  We develop the moment method for Bose-Einstein condensates (BECs) at finite
temperatures that enable us to study collective sound modes from the
hydrodynamic to the collisionless regime. In particular, we investigate
collective excitations in a weakly interacting dilute Bose gas by applying the
moment method to the Zaremba-Nikuni-Griffin equation, which is the coupled
equation of the Boltzmann equation with the generalized Gross-Pitaevskii
equation. Utilizing the moment method, collective excitations in the crossover
regime between the hydrodynamic and collisionless regimes are investigated in
detail. In the crossover regime, the second sound mode loses the weight of the
density response function because of the significant coupling with incoherent
modes, whereas the first sound shows a distinct but broad peak structure. We
compare the result obtained by the moment method with that of the Landau
two-fluid equations and show that the collective mode predicted by the Landau
two-fluid equations well coincides with the result from the moment method even
far from the hydrodynamic regime, whereas clear distinction also emerges in the
relatively higher momentum regime.
