  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.
