  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.
