In this review paper we describe the problem of deriving a Boltzmann equation for a system of $N$ interacting quantum particles, under the appropriate scaling limits. We mainly follow the approach developed by the authors in previous works. From a rigorous viewpoint, only partial results are available, even for short times, so that the complete problem is still open.
"A Short Review on the Derivation of the Nonlinear Quantum Boltzmann Equations." Commun. Math. Sci. 5 (S1) 55 - 71, February 2007.