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Other titles in the Progress in Nonlinear Differential Equations and Their Appli series:
Progress in Nonlinear Differential Equations and Their Appli #67: Vortices in BoseEinstein Condensatesby Amandine Aftalion
Synopses & ReviewsPublisher Comments:Since the first experimental achievement of BoseEinstein condensates (BEC) in 1995 and the award of the Nobel Prize for Physics in 2001, the properties of these gaseous quantum fluids have been the focus of international interest in physics. This monograph is dedicated to the mathematical modelling of some specific experiments which display vortices and to a rigorous analysis of features emerging experimentally. In contrast to a classical fluid, a quantum fluid such as a BoseEinstein condensate can rotate only through the nucleation of quantized vortices beyond some critical velocity. There are two interesting regimes: one close to the critical velocity, where there is only one vortex that has a very special shape; and another one at high rotation values, for which a dense lattice is observed. One of the key features related to superfluidity is the existence of these vortices. We address this issue mathematically and derive information on their shape, number, and location. In the dilute limit of these experiments, the condensate is well described by a mean field theory and a macroscopic wave function solving the socalled GrossPitaevskii equation. The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. We prove existence of solutions that have properties consistent with the experimental observations. Open problems related to recent experiments are presented. The work can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum fluids, and can also complement a graduate seminar in elliptic PDEs or modelling of physical experiments.
Synopsis:This book provides an uptodate approach to the diagnosis and management of endocarditis based on a critical analysis of the recent studies. It is the only uptodate clinically oriented textbook available on this subject. The book is structured in a format that is easy to follow, clinically relevant and evidence based. The author has a special interest in the application of ultrasound in the study of cardiac structure and function.
Synopsis:One of the key issues related to superfluidity is the existence of vortices. In very recent experiments on BoseEinstein 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 BoseEinstein condensate can rotate only through the nucleation of quantized vortices. This monograph is dedicated to the mathematical modeling of these phenomena.
One of the experiments studied focuses on rotating the trap holding the atoms. At low velocity, no modification of the condensate is observed, while beyond some critical value, vortices appear in the system. There are two interesting regimes: one close to the critical velocity where there is only one vortex; and another one at high rotation values, for which a dense lattice is observed. Another experiment that is studied consists of superfluid flow around an obstacle. At low velocity, the flow is stationary; while at larger velocity, vortices are nucleated from the boundary of the obstacle. The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. The mathematical analysis is made in the framework of the GrossPitaevskii 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 modeling of physical experiments. Table of ContentsPreface. The Physical Experiments and Their Mathematical Modelling. The Mathematical Setting: A Survey of the Main Theorems. Twodimensional Model for a Rotating Condensate. Other Trapping Potentials. High Velocity and Quantum Hall Regime. Threedimensional Rotating Condensate. Superfluid Flow around an Obstacle. Further Open Problems. References. Index.
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Related SubjectsScience and Mathematics » Mathematics » Applied Science and Mathematics » Mathematics » Differential Equations Science and Mathematics » Physics » Classical Mechanics Science and Mathematics » Physics » Fluid Mechanics Science and Mathematics » Physics » Math Science and Mathematics » Physics » Relativity Theory 

