Number-conserving rate equation for sympathetic cooling of a boson gas
  We derive a particle number-conserving rate equation for the ground state and for the elementary excitations of a bosonic system which is in contact with a gas of a different species (sympathetic cooling). We use the Giradeau-Arnowitt method and the model derived by Lewenstein et. al. with an additional assumption: the high-excited levels thermalize much faster with the cooling agent than the other levels. Evaporation of particles, know to be important in the initial stages of the cooling process, is explicitly included. 
A.N. Salgueiro