  The thermodynamic entropy of coarse-grained (CG) models stands as one of the
most important properties for quantifying the missing information during the CG
process and for establishing transferable (or extendible) CG interactions.
However, performing additional CG simulations on top of model construction
often leads to significant additional computational overhead. In this work, we
propose a simple hierarchical framework for predicting the thermodynamic
entropies of various molecular CG systems. Our approach employs a decomposition
of the CG interactions, enabling the estimation of the CG partition function
and thermodynamic properties a priori. Starting from the ideal gas description,
we leverage classical perturbation theory to systematically incorporate simple
yet essential interactions, ranging from the hard sphere model to the
generalized van der Waals model. Additionally, we propose an alternative
approach based on multiparticle correlation functions, allowing for systematic
improvements through higher-order correlations. Numerical applications to
molecular liquids validate the high fidelity of our approach, and our
computational protocols demonstrate that a reduced model with simple energetics
can reasonably estimate the thermodynamic entropy of CG models without
performing any CG simulations. Overall, our findings present a systematic
framework for estimating not only the entropy but also other thermodynamic
properties of CG models, relying solely on information from the reference
system.
