Approximate independence of the amplitudes.

Variables $s_j$ do not separate in the above equation for the PDF. Indeed, substituting

$${\cal P}^{(N,a)} = P^{(a)}_{j_1} P^{(a)}_{j_2} \dots P^{(a)}_{j_N} \;$$ (38)

into the discrete version of (38) we see that it turns into zero on the thermodynamic solution with $P^{(a)}_{j} = \omega_j \exp(-\omega_j s_j)$. However, it is not zero for the one-mode PDF $P^{(a)}_{j}$ corresponding to the cascade-type Kolmogorov-Zakharov (KZ) spectrum $n_j^{kz}$, i.e. $P^{(a)}_{j} = (1/n_j^{kz}) \exp(-s_j /n_j^{kz})$ (see next section), nor it is likely to be zero for any other PDF of form (39). This means that, even initially independent, the amplitudes will correlate with each other at the nonlinear time. Does this mean that the existing WT theory, and in particular the kinetic equation, is invalid?

To answer to this question let us differentiate the discrete version of the equation (35) with respect to $\lambda$'s to get equations for the amplitude moments. We can easily see that

$$\partial_t \left(\langle A_{j_1}^2 A_{j_2}^2 \rangle -\lang... ...gle \right) = O({\epsilon}^4) \quad (j_1, j_2 \in {\cal B}_N)$$ (39)

if $\langle A_{j_1}^2 A_{j_2}^2 A_{j_3}^2 \rangle = \langle A_{j_1}^2 \rangle \langle A_{j_2}^2 \rangle \langle A_{j_3}^2 \rangle$ (with the same accuracy) at $t=0$. Similarly, in terms of PDF's

$$\partial_t \left(P^{(2,a)}_{j_1, j_2} (s_{j_1}, s_{j_2}) - ... ...2}) \right) = O({\epsilon}^4) \quad (j_1, j_2 \in {\cal B}_N)$$ (40)

if $P^{(4,a)}_{j_1, j_2, j_3, j_4} (s_{j_1}, s_{j_2}, s_{j_3}, s_{j_4}) = P^{(a)}_{... ...s_{j_1}) P^{(a)}_{j_2}(s_{j_2}) P^{(a)}_{j_3}(s_{j_3}) P^{(a)}_{j_4}(s_{j_4})$ at $t=0$. Here $P^{(4,a)}_{j_1, j_2, j_3, j_4} (s_{j_1}, s_{j_2}, s_{j_3}, s_{j_4})$, $P^{(2,a)}_{j_1, j_2} (s_{j_1}, s_{j_2})$ and $P^{(a)}_{j} (s_{j})$ are the four-mode, two-mode and one-mode PDF's obtained from $\cal P$ by integrating out all but 3,2 or 1 arguments respectively. One can see that, with a ${\epsilon}^2$ accuracy, the Fourier modes will remain independent of each other in any pair over the nonlinear time if they were independent in every triplet at $t=0$.

Similarly, one can show that the modes will remain independent over the nonlinear time in any subset of $M<N$ modes with accuracy $M/N$ (and ${\epsilon}^2$) if they were initially independent in every subset of size $M+1$. Namely

$$P^{(M,a)}_{j_1, j_2, \dots , j_M} (s_{j_1}, s_{j_2}, s_{j_M}) - P... ...^{(a)}_{j_2}(s_{j_2} ) \dots P^{(a)}_{j_M}(s_{j_M} ) = O(M/N) + O({\epsilon}^2)$$

$$\quad (j_1, j_2, \dots, j_M \in {\cal B}_N)$$ (41)

if $P^{(M+1,a)}_{j_1, j_2, \dots, j_{M+1}} = P^{(a)}_{j_1} P^{(a)}_{j_2} \dots P^{(a)}_{j_{M+1}}$ at $t=0$.

Mismatch $O(M/N)$ arises from some terms in the ZS equation with coinciding indices $j$. For $M=2$ there is only one such term in the $N$-sum and, therefore, the corresponding error is $O(1/N)$ which is much less than $O(\epsilon^2)$ (due to the order of the limits in $N$ and $\epsilon$). However, the number of such terms grows as $M$ and the error accumulates to $O(M/N)$ which can greatly exceed $O(\epsilon^2)$ for sufficiently large $M$.

We see that the accuracy with which the modes remain independent in a subset is worse for larger subsets and that the independence property is completely lost for subsets approaching in size the entire set, $M \sim N$. One should not worry too much about this loss because $N$ is the biggest parameter in the problem (size of the box) and the modes will be independent in all $M$-subsets no matter how large. Thus, the statistical objects involving any finite number of modes are factorisable as products of the one-mode objects and, therefore, the WT theory reduces to considering the one-mode objects. This results explains why we re-defined RPA in its relaxed ``essential RPA'' form. Indeed, in this form RPA is sufficient for the WT closure and, on the other hand, it remains valid over the nonlinear time. In particular, only property (40) is needed, as far as the amplitude statistics is concerned, for deriving the 3-wave kinetic equation, and this fact validates this equation and all of its solutions, including the KZ spectrum which plays an important role in WT.

The situation were modes can be considered as independent when taken in relatively small sets but should be treated as dependent in the context of much larger sets is not so unusual in physics. Consider for example a distribution of electrons and ions in plasma. The full $N$-particle distribution function in this case satisfies the Liouville equation which is, in general, not a separable equation. In other words, the $N$-particle distribution function cannot be written as a product of $N$ one-particle distribution functions. However, an $M$-particle distribution can indeed be represented as a product of $M$ one-particle distributions if $M \ll N_D$ where $N_D$ is the number of particles in the Debye sphere. We see an interesting transition from a an individual to collective behaviour when the number of particles approaches $N_D$. In the special case of the one-particle function we have here the famous mean-field Vlasov equation which is valid up to $O(1/N_D)$ corrections (representing particle collisions).