  The first paper of this series [J. Chem. Phys. 158, 034103 (2023)]
demonstrated that excess entropy scaling holds for both fine-grained and
corresponding coarse-grained (CG) systems. Despite its universality, a more
exact determination of the scaling relationship was not possible due to the
semi-empirical nature. In this second paper, an analytical excess entropy
scaling relation is derived for bottom-up CG systems. At the single-site CG
resolution, effective hard sphere systems are constructed that yield
near-identical dynamical properties as the target CG systems by taking
advantage of how hard sphere dynamics and excess entropy can be analytically
expressed in terms of the liquid packing fraction. Inspired by classical
equilibrium perturbation theories and recent advances in constructing hard
sphere models for predicting activated dynamics of supercooled liquids, we
propose a new approach for understanding the diffusion of molecular liquids in
the normal regime using hard sphere reference fluids. The proposed "fluctuation
matching" is designed to have the same amplitude of long wavelength density
fluctuations (dimensionless compressibility) as the CG system. Utilizing the
Enskog theory to derive an expression for hard sphere diffusion coefficients, a
bridge between the CG dynamics and excess entropy is then established. The CG
diffusion coefficient can be roughly estimated using various equations of the
state, and an accurate prediction of accelerated CG dynamics at different
temperatures is also possible in advance of running any CG simulation. By
introducing another layer of coarsening, these findings provide a more rigorous
method to assess excess entropy scaling and understand the accelerated CG
dynamics of molecular fluids.
