Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Evolution of the Boson gas at zero temperature: Mean-field limit and second-order correction


Authors: Manoussos Grillakis, Matei Machedon and Dionisios Margetis
Journal: Quart. Appl. Math. 75 (2017), 69-104
MSC (2010): Primary 35Q40, 82C10; Secondary 81V45, 70F10
DOI: https://doi.org/10.1090/qam/1455
Published electronically: August 24, 2016
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Abstract: A large system of $ N$ integer-spin atoms, called Bosons, manifests one of the most coherent macroscopic quantum states known to date, the ``Bose-Einstein condensate'', at extremely low temperatures. As $ N\to \infty $, this system is usually described by a mean-field limit: a single-particle wave function, the condensate wave function, that satisfies a nonlinear Schrödinger-type equation. In this expository paper, we review kinetic aspects of the mean-field Boson evolution. Furthermore, we discuss recent advances in the rigorous study of second-order corrections to this mean-field limit. These corrections originate from the quantum-kinetic mechanism of pair excitation, which lies at the core of pioneering works in theoretical physics including ideas of Bogoliubov, Lee, Huang, Yang and Wu. In the course of our exposition, we revisit the formalism of Fock space, which is indispensable for the analysis of pair excitation.


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Additional Information

Manoussos Grillakis
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: mng@math.umd.edu

Matei Machedon
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: mxm@math.umd.edu

Dionisios Margetis
Affiliation: Department of Mathematics, and Institute for Physical Science and Technology, and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, Maryland 20742
Email: dio@math.umd.edu

DOI: https://doi.org/10.1090/qam/1455
Keywords: Quantum dynamics, weakly interacting Bosons, mean-field limit, Bogoliubov transformation, pair excitation, non-linear Schr\"odinger equation
Received by editor(s): June 29, 2016
Published electronically: August 24, 2016
Additional Notes: The third author was partly supported by the National Science Foundation through Grant DMS-1517162.
Article copyright: © Copyright 2016 Brown University

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