Equivalence of thermodynamic persistent current for constant number of particles and dynamic response in mesoscopic metal rings

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Abstract
We compare dynamic and thermodynamic persistent currents in a mesoscopic metal ring threaded by a magnetic flux in the regime 1≲l/L≲M (l is the elastic mean free path, L is the circumference of the ring, and M is the number of transverse channels). It is shown by explicit calculation that the thermodynamic current for a fixed number of particles is identical to the dynamic current obtained from the Kubo formula for fixed chemical potential. Thus, thermodynamic averaging in a canonical ensemble is equivalent to calculating the dynamic response in a grand-canonical one.