2010/09/04 by László Erdős, Laszlo Erdos, Erdos, Laszlo · 1 voice · 1 citation
Mathematics · Physics and Astronomy · #60J65 #81T18 #82C10 #82C44 #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #math-ph #math.MP #msc:60J65 #msc:81T18 #msc:82C10 #msc:82C44
paper · pdf · doi:10.48550/arxiv.1009.0843
92 pages, 28 figures
arxiv created 2010/09/04 · openalex publication_date 2010/09/04 · arxiv published 2010/09/04 · arxiv updated 2010/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Schrödinger 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, 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.