Theory of the sign of multi-probe conductances for normal and superconducting materials

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Abstract
We examine equivalent forms of the current-voltage relationship applicable to mesoscopic superconductors and use one of these to derive general conditions for the sign of the conductance in a four-probe measurement. For disordered systems, with well separated voltage probes, where the normal-state conductance is positive, we predict that the sign of the longitudinal four-probe conductance of superconducting wires can be reversed by varying an applied magnetic field, and that at certain critical values of the field, the conductance passes through a singularity.