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Next: One-mode statistics Up: Setting the stage II: Previous: Wavefields with long spatial

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

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

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

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

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

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


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

next up previous
Next: One-mode statistics Up: Setting the stage II: Previous: Wavefields with long spatial
Dr Yuri V Lvov 2007-01-23