  Recently, we introduced Relative Resolution as a hybrid formalism for fluid
mixtures [1]. The essence of this approach is that it switches molecular
resolution in terms or relative separation: While nearest neighbors are
characterized by a detailed fine-grained model, other neighbors are
characterized by a simplified coarse-grained model. Once the two models are
analytically connected with each other via energy conservation, Relative
Resolution can capture the structural and thermal behavior of (nonpolar)
multi-component and multi-phase systems across state space. The current work is
a natural continuation of our original communication [1]. Most importantly, we
present the comprehensive mathematics of Relative Resolution, basically casting
it as a multipole approximation at appropriate distances; the current set of
equations importantly applies for all systems (e.g, polar molecules). Besides,
we continue examining the capability of our multiscale approach in molecular
simulations, importantly showing that we can successfully retrieve not just the
statics but also the dynamics of liquid systems. We finally conclude by
discussing how Relative Resolution is the inherent variant of the famous
"cell-multipole" approach for molecular simulations.
