Department of Mathematics

University of Arizona

**Abstract**: A one-dimensional Ginzburg-Landau model describing a
superconducting closed thin wire with arbitrary cross-section subject to a
large applied magnetic field is derived from the three-dimensional
Ginzburg-Landau energy in the spirit of $\Gamma$-convergence. Our result proves
the validity of the formal result of Richardson and Rubinstein revealing the
double limit of large field and thin domain. An additional magnetic potential
related to the applied field is found in the limiting functional which yields a
parabolic background for the oscillatory phase transition curve between the
normal and superconducting states.