The Ginzburg-Landau order parameter near the second critical field

TL;DR

This paper analyzes the behavior of the Ginzburg-Landau order parameter near the second critical magnetic field, providing a leading order approximation of its norm in small regions as the Ginzburg-Landau parameter grows large.

Contribution

It offers a new asymptotic approximation for the order parameter in the high Ginzburg-Landau limit near the second critical field.

Findings

Leading order approximation of the order parameter's L2-norm in small squares.

Asymptotic behavior characterized as the Ginzburg-Landau parameter tends to infinity.

Insights into the distribution of superconducting electrons near the second critical field.

Abstract

In Ginzburg-Landau Theory of superconductivity, the density and location of the superconducting electrons are measured by a complex-valued wave function, the order parameter. In this paper, when the intensity of the applied magnetic field is close to the second critical field, and when the order parameter minimizes the Ginzburg-Landau functional defined over a two dimensional domain, the leading order approximation of its $L^2$-norm in `small' squares is given as the Ginzburg-Landau parameter tends to infinity.