Dynamic response and thermodynamics of mesoscopic metal rings at constant particle number.

We show by means of simple exact manipulations that the thermodynamic persistent current I(phi,N) in a mesoscopic metal ring threaded by a magnetic flux phi at constant particle number N agrees even beyond linear response with the dynamic current I-dy(phi, N) that is defined via the response to a time-dependent flux in the limit that the frequency of the flux vanishes. However, it is impossible to express the disorder average of I-dy(phi, N) in terms of conventional Green's functions at flux-independent chemical potential, because the part of the dynamic response function that involves two retarded and two advanced Green's functions is not negligible. Therefore the dynamics cannot be used to map a canonical average onto a more tractable grand-canonical one. We also calculate the zero frequency limit of the dynamic current at constant chemical potential beyond linear response and show that it is fundamentally different from any thermodynamic derivative.