Under appropriate geometry, magnetic field, and temperature, numerical minimization of the free energy predicts a stable superconducting phase where part of the sample is normal, so that the magnetic Aux is not enclosed by the superconducting part. This phase mediates between the normal phase and the superconducting phase which has been usually considered. For one point in the filed-temperature plane, it has been proven analytically that this intermediate phase minimizes the free energy. Near the transition, even when the doubly connected phase is stable, the order parameter tries to mimic that of a simply connected phase.
Many analyses based on the time-dependent Ginzburg-Landau model are not consistent with statistical mechanics, because thermal fluctuations are not taken correctly into account. We use the fluctuation-dissipation theorem in order to establish the appropriate size of the Langevin terms, and thus ensure the required consistency. Fluctuations of the electromagnetic potential are essential, even when we evaluate quantities that do not depend directly on it. Our method can be cast in gauge-invariant form. We perform numerous tests, and all the results are in agreement with statistical mechanics. We apply our method to evaluate paraconductivity of a superconducting wire. The Aslamazov-Larkin result is recovered as a limiting situation. Our method is numerically stable and the nonlinear term is easily included. We attempt a comparison between our numerical results and the available experimental data. Within an appropriate range of currents, phase slips occur, but we found no evidence for thermally activated phase slips. We studied the behavior of a moderate constriction. A constriction pins and enhances the occurrence of phase slips.
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