Boltzmann equation and Hamiltonian

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Main Question or Discussion Point

Hello! I read today, in the context of DM, about the Boltzmann equation: $$L[f]=C[f]$$ where $L[f]$ is the Liouville operator (basically $\frac{df}{dt}$), with $f(x,v,t)$ being the phase-space distribution of the system and $C[f]$ being the collision operator. I am a bit confused about how should I think about this in general. When you calculate $L[f]$ you use the Hamiltonian, which includes the potential energy, hence all the interactions of the system. Then, what does $C[f]$ account for? Initially I thought that it is used when you have an external system interacting with the original one. But in the context of DM, it seems that it can be a self interaction (DM annihilation for example). So, now I assume that you don't really put all your interactions in the Hamiltonian, but I am not sure. Do you just put the gravity in the case of DM? And treat other interactions (weak force for example) as part of the collision operator? And in general, what you add to the Hamiltonian and what to the collision operator? Thank you!

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