The collective oscillations of a condensate at zero temperature are well described by the solutions of the linearized time-dependent Gross–Pitaevskii (GP) equation of motion for the condensate wavefunction Φ(r, t). At finite temperatures, however, the condensate dynamics is modified by interactions with the noncondensate atoms that comprise the thermal cloud. To account for these interactions in detail involves a sophisticated numerical analysis, which will be described in Chapter 11. However, some qualitative understanding of the effect of collisions between the condensate and noncondensate components can be gained by treating the thermal cloud within an approximation that ignores its dynamics. This approximation, referred to as the static thermal cloud approximation, is the topic of the present chapter. As explained in more detail below, it is defined by the assumption that the condensate moves in the presence of a thermal cloud that remains in a state of thermal equilibrium. Thus, if the condensate is induced to oscillate, it initially departs from equilibrium with the thermal cloud, but collisions lead to a damping of the condensate oscillation and ultimately equilibrate the two components. This collisional damping is in addition to the usual Landau and Beliaev damping, which is present even in the “collisionless” regime.

The approximate version of the fully coupled ZNG equations to be discussed here provides the simplest finite-temperature extension of the theory of condensate dynamics based on the usual GP equation. The extent to which the treatment gives a reasonable first approximation will be examined in Chapter 11. It will be shown that the static thermal cloud approximation does provide a qualitative understanding of the damping of modes in which the condensate is the main participant.