entire Brillouin zone, as given in (\[eq:bog\_distr\]), and 2) there are regions in phase space for which energy and momentum are conserved according to (\[eq:E\_cons\]), which allows the Bogoliubov modes to scatter and redistribute (quasi)resonantly.

The Truncated Wigner Method {#sec:TWA}
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A possible approach to compute corrections beyond Bogoliubov and quantify the observable signatures of the Beliaev-Landau processes is the so-called Truncated Wigner approximation (TWA) [@gardiner_quantum_noise; @vogel_twa; @castin_TWA; @polkovnikov_phase_space]. This approach is based on a one-to-one mapping of the quantum master equation for the density matrix (\[eq:q\_master\]) onto a partial differential equation for the corresponding Wigner distribution. The resulting Fokker-Planck equation can be efficiently simulated if the terms with a third-order derivative are neglected. Since these terms are proportional to the single-particle interaction constant $U$, one expects this approximation to be accurate for sufficiently weak values of $U$ [@Iacopo_QFL]. This leads to a stochastic differential equation for a classical field $\varphi_j(t)$ $$\label{eq:TWA}
\begin{split}
i d\varphi_j(t) = \Big[-\Big(\delta + i\frac{\gamma}{2}\Big)\varphi_j(t) -J\big(\varphi_{j+1}(t) + \varphi_{j-1}(t)\big)\\
 + U\big(|\varphi_j(t)|^2-1\big)\varphi_j(t) + \Omega_j(t) \Big]dt + \sqrt{ \frac{\gamma}{2}} dW_j(t),
\end{split}$$ where the stochastic Wiener increment $dW_j(t)$ is white Gaussian noise with variance $\big\langle dW^\ast_j(t) dW_{j'}(t) \big\rangle = \delta_{j,j'} dt$ and a random phase. Average values of the field 