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

where is a set of parameters, and .

where . This expression can be verified by considering mean of a function using the averaging rule (17) and expanding in the angular harmonics (basis functions on the unit circle),

(22) |

where are indices enumerating the angular harmonics. Substituting this into (17) with PDF given by (24) and taking into account that any nonzero power of 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,

where stands for inverse the Laplace transform with respect to all parameters and are the angular harmonics indices.

Note that we could have defined for all real 's in which case obtaining would involve finding the Mellin transform of with respect to all 's. We will see below however that, given the random-phased initial conditions, will remain zero for all non-integer '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 's.

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

where

(25) |

is an -mode generating function for the amplitude statistics. Here, the Kronecker symbol ensures independence of the PDF from the phase factors . As a first step in validating the RPA property we will have to prove that the generating functional remains of form (27) up to and corrections over the nonlinear time provided it has this form at .