On the representability problem and the physical meaning of coarse-grained models

In coarse-grained (CG) models where certain fine-grained (FG, i.e., atomistic resolution) observables are not directly represented, one can nonetheless identify indirect the CG observables that capture the FG observable's dependence on CG coordinates. Often, in these cases it appears that a CG observable can be defined by analogy to an all-atom or FG observable, but the similarity is misleading and significantly undermines the interpretation of both bottom-up and top-down CG models. Such problems emerge especially clearly in the framework of the systematic bottom-up CG modeling, where a direct and transparent correspondence between FG and CG variables establishes precise conditions for consistency between CG observables and underlying FG models. Here we present and investigate these representability challenges and illustrate them via the bottom-up conceptual framework for several simple analytically tractable polymer models. The examples provide special focus on the observables of configurational internal energy, entropy, and pressure, which have been at the root of controversy in the CG literature, as well as discuss observables that would seem to be entirely missing in the CG representation but can nonetheless be correlated with CG behavior. Though we investigate these problems in the framework of systematic coarse-graining, the lessons apply to top-down CG modeling also, with crucial implications for simulation at constant pressure and surface tension and for the interpretations of structural and thermodynamic correlations for comparison to experiment.