Relative entropy as a universal metric for multiscale errors

We show that the relative entropy, S-rel, suggests a fundamental indicator of the success of multiscale studies, in which coarse-grained (CG) models are linked to first-principles (FP) ones. We demonstrate that S-rel inherently measures fluctuations in the differences between CG and FP potential energy landscapes, and develop a theory that tightly and generally links it to errors associated with coarse graining. We consider two simple case studies substantiating these results, and suggest that S-rel has important ramifications for evaluating and designing coarse-grained models.