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Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive the self-consistent Hartree-Fock-Bogoliubov (HFB) collisionless kinetic equations and the associated equation of motion for the condensate wavefunction for a trapped Bose-condensed gas. Our work generalizes earlier work by Kane and Kadanoff (KK) for a uniform Bose gas. We include the off-diagonal (anomalous) pair correlations, and thus we have to introduce an off-diagonal distribution function in addition to the normal (diagonal) distribution function. This results in two coupled kinetic equations. If the off-diagonal distribution function can be neglected as a higher-order contribution, we obtain the semi-classical kinetic equation recently used by Zaremba, Griffin and Nikuni (based on the simpler Popov approximation). We discuss the static local equilibrium solution of our coupled HFB kinetic equations within the semiclassical approximation. We also verify that a solution is the rigid in-phase oscillation of the equilibrium condensate and non-condensate density profiles, oscillating with the trap frequency.

Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive kinetic equations for the non-condensate atoms at finite temperatures which include the effect of binary collisions between atoms. The effect of collisions is included using the second-order self-energy given by the Beliaev (gapless) approximation. We limit our discussion to finite temperatures where we can use the single-particle Hartree-Fock spectrum for the excited atoms. In this limit, we can neglect the off-diagonal propagators (g 12 and g 21 ). As expected, this leads to the kinetic equations and collision integrals used in recent work by Zaremba, Nikuni, and Griffin (ZNG) [1]. We also derive a consistent equation of motion for the condensate wavefunction, involving a finite-temperature generalization of the well-known Gross-Pitaevskii equation which includes a dissipative term, as well as the mean field of the non-condensate.

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