  Applying an excess entropy scaling formalism to the coarse-grained (CG)
dynamics of liquids, we discovered that missing rotational motions during the
CG process are responsible for artificially accelerated CG dynamics. In the
context of the dynamic representability between the fine-grained (FG) and CG
dynamics, this work introduces the well-known Stokes-Einstein and
Stokes-Einstein-Debye relations to unravel the rotational dynamics underlying
FG trajectories, thereby allowing for an indirect evaluation of the effective
rotations based only on the translational information at the reduced CG
resolution. Since the representability issue in CG modeling limits a direct
evaluation of the shear stress appearing in the Stokes-Einstein and
Stokes-Einstein-Debye relations, we introduce a translational relaxation time
as a proxy to employ these relations, and we demonstrate that these relations
hold for the ambient conditions studied in our series of work. Additional
theoretical links to our previous work are also established. First, we
demonstrate that the effective hard sphere radius determined by the classical
perturbation theory can approximate the complex hydrodynamic radius value
reasonably well. Also, we present a simple derivation of an excess entropy
scaling relationship for viscosity by estimating the elliptical integral of
molecules. In turn, since the translational and rotational motions at the FG
level are correlated to each other, we conclude that the "entropy-free" CG
diffusion only depends on the shape of the reference molecule. Our results and
analyses impart an alternative way of recovering the FG diffusion from the CG
description by coupling the translational and rotational motions at the
hydrodynamic level.
