Dynamics of trapped bose gases at finite temperatures

Starting from an approximate microscopic model of a trapped Bose-condensed gas at finite temperatures, we derive an equation of motion for the condens ate wavefunction and a quantum kinetic equation for the distribution functi on for the excited atoms. The kinetic equation is a generalization of our e arlier work in that collisions between the condensate and non-condensate (C -12) are now included, in addition to collisions between the excited atoms as described by the Uehling-Uhlenbeck (C-22) collision integral. The contin uity equation for the local condensate density contains a source term Gamma (12) which is related to the C-12 collision term. If we assume that the C-2 2 collision rate is sufficiently rapid to ensure that the non-condensate di stribution function can be approximated by a local equilibrium Bose distrib ution, the kinetic equation can be used to derive hydrodynamic equations fo r the non-condensate. The Gamma(12) source terms appearing in these equatio ns play a key role in describing the equilibration of the local chemical po tentials associated with the condensate and non-condensate components. We g ive a detailed study of these hydrodynamic equations and show how the Landa u two-fluid equations emerge in the frequency domain omega tau(mu) much les s than 1, where tau(mu) is a characteristic relaxation time associated with C-12 collisions. More generally, the lack of complete local equilibrium be tween the condensate and non-condensate is shown to give rise to a new rela xational mode which is associated with the exchange of atoms between the tw o components. This new mode provides an additional source of damping in the hydrodynamic regime. Our equations are consistent with the generalized Koh n theorem for the center of mass motion of the trapped gas even in the pres ence of collisions. Finally, we formulate a variational solution of the equ ations which provides a very convenient and physical way of estimating norm al mode frequencies. In particular, we use relatively simple trial function s within this approach to work out some of the monopole, dipole and quadrup ole oscillations for an isotropic trap.