Ginzburg-Landau Phase Transition Theory and Superconductivity

ISBN-13: 9783764364861 / Angielski / Twarda / 2000 / 384 str.

K. -H Hoffmann; K. H. Hofmann; Q. Tang

Opis

The theory of complex Ginzburg-Landau type phase transition and its applica tions to superconductivity and superfluidity has been a topic of great interest to theoretical physicists and has been continuously and persistently studied since the 1950s. Today, there is an abundance of mathematical results spread over numer ous scientific journals. However, before 1992, most of the studies concentrated on formal asymptotics or linear analysis. Only isolated results by Berger, Jaffe and Taubes and some of their colleagues touched the nonlinear aspects in great detail. In 1991, a physics seminar given by Ed Copeland at Sussex University inspired Q. Tang, the co-author of this monograph, to study the subject. Independently in Munich, K.-H. Hoffmann and his collaborators Z. Chen and J. Liang started to work on the topic at the same time. Soon it became clear that at that time, groups of mathematicians at Oxford and Virginia Tech had already studied the subject for a couple of years. They inspired experts in interface phase transition problems and their combined effort established a rigorous mathematical framework for the Ginzburg-Landau system. At the beginning Q. Tang collaborated with C.M. Elliott and H. Matano."

Spis treści

1 Introduction.- 1.1 Brief history.- 1.1.1 Meissner effect – diamagnetism.- 1.1.2 The London equation and the penetration depth.- 1.1.3 The coherence length.- 1.1.4 Classification of superconductors.- 1.1.5 Vortices.- 1.1.6 Summary.- 1.2 The G-L phenomenological theory.- 1.2.1 The free energy and the G-L equations.- 1.2.2 Rescaling and the values of the constants.- 1.2.3 Gauge invariance.- 1.3 Some considerations arising from scaling.- 1.3.1 The two characteristic lengths ?(T) and À(T).- 1.3.2 The validity of the G-L theory.- 1.4 The evolutionary G-L system – 2-d case.- 1.4.1 The system.- 1.4.2 Mathematical scaling.- 1.4.3 The G-L functional as a Lyapunov functional.- 1.4.4 Gauge invariance.- 1.4.5 A uniform bound on |?|.- 1.5 Exterior evolutionary Maxwell system.- 1.5.1 Review of the Maxwell system.- 1.5.2 The G-L superconductivity model.- 1.5.3 The setting of the problem.- 1.6 Exterior steady-state Maxwell system.- 1.7 Surface energy, superconductor classification.- 1.7.1 The sign of ?ns when ? ? 1.- 1.7.2 The sign of ?ns when ? ? 1.- 1.7.3 The case $$\mathcal = 1/\sqrt 2 $$.- 1.7.4 Conclusion.- 1.8 Difference between 2-d and 3-d models.- 1.9 Bibliographical remarks.- 2 Mathematical Foundation.- 2.1 Co-dimension one phase transition problems.- 2.1.1 Steady state problems.- 2.1.2 Evolutionary problems.- 2.1.3 Long time behaviour.- 2.2 Co-dimension two phase transition problems.- 2.2.1 Steady state problems on bounded domains.- 2.2.2 Steady state problems on ?2.- 2.2.3 Evolutionary problems.- 2.2.4 Long time behaviour.- 2.3 Mathematical description of vortices in ?2.- 2.4 Asymptotic methods for describing vortices in ?2.- 2.4.1 Steady state case in ?2.- 2.4.2 Evolutionary case in ?2 - Introduction•.- 2.4.3 Evolutionary case in ?2 - far field expansion:.- 2.4.4 Evolutionary case in ?2 - local structure of the far fieldsolution near a vortex.- 2.4.5 Evolutionary case in ?2 - Core expansion.- 2.4.6 Evolutionary case in ?2 - Matching of the core and far fieldexpansions.- 2.4.7 Vortex motion equation.- 2.5 Asymptotic methods for describing vortices in ?3.- 2.5.1 Steady state case in ?3.- 2.5.2 Evolutionary case in ?3.- 2.6 Bibliographical remarks.- 3 Asymptotics Involving Magnetic Potential.- 3.1 Basic facts concerning fluid vortices.- 3.2 Asymptotic analysis.- 3.2.1 2-D steady state case.- 3.2.2 Evolutionary case.- 3.2.3 Far field.- 3.2.4 Core region.- 3.3 Asymptotic analysis of densely packed vortices.- 3.3.1 Outer region – a mean field model.- 3.3.2 Intermediate region.- 3.3.3 Core region.- 3.4 Bibliographical remarks.- 4 Steady State Solutions.- 4.1 Existence of steady state solutions.- 4.1.1 The outside field is a given function, 2-d case.- 4.1.2 The outside field is governed by the Maxwell system, 3-d case.- 4.2 Stability and mapping properties of solutions.- 4.2.1 Non-existence of local maxima.- 4.2.2 Boundedness of the order parameter.- 4.2.3 Constant solutions and mixed state solutions.- 4.3 Co-dimension two vortex domain.- 4.4 Breakdown of superconductivity.- 4.5 A linearized problem.- 4.6 Bibliographical remarks.- 5 Evolutionary Solutions.- 5.1 2-d solutions with given external field.- 5.1.1 Mathematical setting.- 5.1.2 Existence and uniqueness of solutions.- 5.1.3 Proof of Theorem 1.2.- 5.1.4 Proof of Theorem 1.1.- 5.2 Existence of 3-d evolutionary solutions.- 5.3 The existence of an ?-limit set as t ? ?.- 5.4 An abstract theorem on global attractors.- 5.5 Global atractor for the G-L sstem.- 5.6 Physical bounds on the global attractor.- 5.7 The uniqueness of the long time limit of the evolutionary G-L so-lutions.- 5.8 Bibliographical remarks.- 6 Complex G-L Type Phase Transition Theory.- 6.1 Existence and basic properties of solutions.- 6.2 BBH type upper bound for energy of minimizers.- 6.3 Global estimates.- 6.4 Local estimates.- 6.5 The behaviour of solutions near vortices.- 6.6 Global ?-independent estimates.- 6.7 Convergence of the solutions as ? ? 0.- 6.8 Main results on the limit functions.- 6.9 Renormalized energies.- 6.10 Bibliographical remarks.- 7 The Slow Motion of Vortices.- 7.1 Introduction.- 7.2 Preliminaries.- 7.3 Estimates from below for the mobilities.- 7.4 Estimates from above for the mobilities.- 7.5 Bibliographical remarks.- 8 Thin Plate/Film G-L Models.- 8.1 The outside Maxwell system – steady state case.- 8.1.1 The energy bound.- 8.1.2 Convergence properties of the resealed variables.- 8.1.3 Passing to the limit.- 8.2 The outside field is given – evolutionary case.- 8.2.1 Existence and uniqueness of solutions.- 8.2.2 The limit when ? ? 0.- 8.2.3 Some estimates.- 8.2.4 The convergence.- 8.3 The outside field is given – formal analysis.- 8.3.1 Variational formulation.- 8.3.2 Formal asymptotic analysis when ? ? 0.- 8.4 Bibliographical remarks.- 9 Pinning Theory.- 9.1 Local Pohozaev-type identity.- 9.2 Estimate the energy of minimizers.- 9.3 Local estimates.- 9.4 Global Estimates.- 9.5 Convergence of solutions and the term $$ \frac {{\varepsilon ^2 }}\int_\Omega {(\left| {\psi _\varepsilon } \right|^2 - 1)^2 } $$.- 9.6 Properties of ?*, A*.- 9.7 Renormalized energy.- 9.8 Pinning of vortices in other circumstances.- 9.8.1 G-L model subject to thermo-perturbation or large horizon-tal field.- 9.8.2 An anisotropic G-L model.- 9.8.3 A thin film G-L model.- 9.9 Bibliographical remarks.- 10 Numerical Analysis.- 10.1 Introduction.- 10.2 Discretization.- 10.2.1 Weak formulation.- 10.2.2 Discretization.- 10.3 Stability estimates.- 10.4 Error estimates.- 10.5 A numerical example.- 10.6 Discretization using variable step length.- 10.7 A dual problem.- 10.7.1 Stability estimates.- 10.7.2 Error representation formula.- 10.8 A posteriori error analysis.- 10.8.1 Residuals.- 10.8.2 Proof of Theorem 4.1.- 10.9 Numerical implementation.- 10.9.1 Comparison of the schemes.- 10.10 Bibliographical remarks.- References.