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
MathSciNet review:
3580096
Full-text PDF
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Additional Information
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
MR Author ID:
77045
Email:
mng@math.umd.edu
Matei Machedon
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
MR Author ID:
117610
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
Keywords:
Quantum dynamics,
weakly interacting Bosons,
mean-field limit,
Bogoliubov transformation,
pair excitation,
non-linear Schrödinger 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