Collective fast neutrino flavor conversions in an 1D box: (I) initial condition and long-term evolution

We perform numerical simulations of fast collective neutrino flavor conversions in an one-dimensional box mimicking a system with the periodic boundary condition in one spatial direction and translation symmetry in the other two dimensions. We evolve the system over several thousands of the characteristic timescale (inverse of the interaction strength) with different initial $\bar\nu_e$ to $\nu_e$ number density ratios and different initial seed perturbations. We find that small scale structures are formed due to the interaction of the flavor waves. This results in nearly flavor depolarization in a certain neutrino phase space, when averaged over the entire box. Specifically, systems with initially equal number of $\nu_e$ and $\bar\nu_e$ can reach full flavor depolarization for the entire neutrino electron lepton number ($\nu$ELN) angular spectra. For systems with initially unequal $\nu_e$ and $\bar\nu_e$, flavor depolarization can only be reached in one side of the $\nu$ELN spectra, dictated by the net neutrino $e-x$ lepton number conservation. Quantitatively small differences depending on the initial perturbations are also found when different perturbation seeds are applied. Our numerical study here provides new insights for efforts aiming to include impact of fast flavor conversions in astrophysical simulations while calls for better analytical understanding accounting for the evolution of fast flavor conversions.