Multiscale simulations are commonly used to study the properties of complex systems with multiple length and time scales . These methods often derive simplified “coarse-grained” (CG) models via detailed “first-principles” (FP) ones, in turn enabling longer and larger computational studies. Nevertheless, it has been challenging to rigorously and systematically connect the approximations made in given CG systems with errors in their predictions of corresponding FP properties. Recently, we introduced a solution for this problem based on an informatic property termed the relative entropy, a metric which measures the deviations of the particular CG ensemble probabilities off the reference FP ones .
Here, we use the relative entropy approach to develop a comprehensive framework for coarse-graining. Reminiscent of (conventional) thermal physics, this multiscale theory most notably demonstrates that the relative entropy consistently signals various multiscale errors (i.e., differences between CG and FP properties) . We gain physical insight for this attribute of the relative entropy via an analytical reformulation of the informatic property, showing that the relative entropy formally compares the potential energy landscapes of the FP and CG ensembles by measuring the fluctuations in their differences . These findings suggest that minimization of the relative entropy attains the practical aim of striving to equate the properties of the FP and CG systems. Furthermore, the reformulation enables an efficient numerical computation of the relative entropy, and we present a novel algorithm based on it . We also analytically show that relative entropy minimization generates the working principles of other parameterization methodologies (e.g., force-matching and inverse-Boltzmann) . We exemplify this universal framework on a case study involving the mean-field treatment of the Ising model. We thus find that the relative entropy presents an effective tool for multiscale systems.
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