| Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive the equations of motion for the dynamics of a trapped dilute Bose-condensed atomic gas at finite temperatures. These include an equation of motion for the macroscopic wavefunction describing the atoms in the condensate, as well as a Boltzmann-like kinetic equation for the single-particle distribution function describing the atoms thermally excited out of the condensate. We derive these equations using several self-energy approximations for the non-condensate atoms. At high enough temperatures, one can work within the first order Hartree-Fock-Bogoliubov approximation for the Beliaev self-energy functions and derive a "collisionless" kinetic equation. We then extend this analysis to include the second-order self-energy contributions associated with two-body collisions, leading to a Boltzmann-like equation for Bose atoms, with the collision integrals included. This equation, combined with the generalized Gross-Pitaevskii equation of motion for the condensate atoms, has been extensively used by Zaremba, Nikuni and Griffin. The role of collisions which transfers atoms between the condensate and non-condensate component is emphasized. The major result of this thesis is a generalization of previous studies to very low temperatures, where the non-condensate thermal cloud atoms are treated in the so-called Bogoliubov-Popov approximation. We derive a kinetic equation for the resulting quasiparticles (instead of atoms) and show how the collision integrals are modified by Bogoliubov coherence factors. A detailed comparison is made with previous discussion of the non-equilibrium behaviour of uniform interacting Bose gases, including the work of Kirkpatrick and Dorfman as well as Eckern. |
