  We present a hydrodynamic theory describing pair diffusion in systems with
periodic boundary conditions, thereby generalizing earlier work on
self-diffusion [D\"unweg and Kremer, J. Chem. Phys. 1993, 99, 6983-6997; Yeh
and Hummer, J. Phys. Chem. B 2004, 108, 15873-15879]. Its predictions are
compared to Molecular Dynamics simulations for a liquid carbonate electrolyte
and two ionic liquids, for which we characterize the correlated motion between
distinct ions. Overall, we observe good agreement between theory and simulation
data, highlighting that hydrodynamic interactions universally dictate ion
correlations. However, when summing over all ion pairs in the system to obtain
the cross-contributions to the total cationic or anionic conductivity, the
hydrodynamic interactions between ions with like and unlike charges largely
cancel. Consequently, significant conductivity contributions only arise from
deviations from a hydrodynamic flow field of an ideal fluid, that is, from the
local electrolyte structure as well as from relaxation processes in the
subdiffusive regime. In case of ionic liquids, the momentum-conservation
constraint additionally is vital, which we study by employing different ionic
masses in the simulations. Our formalism will likely also be helpful to
estimate finite-size effects of the conductivity or of Maxwell-Stefan
diffusivities in simulations.
