Vortices in Bose-Einstein Condensates (eBook)

Amandine Aftalion (Autor)

eBook Download: PDF

2007 | 2006
XII, 203 Seiten
Birkhauser Boston (Verlag)
978-0-8176-4492-5 (ISBN)

Infos

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One of the key issues related to superfluidity is the existence of vortices. In very recent experiments on Bose-Einstein condensates, vortices have been observed by rotating the trap holding the atoms. In contrast to a classical fluid for which the equilibrium velocity corresponds to solid body rotation, a quantum fluid such as a Bose-Einstein condensate can rotate only through the nucleation of quantized vortices. This monograph is dedicated to the mathematical modelling of these phenomena.The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. The mathematical analysis is made in the framework of the Gross Pitaevskii energy. Results are presented and open problems related to recent experiments are explained.The work can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum condensates, and can also complement a graduate seminar in elliptic PDEs or modelling of physical experiments.

Inhalt

Preface 6
Contents 9
1 The Physical Experiments and Their Mathematical Modelling 13
1.1 A hint on the experiments 13
1.2 The mathematical framework 17
2 The Mathematical Setting: A Survey of the Main Theorems 30
2.1 Small e problem 30
2.2 Vortex lattice 35
2.3 Flow around an obstacle 38
3 Two-Dimensional Model for a Rotating Condensate 40
3.1 Main results 41
3.2 Preliminaries 46
3.3 Bounded number of vortices 49
3.4 Refined structure of vortices 57
3.5 Lower bound 66
3.6 Upper bound 72
3.7 Final expansion and properties of vortices 77
3.8 Open Questions 86
4 Other Trapping Potentials 89
4.1 Non radial harmonic potential 90
4.2 Quartic potential 92
4.3 Open questions 108
5 High-Velocity and Quantum Hall Regime 109
5.1 Introduction 110
5.2 Regular lattice 117
5.3 Distorted lattice 121
5.4 Infinite number of zeros 127
5.5 Other trapping potentials 129
5.6 Open questions 129
6 Three-Dimensional Rotating Condensate 132
6.1 Numerical simulations 133
6.2 Formal derivation of the reduced energy E[. ] 135
6.3 G convergence results 141
6.4 Single Vortex line, study of E[. ] 151
6.5 A few open questions 163
7 Superfluid Flow Around an Obstacle 166
7.1 Mathematical setting 167
7.2 Proof of Theorem 7.1 176
7.3 Proof of Theorem 7.2 189
8 Further Open Problems 203
8.1 Setting in the whole space for the Thomas–Fermi regime 203
8.2 Other scalings 204
8.3 Other models 205
References 207
Index 212

Produkt-Details

• Erscheint lt. Verlag: 14.10.2007
• Reihe/Serie: Progress in Nonlinear Differential Equations and Their Applications
• Zusatzinfo: XII, 203 p. 18 illus.
• Verlagsort: Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: Condensed Matter • Mathematical Modeling • Partial differential equations • Potential • Superfluidity • Vortices
• ISBN-10: 0-8176-4492-X / 081764492X
• ISBN-13: 978-0-8176-4492-5 / 9780817644925