On the dynamics of finite temperature trapped Bose gases

Abstract

The system that describes the dynamics of a Bose–Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators C12 and C22, which describe collisions of the condensate and the non-condensate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only a collision operator similar to C22, which possesses a blow-up positive radial solution with respect to the L∞ norm (cf. [29]). On the other hand, at the very low temperature regime (only a portion of the transition temperature TBEC), the system can be simplified into an equation of C12, with a different (much higher order) transition probability, which has a unique global classical positive radial solution with weighted L1 norm (cf. [3]). In our model, we first decouple the QB, which contains C12+C22, and the NLS equations, then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in [29], due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in [29]. A new theory is then supplied.