Let us consider the initial fields
which are of the RPA type as defined above. We will perform
averaging over the statistics of the initial fields
in order to obtain an evolution equations, first for and
then for the multi-mode PDF.
Let us introduce a graphical classification of the above terms which
will allow us to simplify the statistical averaging and to understand
which terms are dominant. We will only consider here contributions from
and which will allow us to understand the basic method.
Calculation of the rest of the terms, , and ,
follows the same principles and can be found in [24].
First,
The linear in
terms are represented by which, upon using
(34), becomes

Let us introduce some graphical notations for a simple classification of different contributions to this and to other (more lengthy) formulae that will follow. Combination will be marked by a vertex joining three lines with in-coming and out-coming and directions. Complex conjugate will be drawn by the same vertex but with the opposite in-coming and out-coming directions. Presence of and will be indicated by dashed lines pointing away and toward the vertex respectively.

Let us average over all the independent phase factors in the set . Such averaging takes into account the statistical independence and uniform distribution of variables . In particular, , and . Further, the products that involve odd number of 's are always zero, and among the even products only those can survive that have equal numbers of 's and 's. These 's and 's must cancel each other which is possible if their indices are matched in a pairwise way similarly to the Wick's theorem. The difference with the standard Wick, however, is that there exists possibility of not only internal (with respect to the sum) matchings but also external ones with 's in the pre-factor .

Obviously, non-zero contributions can only arise for terms in which
all 's cancel out either via internal mutual couplings within
the sum or via their external couplings to the 's in the
-product. The internal couplings will indicate by joining the
dashed lines into loops whereas the external matching will be shown as
a dashed line pinned by a blob at the end. The number of blobs in
a particular graph will be called the *valence* of this graph.

Note that there will be no
contribution from the internal couplings between the incoming and the
out-coming lines of the same vertex because, due to the
-symbol, one of the wavenumbers is 0 in this case, which means
^{5}that . For we have

and

which correspond to the following expressions,

and

Because of the -symbols involving 's, it takes very special combinations of the arguments in for the terms in the above expressions to be non-zero. For example, a particular term in the first sum of (53) may be non-zero if two 's in the set are equal to 1 whereas the rest of them are 0. But in this case there is only one other term in this sum (corresponding to the exchange of values of and ) that may be non-zero too. In fact, only utmost two terms in the both (53) and (54) can be non-zero simultaneously. In the other words, each external pinning of the dashed line removes summation in one index and, since all the indices are pinned in the above diagrams, we are left with no summation at all in i.e. the number of terms in is with respect to large . We will see later that the dominant contributions have terms. Although these terms come in the order, they will be much greater that the terms because the limit must always be taken before .

Let us consider the first of the
-terms, . Substituting
(34) into (48), we have

(49) |

where

(50) |

Here the graphical notation for the interaction coefficients and the amplitude is the same as introduced in the previous section and the dotted line with index indicates that there is a summation over but there is no amplitude in the corresponding expression.

Let us now perform the phase averaging which corresponds to the internal and external
couplings of the dashed lines. For
we have

where

We have not written out the third term in (57) because it is just a complex conjugate of the second one. Observe that all the diagrams in the first line of (57) are with respect to large because all of the summations are lost due to the external couplings(compare with the previous section). On the other hand, the diagram in the second line contains two purely-internal couplings and is therefore . This is because the number of indices over which the summation survives is equal to the number of purely internal couplings. Thus, the zero-valent graphs are dominant and we can write

(52) |

Now we are prepared to understand a general rule: the dominant contribution
always comes from the graphs with minimal valence, because each external
pinning reduces the number of summations by one. The minimal valence of
the graphs in
is one and, therefore,
is order times smaller than
.
On the other hand,
is of the same order as
, because it contains zero-valence graphs. We have

Summarising, we have

(54) |

Thus, we considered in detail the different terms involved in and we found that the dominant contributions come from the zero-valent graphs because they have more summation indices involved. This turns out to be the general rule in both three-wave and four-wave cases and it allows one to simplify calculation by discarding a significant number of graphs with non-zero valence. Calculation of terms to can be found in [24] and here we only present the result,

(55) |

(i.e. is order times smaller than or ) and

(56) |

Using our results for in (46) and (45) we have

Here partial derivatives with respect to appeared because of the factors. This expression is valid up to and corrections. Note that we still have not used any assumption about the statistics of 's. Let us now limit followed by (we re-iterate that this order of the limits is essential). Taking into account that , and and, replacing by we have

Here variational derivatives appeared instead of partial derivatives because of the limit.

Now we can observe that the evolution equation for does not involve which means that if initial contains factor it will be preserved at all time. This, in turn, means that the phase factors remain a set of statistically independent (of each each other and of 's) variables uniformly distributed on . This is true with accuracy (assuming that the -limit is taken first, i.e. ) and this proves persistence of the first of the ``essential RPA'' properties. Similar result for a special class of three-wave systems arising in the solid state physics was previously obtained by Brout and Prigogine [20]. This result is interesting because it has been obtained without any assumptions on the statistics of the amplitudes and, therefore, it is valid beyond the RPA approach. It may appear useful in future for study of fields with random phases but correlated amplitudes.