The collision term in the Boltzmann equation is due to microscopic scattering of particles. In the simplest derivations of the equation from quantum field theory (rather than from plausibility arguments), it just accounts for 2-particle scattering in the Born approximation. But there are more complete derivations that include more complete scattering information, either higher order terms or multiparticle scattering. For example,. http://dx.doi.org/10.1103/PhysRevD.73.025005 includes multiparticle scattering in the tree approximation.

I believe that I had seen long ago a derivation that produced the complete scattering behavior (i.e., involved the full multiparticle S-matrix or T-matrix), but I don't remember the details or where I read this. It probably was a derivation from nonrelativistic QFT.

Can anyone help me to locate a suitable reference?