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-Enstein 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 $1\leftrightarrow 2$ interactions between the condensate and the excited atoms. The other two describe the $2\leftrightarrow 2$ and $1\leftrightarrow 3$ interactions between the excited atoms themselves.