(359d) The Relative Entropy as An Indicator of Multiscale Errors

Multiscale simulations are commonly used to study the properties of complex systems with multiple length and time scales [1]. 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. Here, we show that the relative entropy provides a solution for this problem. Recently, we introduced a fundamental theory for multiscale physics based on this informatic property; its main aspect is that the relative entropy measures the deviations of the particular CG ensemble probabilities off the reference FP ones [2]. Thus, relative entropy minimization provides a convenient route for multiscale procedures, and we have shown both that it recovers a number of existing relations for inverse problems in statistical physics and that it can be effectively used in the development of numerical algorithms for coarse-graining [2].

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) [3]. 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 [3]. 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).

To demonstrate the theories and methodologies described above, we apply these efforts onto several systems. We show that the relative entropy provides new perspectives on two classic models in statistical mechanics (ideal gas and lattice gas), predicting the breakdown of model assumptions [3]. Utilizing a spherically-symmetric water model that has been already optimized via the original relative entropy approach [4], we also use this framework to investigate the hydrophobic interaction, focusing on its subtle behavior across multiple scales [5]. All these results suggest that the relative entropy presents an effective tool for multiscale systems of many kinds, from plain analytical approaches to realistic computational studies.

1. Voth, G.A., Introduction: Coarse-Graining in Molecular Modeling and Simulation. Journal of Chemical Theory and Computation, 2006. 2(3): p. 463-463.

2. Shell, M.S., The relative entropy is fundamental to multiscale and inverse thermodynamic problems. The Journal of Chemical Physics, 2008. 129(14): p. 144108-7.

3. Chaimovich, A. and M.S. Shell, Relative entropy: A universal metric for multiscale errors. Submitted, 2010.

4. Chaimovich, A. and M.S. Shell, Anomalous waterlike behavior in spherically-symmetric water models optimized with the relative entropy. Physical Chemistry Chemical Physics, 2009. 11(12): p. 1901-1915.

5. Hammer, M.U., et al., The Search for the hydrophobic force law. Accepted, 2010.