Einstein's kinetic theory of the Brownian motion, based upon light water
molecules continuously bombarding a heavy pollen, provided an explanation of
diffusion from the Newtonian mechanics. Since the discovery of quantum
mechanics it has been a challenge to verify the emergence of diffusion from the
Schr\"odinger equation.

The first step in this program is to verify the linear Boltzmann equation as
a certain scaling limit of a Schr\"odinger equation with random potential. In
the second step, one considers a longer time scale that corresponds to
infinitely many Boltzmann collisions. The intuition is that the Boltzmann
equation then converges to a diffusive equation similarly to the central limit
theorem for Markov processes with sufficient mixing. In these lecture notes
(prepared for the Les Houches summer school in 2010 August) we present the
mathematical tools to rigorously justify this intuition. The new material
relies on joint papers with H.-T. Yau and M. Salmhofer.