Equation for the multi-mode PDF

Taking the inverse Laplace transform of (64) we have the following equation for the PDF,

$$\dot {\cal P} = - \int {\delta {\cal F}_j \over \delta s_j} \, dk_j,$$ (59)

where ${\cal F}_j$ is a flux of probability in the space of the amplitude $s_j$,

$$-{{\cal F}_j \over 4 \pi {\epsilon}^2 s_j} = \int \big\{ (\vert V_{mn}^{j}\vert^2 \delta(\omega_{mn}^{j}) \del... ...(\omega_{jm}^{n}) \delta_{j+m}^{n} ) s_n s_m {\delta {\cal P} \over \delta s_j}$$

$$+2 {\cal P} ( \vert V_{jm}^{n}\vert^2 \delta(\omega_{jm}^{n}) \de... ...+m}^{n} - \vert V_{mn}^{j}\vert^2 \delta(\omega_{mn}^{j}) \delta_{m+n}^{j} )s_m$$

$$+2 (\vert V_{jm}^{n}\vert^2 \delta(\omega_{jm}^{n}) \delta_{j+m}^... ...lta_{m+n}^{j} ) s_n s_m {\delta {\cal P} \over \delta s_m} \big\} \, dk_m dk_n.$$ (60)

This equation is identical to the one originally obtained by Peierls [19] and later rediscovered by Brout and Prigogine [20] in the context of the physics of anharmonic crystals,

$$\dot {\cal P} = {16 \pi {\epsilon}^2 } \int \vert V_{mn}^{j}\vert... ...s_n \left[{\delta \over \delta s} \right]_3 {\cal P} \right) \, dk_j dk_m dk_n,$$ (61)

where

$$\left[{\delta \over \delta s} \right]_3 = {\delta \over \delta s_j} - {\delta \over \delta s_m} -{\delta \over \delta s_n}.$$ (62)

Zaslavski and Sagdeev [21] were the first to study this equation in the WT context. However, the analysis of [19,20,21] was restricted to the interaction Hamiltonians of the ``potential energy'' type, i.e. the ones that involve only the coordinates but not the momenta. This restriction leaves aside a great many important WT systems, e.g. the capillary, Rossby, internal and MHD waves. Our result above indicates that the Peierls equation is also valid in the most general case of 3-wave systems. Note that Peierls form (67) - (68) looks somewhat more elegant and symmetric than (65) - (66). However, form (65) - (66) has advantage because it is in a continuity equation form. Particularly for steady state solutions, one can immediately integrate it once and obtain,

$${\cal F}\{s(k_l)\} = \hbox{curl}_s A = \int {\epsilon}_{lnm} {\delta A\{s(k_n)\} \over \delta s(k_m)} \, dk_m dk_n,$$ (63)

where $A$ is an arbitrary functional of $s(k)$ and ${\epsilon}_{lnm}$ is the antisymmetric tensor,

$${\epsilon}_{lnm} = 1 \hbox{for} k_l > k_m > k_n \; \; \hbox{and cyclic permutations of} \;\;\; l,m,n,$$

$${\epsilon}_{lnm} = -1 \hbox{for all other} k_l \ne k_m \ne k_n,$$

$${\epsilon}_{lnm} = 0 \hbox{if } k_l=k_m , k_l= k_n \; \hbox{or} \; k_n=k_m.$$

In the other words, probability flux $F$ can be an arbitrary solenoidal field in the functional space of $s(k)$. One can see that (69) is a first order equation with respect to the $s$-derivative. Special cases of the steady solutions are the zero-flux and the constant-flux solutions which, as we will see later correspond to a Gaussian and intermittent wave turbulence respectively.

Here we should again emphasise importance of the taken order of limits, $N \to \infty$ first and ${\epsilon}\to 0$ second. Physically this means that the frequency resonance is broad enough to cover great many modes. Some authors, e.g. [19,20,21], leave the sum notation in the PDF equation even after the ${\epsilon}\to 0$ limit taken giving $\delta(\omega_{jm}^{n})$. One has to be careful interpreting such formulae because formally the RHS is nill in most of the cases because there may be no exact resonances between the discrete $k$ modes (as it is the case, e.g. for the capillary waves). In real finite-size physical systems, this condition means that the wave amplitudes, although small, should not be too small so that the frequency broadening is sufficient to allow the resonant interactions. Our functional integral notation is meant to indicate that the $N \to \infty$ limit has already been taken.