmonics. Substituting this into (17) with PDF given by (24) and taking into account that any nonzero power of $\xi_l$ will give zero after the integration over the unit circle, one can see that LHS=RHS, i.e. that (24) is correct. Now we can easily represent (24) in terms of the generating functional,

$${\cal P}^{(N)} \{s, \xi \} = \hat {\cal L}_\lambda^{-1} \sum_{\{\... ...t( Z^{(N)} \{\lambda, \mu\} \, \prod_{l \in {\cal B}_N } \xi_l^{-\mu_l} \right)$$ (23)

where $\hat {\cal L}_\lambda^{-1}$ stands for inverse the Laplace transform with respect to all $\lambda_l$ parameters and $\{\mu \} \equiv \{ \mu_l \in {\cal Z}; {l \in {\cal B}_N } \}$ are the angular harmonics indices.

Note that we could have defined $Z$ for all real $\mu_l$'s in which case obtaining $P$ would involve finding the Mellin transform of $Z$ with respect to all $\mu_l$'s. We will see below however that, given the random-phased initial conditions, $Z$ will remain zero for all non-integer $\mu_l$'s. More generally, the mean of any quantity which involves a non-integer power of a phase factor will also be zero. Expression (26) can be viewed as a result of the Mellin transform for such a special case. It can also be easily checked by considering the mean of a quantity which involves integer powers of $\psi_l$'s.

By definition, in RPA fields all variables $A_l$ and $\psi_l$ are statistically independent and $\psi_l$'s are uniformly distributed on the unit circle. Such fields imply the following form of the generating functional

$$Z^{(N)} \{\lambda, \mu \} = Z^{(N,a)} \{\lambda \} \, \prod_{l \in {\cal B}_N } \delta(\mu_l),$$ (24)

where

$$Z^{(N,a)} \{\lambda \} =\langle \prod_{l \in {\cal B}_N } e^{\lambda_l A_l^2} \rangle = Z^{(N)} \{\lambda, \mu\}\vert _{\mu=0}$$ (25)

is an $N$-mode generating function for the amplitude statistics. Here, the Kronecker symbol $\delta(\mu_l)$ ensures independence of the PDF from the phase factors $\psi_l$. As a first step in validating the RPA property we will have to prove that the generating functional remains of form (27) up to $1/N$ and $O({\epsilon}^2)$ corrections over the nonlinear time provided it has this form at $t=0$.