Case of thermodynamic failure in the Ginzburg-Landau approach to fluctuation superconductivity

The Ginzburg-Landau approach postulates an energy density, together with an interpretation for the supercurrent, and invokes Ohm's law. We consider quasi -one-dimensional nonuniform superconducting loops, either smooth or piecewise uniform, that enclose a magnetic flux, above the critical temperature. We evaluate the averages of the current and of the power per unit length delivered by the electric field due to thermal fluctuations. We consider three averages: the canonical ensemble average, the canonical ensemble in the reciprocal space, and the time average using a time -dependent model. All the evaluations imply that heat is imparted to part of the loop and removed in the other part, despite the assumption that the loop is at uniform temperature.