Conference ID: 937-594-57325
PIN Code: 104126
According to Bohr’s correspondence principle in quantum mechanics, the evolution of a particle whose action is very large when compared to Planck’s constant can be approximately described by Newton’s laws of classical mechanics. In this asymptotic regime, the quantum density describing the state of the particle, which is an operator on a Hilbert space, « converge » to a « distribution function » on the particle classical phase space. The purpose of this talk is to explain how some ideas of optimal transport can be used to « metrize » the set of quantum densities, and to estimate the difference between quantum densities and distribution functions by a quantity which is nicely propagated by the quantum dynamics, and converges to the Euclidean distance between phase space points in the classical limit. Applications include the uniformity of the mean-field limit in quantum mechanics, and uniform uniform error bounds for time splitting methods for the von Neumann equation in the semiclassical regime. (Based on works in collaboration with Shi Jin, Clément Mouhot and Thierry Paul).
François Golse is currently Professor of Mathematics at Ecole Polytechnique (Paris area, France). His research interests are partial differential equations and mathematical physics, especially kinetic models and their connection with fluid dynamics, and, more recently, the quantum dynamics of large particle systems in the semiclassical and mean-field regimes. Together with Laure Saint-Raymond, he has been awarded the first PDE prize of the Society for Industrial and Applied Mathematics (SIAM) in 2006. He has given the 1993 Peccot Lectures at Collège de France (Paris), and the 2010 Harold Grad Lecture in the 27th International Symposium on Rarefied Gas Dynamics (Pacific Grove, USA). He has been a plenary speaker at the 2004 European Congress of Mathematics (Stockholm), and an invited speaker at the 2006 International Congress of Mathematicians (Madrid).