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 $1\leftrightarrow 2$ 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.

中文翻译：

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Bose-Einstein凝聚物的Bogoliubov激发的Boltzmann型碰撞算子：统一框架

从在玻色-爱因斯坦凝聚物存在下弱相互作用的玻色气体的哈密顿的波哥廖布夫对角化开始，我们得出了波哥廖布夫激发的动力学方程。在不丢掉任何换向器的情况下，我们发现了三个碰撞过程。其中之一描述了$\mathrm{1个}\leftrightarrow 2$冷凝物和激发原子之间的相互作用。另外两个描述$2\leftrightarrow 2$ 和 $\mathrm{1个}\leftrightarrow 3$ 激发原子之间的相互作用。