  A theory describing the operation of a superconducting nanowire quantum
interference device (NQUID) is presented. The device consists of a pair of
thin-film superconducting leads connected by a pair of topologically parallel
ultra-narrow superconducting wires. It exhibits intrinsic electrical
resistance, due to thermally-activated dissipative fluctuations of the
superconducting order parameter. Attention is given to the dependence of this
resistance on the strength of an externally applied magnetic field aligned
perpendicular to the leads, for lead dimensions such that there is essentially
complete and uniform penetration of the leads by the magnetic field. This
regime, in which at least one of the lead dimensions lies between the
superconducting coherence and penetration lengths, is referred to as the
mesoscopic regime. The magnetic field causes a pronounced oscillation of the
device resistance, with a period not dominated by the Aharonov-Bohm effect
through the area enclosed by the wires and the film edges but, rather, in terms
of the geometry of the leads, in contrast to the well-known Little-Parks
resistance of thin-walled superconducting cylinders. A theory, encompassing
this phenomenology, is developed through extensions, to the setting of parallel
superconducting wires, of the Ivanchenko-Zil'berman-Ambegaokar-Halperin theory
for the case of short wires and the Langer-Ambegaokar-McCumber-Halperin theory
for the case of longer wires. It is demonstrated that the NQUID acts as a probe
of spatial variations in the superconducting order parameter.
