Equation for the PDF

Taking the inverse Laplace transform of (35) we have the following equation for the PDF,

$$\dot {\cal P} = - \int {\delta F_j \over \delta s_j} \, dk_j,$$ (35)

where $F_j$ is a flux of probability in the space of the amplitude $s_j$,

$$F_j = 4 \pi {\epsilon}^2 \int \big\{ (\vert V_{mn}^{j}\vert^2 \delta(\o... ...ft[ s_n s_m {\cal P} - {\delta \over \delta s_j} (s_j s_n s_m {\cal P}) \right]$$

$$-2 {\cal P} \left[\vert V_{mn}^{j}\vert^2 \delta(\omega_{mn}^{j})... ...{n}\vert^2 \delta(\omega_{jm}^{n}) \delta_{j+m}^{n} (s_j s_m - s_j s_n) \right]$$

$$-2 (\vert V_{jm}^{n}\vert^2 \delta(\omega_{jm}^{n}) \delta_{j+m}^... ...+n}^{j} ) {\delta \over \delta s_m} (s_j s_n s_m {\cal P}) \big\} \, dk_m dk_n.$$ (36)

This expression can be simplified to

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

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

$$+2 (\vert V_{jm}^{n}\vert^2 \delta(\omega_{jm}^{n}) \delta_{j+m}^... ...lta_{m+n}^{j} ) s_n s_m {\delta {\cal P} \over \delta s_m} \big\} \, dk_m dk_n.$$ (37)

This equation is identical to the one originally obtained by Peierls [15] and later rediscovered by Brout and Prigogine [16] in the context of the physics of anharmonic crystals. Zaslavski and Sagdeev [17] were the first to study this equation in the WT context. However, the analysis of [15,16,17] 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.

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. [15,16,17], 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.