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Significance Physical phenomena can often be described by surprisingly few order parameters. Unfortunately, it is challenging to identify these essential degrees of freedom. Here we develop a statistical physics framework for exploring the landscape of order parameters, or coarse-grained representations, for a microscopic protein model. We employ Monte Carlo methods to statistically characterize this landscape. We define metrics assessing the intrinsic quality of each representation for preserving the configurational information and large-scale motions of the underlying microscopic model. Interestingly, these metrics are anticorrelated in low-resolution representations. Moreover, below a critical resolution, a phase transition qualitatively distinguishes superior and inferior representations. Finally, we relate our work to recent approaches for clustering graphs and detecting communities in networks. 

Low resolution coarse-grained (CG) models provide remarkable com- putational and conceptual advantages for simulating soft materials. In principle, bottom-up CG models can reproduce all structural and thermodynamic properties of atomically detailed models that can be observed at the resolution of the CG model. This review discusses recent progress in developing theory and computational methods for achieving this promise. We first briefly review variational approaches for parameterizing interaction potentials and their relationship to ma- chine learning methods. We then discuss recent approaches for si- multaneously improving both the transferability and thermodynamic properties of bottom-up models by rigorously addressing the density- and temperature-dependence of these potentials. We also briefly dis- cuss exciting progress in modeling high resolution observables with low- resolution CG models. More generally, we highlight the essential role of the bottom-up framework not only for fundamentally understand- ing the limitations of prior CG models, but also for developing robust computational methods that resolve these limitations in practice.