Generalized Boltzmann equation for a trapped Bose-condensed gas using the Kadanoff-Baym formalism

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 ($\tilde{g}_{12}$ and $\tilde{g}_{21}$). 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.