Boltzmann-type collision operators for Bogoliubov excitations of Bose-Einstein condensates: A unified framework.

Starting from the Bogoliubov diagonalization for the Hamiltonian of a weakly interacting Bose gas under the presence of a Bose-Einstein condensate, we derive the kinetic equation for the Bogoliubov excitations. Without dropping any of the commutators, we find three collisional processes. One of them describes the 1↔2 interactions between the condensate and the excited atoms. The other two describe the 2↔2 and 1↔3 interactions between the excited atoms themselves.