Convergence of the quantum Boltzmann map

Abstract

We consider a non-linear map on the space of density matrices, which we call the Boltzmann map τ. It is the composition of a doubly stochastic mapT on the space ofn-body states, and the conditional expectation onto the one-body space. WhenT is ergodic, then the iterates of τ take any initial state to the uniform distribution. If the energy levels are equally spaced, andT conserves energy and is ergodic on each energy shell, then iterates of τ take any initial state of finite energy to a canonical distribution.