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Einstein's kinetic theory of Brownian motion, based upon light water molecules continuously bombarding a heavy pollen, provided an explanation of diffusion from Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from Schrödinger's equation. The first step in this program is to verify the linear Boltzmann equation as a certain scaling limit of a Schrödinger equation with random potential. In the second step, a longer time scale that corresponds to infinitely many Boltzmann collisions is considered. 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 this chapter the mathematical tools to justify this intuition rigorously is presented. This new material relies on joint papers with H. -T. Yau and M. Salmhofer.

*Keywords: *
quantum Brownian Motion;
Boltzmann Equation;
quantum diffusion

*Chapter.*
*44044 words.*
*Illustrated.*

*Subjects: *
Atomic, Molecular, and Optical Physics

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