Given a distribution of atomic configurations, what energy function best reproduces them under canonical conditions? Alternatively, what are the effective van der Waals or other coarse-grained parameters that optimally mimic a detailed model? Such questions led to the pioneering reverse Monte Carlo technique of Lyubartsev and Laaksonen [1], which has been a critical part of coarse-graining atomistic models with effective pair-potentials. Here, we provide a needed rigorous derivation and general framework for measuring and optimizing the degree of overlap between two configurational ensembles. We show that the exponential of the negative relative entropy between ensembles A and B, Srel = sumi pA(i) ln pA(i)/pB(i), gives the probability that one “quanta” of ensemble B correctly reproduces the configurational probabilities of ensemble A. The relative entropy is well-established in information theory and, along those lines, was employed by Wu and Kofke to optimize sampling in free energy calculations [2]. We show that the relative entropy is a remarkably general framework that can be used to derive the variational mean-field equation, the reverse MC method, and in general an algorithm for optimizing any force field or ensemble parameter (e.g, temperature, density). We demonstrate this latter approach using several test systems and employing an efficient flat histogram algorithm for computation of the free energies [3]. We then show how ensemble optimization can be used to understand water's anomalies; that is, we calculate the ensemble overlap between water and a simple fluid (Lennard Jones) to show where it maximally deviates from simple-liquid behavior.

[1] A. P. Lyubartsev and A. Laaksonen, Phys. Rev. E 52, 3730 (1995).

[2] D. Wu and D. A. Kofke, J. Chem. Phys. 123, 054103 (2005); J. Chem. Phys. 123, 084109 (2005).

[3] M. S. Shell, P. G. Debenedetti, and A. Z. Panagiotopoulos, J. Chem. Phys. 119, 9406 (2003).