  We revisit the statistical mechanics of charge fluctuations in capacitors. In
constant-potential classical molecular simulations, the atomic charge of
electrode atoms are treated as additional degrees of freedom which evolve in
time so as to satisfy the constraint of fixed electrostatic potential for each
configuration of the electrolyte. The present work clarifies the role of the
overall electroneutrality constraint, as well as the link between the averages
computed within the Born-Oppenheimer approximation and that of the full
constant-potential ensemble. This allows us in particular to derive a complete
fluctuation-dissipation relation for the differential capacitance, that
includes a contribution from the charge fluctuations (around the charges
satisfying the constant-potential and electroneutrality constraints) also
present in the absence of an electrolyte. We provide a simple expression for
this contribution from the elements of the inverse of the matrix defining the
quadratic form of the fluctuating charges in the energy. We then illustrate
numerically the validity of our results, and recover the expected result for an
empty capacitor with structureless electrodes at large inter-electrode
distances. By considering a variety of liquids between graphite electrodes, we
confirm that this contribution to the total differential capacitance is small
compared to that induced by the thermal fluctuations of the electrolyte.
