... taken.¹

It is easily to extend the analysis to the infinite Fourier space, $k_{max} = \infty$. In this case, the full joint PDF would still have to be defined as a $N \to \infty$ limit of an $N$-mode PDF, but this limit would have to be taken in such a way that both $k_{max}$ and the density of the Fourier modes tend to infinity simultaneously.

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... scales.²

Hereafter we omit superscript ${(N)}$ in the $N$-mode objects if it does not lead to a confusion.

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... respectively. ³

This technique provides a useful classification method but not a complete mathematical description of the terms involved.

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... means⁴

In the present paper we consider only spatially homogeneous wave turbulence fields. In spatially homogeneous fields, due to momentum conservation, there is no coupling to the zero mode $k=0$ because such coupling would violate momentum conservation. Therefore if one of the arguments of the interaction matrix element $V$ is equal to zero, the matrix element is identically zero. That is to say that for any spatially homogeneous wave turbulence system $V^{k=0}_{k_1 k_2} = V^{k}_{k_1=0 k_2} = V^{k}_{k_1 k_2=0} =0.$

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... factors.⁵

There is of course also a possibility that $k$ couples simultaneously to both indices in a pair, but this contribution contains $\sim 1/N$ less terms and, therefore, should be ignored.

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... graphs. ⁶

The only possibility of the double internal coupling would be in the last graph via joining $m$ with $\nu$ and $\kappa$ with $n$, but this would mean $j=0$ because of the $\delta$-symbols and, therefore, this term is nill.

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