A derivation of the Boltzmann equation from quantum many-body dynamics

We start by introducing a statistical model for the initial data of an N𝑁-body Schrödinger equation, meant to represent a scaled version of an N𝑁-particle quantum system with unit-order velocities and interparticle separations. The statistical model yields the expected functional form and scale of the corresponding BBGKY densities. This motivates a general a priori assumption on the Sobolev space norms of the BBGKY densities, which includes quasi-free states. Under this assumption, we prove that the Wigner transformed densities converge to the Boltzmann hierarchy with quadratic collision kernel and quantum scattering cross section. The proof of convergence uses a framework previously applied to the derivation of Bose Einstein condensate from an N𝑁-body model, and involves exploiting uniform bounds to obtain compactness and weak convergence. The remaining step is to prove the uniqueness of limits, which is performed using the Hewitt-Savage theorem and an extension of the Klainerman-Machedon board game. Our derivation is optimal with respect to regularity considerations. This is joint work with Xuwen Chen, University of Rochester.