In contrast with Chapter 2, in this chapter we include the dynamics of the thermal cloud. As noted in Chapter 1, we treat the noncondensate atoms using the simplest microscopic model approximation that captures the important physics. In particular, we consider only temperatures high enough (T ≥ 0.4TBEC) that the noncondensate atoms can be described by a particle-like Hartree–Fock (HF) spectrum. To extend the analysis to very low temperatures is in principle straightforward (see Chapter 7). However, the details are more complicated since the excitations of the thermal cloud take on a collective aspect (i.e. a Bogoliubov-type quasiparticle spectrum must be used). In trapped Bose gases, the HF single-particle spectrum gives a good approximation down to much lower temperatures than in the case of uniform Bose gases, as first emphasized by Giorgini et al. (1997).

In Section 3.1, we derive a generalized form of the Gross–Pitaevskii equation for the Bose order parameter Φ(r, t) that is valid at finite temperatures. It involves terms that are coupled to the noncondensate component (the thermal cloud) and thus its solution in general requires one to know the equations of motion for the dynamics of the noncondensate atoms. In Section 3.2 we restrict ourselves to finite temperatures high enough that the noncondensate atoms can be described by a quantum kinetic equation for the single-particle distribution function f(p, r, t). A detailed microscopic derivation of this kind of kinetic equation is given in Chapters 6 and 7 using the Kadanoff–Baym Green's function formalism.

A characteristic feature of a Bose-condensed gas is that the kinetic equation governing f(p, r, t) involves a collision integral C12[f, Φ] describing collisions between condensate and noncondensate atoms.