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We develop the moment method for Bose-Einstein condensates (BECs) at finitetemperatures that enable us to study collective sound modes from thehydrodynamic to the collisionless regime. In particular, we investigatecollective excitations in a weakly interacting dilute Bose gas by applying themoment method to the Zaremba-Nikuni-Griffin equation, which is the coupledequation of the Boltzmann equation with the generalized Gross-Pitaevskiiequation. Utilizing the moment method, collective excitations in the crossoverregime between the hydrodynamic and collisionless regimes are investigated indetail. In the crossover regime, the second sound mode loses the weight of thedensity response function because of the significant coupling with incoherentmodes, whereas the first sound shows a distinct but broad peak structure. Wecompare the result obtained by the moment method with that of the Landautwo-fluid equations and show that the collective mode predicted by the Landautwo-fluid equations well coincides with the result from the moment method evenfar from the hydrodynamic regime, whereas clear distinction also emerges in therelatively higher momentum regime.

Collective excitations of a Bose-condensed gas: Fate of second sound in the crossover regime between hydrodynamic and collisionless regimes