Some mathematical problems in the Ginzburg-Landau theory of superconductivity
Bibliographic record
Abstract
In agreement with the Landau theory of phase transitions, a superconductor is described macroscopically by the Ginzburg-Landau equations (1950).These are nonlinear partial differential equations for a cornplex-valued function.ii, (the order parameter), and a vector-field -4 (the vector potential).The equations contain a parame ter. A. which determines whet her t hey describe a superconductor of the first kind ( A < l ) , or of the second kind ( A > 1).It was observed by Abrikosov (1957) that a highly-symmetric family of solutions known as n-vortices plays a central role in the theory.These solutions are classified by their integer topological degree, n E Z.The principal goal of this thesis is to establish the stability properties of the n-vortex.The stability question is studied for three types of evolution equations: a gradient flow, a nonlinear wave equation, and a nonlinear Schrodinger equation.Our main result determines the dependence of the stability of nvortices on the topological degree, n, and on the parameter, A. Specifically, we prove that for X < 1, al1 vortices are stable, while for X > 1, n-vortices are stable if n = rtl and unstable if jnl 2 2. Previous work on vortex sta-bility (Taubes (1980), Stuart (1994)) has focussed on the special case X = 1, in which the Ginzbug-Landau equations reduce to the kst-order Bogomolnyi equations.In particular, our result resolves a long-standing conjecture: first rigorously formulated by Jaffe and Taubes (1980).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".