of these collisions are quasiparticles with smaller momenta, which explains the massive pileup in the momentum distribution around $q_\text{min}$, at the cost of a strong negative dip around $k_\text{max}$. Of course, the TWA-simulated momentum distribution recovers relatively well to the Bogoliubov result for all $k$ values outside of the region $[q_\text{min},k_\text{max}]$ for which there are no resonant Beliaev-Landau scatterings possible. In Appendix \[app:TWA\] we include a simulation of a model without energy-conserving Beliaev-Landau channels and we conclude that in this case the occupation of all modes is positive and much better convergence to the Bogoliubov result is achieved. While this inaccuracy of the TWA is not expected to affect the predictions for dynamical Casimir and Hawking emission [@koghee2014dynamical; @Hawking; @Hawking2] that are at the level of Bogoliubov theory, special care will be needed in the more advanced study of back-reaction effects in analog models of gravity [@balbinot_br].

As far as we we know, there is no simple solution to this intrinsic problem of the TWA. Note that related problems with the TWA are known also in the conservative case of ultracold atomic Bose gases [@castin_TWA]. The equipartition theorem for the classical fields states in fact that the momentum distribution should 