  This is based on 4 lectures given at the 13th Australian Physics Summer
School, Australia National University, Canberra, Jan 17-28, 2000. The main
topic is the theory of collective modes in a trapped Bose gas at finite
temperatures. A generalized Gross-Pitaevskii equation is derived at finite
temperatures, which is used to discuss a new mechanism for damping in the
collisionless region arising from interactions with a static thermal cloud of
non-condensate atoms. Next, introducing a kinetic equation for the thermal
cloud, we derive two-fluid equations of motion for the condensate and
non-condensate components in the collision-dominated hydrodynamic region. We
show that these are precisely the equivalent of the Landau two-fluid equations
in the limit that the two components are in diffusive local equilibrium.
However, our equations also predict the existence of a new zero frequency
relaxational mode, in addition to the usual Landau hydrodynamic modes (such as
first and second sound). The special importance and simplicity of two-fluid
hydrodynamics is stressed.
