The ability to generate accurate coarse-grained models from reference fully atomic (or otherwise "first-principles") ones has become an important component in modeling the behavior of complex molecular systems with large length and time scales. We recently proposed a novel coarse-graining approach based upon variational minimization of a configuration-space functional called the relative entropy, S(rel), that measures the information lost upon coarse-graining. Here, we develop a broad theoretical framework for this methodology and numerical strategies for its use in practical coarse-graining settings. In particular, we show that the relative entropy offers tight control over the errors due to coarse-graining in arbitrary microscopic properties, and suggests a systematic approach to reducing them. We also describe fundamental connections between this optimization methodology and other coarse-graining strategies like inverse Monte Carlo, force matching, energy matching, and variational mean-field theory. We suggest several new numerical approaches to its minimization that provide new coarse-graining strategies. Finally, we demonstrate the application of these theoretical considerations and algorithms to a simple, instructive system and characterize convergence and errors within the relative entropy framework.

Recent efforts have attempted to understand many of liquid water's anomalous properties in terms of effective spherically-symmetric pairwise molecular interactions entailing two characteristic length scales (so-called "core-softened" potentials). In this work, we examine the extent to which such simple descriptions of water are representative of the true underlying interactions by extracting coarse-grained potential functions that are optimized to reproduce the behavior of an all-atom model. To perform this optimization, we use a novel procedure based upon minimizing the relative entropy, a quantity that measures the extent to which a coarse-grained configurational ensemble overlaps with a reference all-atom one. We show that the optimized spherically-symmetric water models exhibit notable variations with the state conditions at which they were optimized, reflecting in particular the shifting accessibility of networked hydrogen bonding interactions. Moreover, we find that water's density and diffusivity anomalies are only reproduced when the effective coarse-grained potentials are allowed to vary with state. Our results therefore suggest that no state-independent spherically-symmetric potential can fully capture the interactions responsible for water's unique behavior; rather, the particular way in which the effective interactions vary with temperature and density contributes significantly to anomalous properties.

After nearly 30 years of research on the hydrophobic interaction, the search for the hydrophobic force law is still continuing. Indeed, there are more questions than answers, and the experimental data are often quite different for nominally similar conditions, as well as, apparently, for nano-, micro-, and macroscopic surfaces. This has led to the conclusion that the experimentally observed force–distance relationships are either a combination of different ‘fundamental’ interactions, or that the hydrophobic force-law, if there is one, is complex – depending on numerous parameters. The only unexpectedly strong attractive force measured in all experiments so far has a range of D ≈ 100–200 Å, increasing roughly exponentially down to ~ 10–20 Å and then more steeply down to adhesive contact at D = 0 or, for power-law potentials, effectively at D ≈ 2 Å. The measured forces in this regime (100–200 Å) and especially the adhesive forces are much stronger, and have a different distance-dependence from the continuum VDW force (Lifshitz theory) for non-conducting dielectric media. We suggest a three-regime force-law for the forces observed between hydrophobic surfaces: In the first, from 100–200 Å to thousands of ångstroms, the dominating force is created by complementary electrostatic domains or patches on the apposing surfaces and/or bridging vapour cavities; a ‘pure’ but still not well-understood ‘long-range hydrophobic force’ dominates the second regime from ~ 150 to ~ 15 Å, possibly due to an enhanced Hamaker constant associated with the ‘proton-hopping’ polarizability of water; while below ~ 10–15 Å to contact there is another ‘pure short-range hydrophobic force’ related to water structuring effects associated with surface-induced changes in the orientation and/or density of water molecules and H-bonds at the water–hydrophobic interface. We present recent SFA and other experimental results, as well as a simplified model for water based on a spherically-symmetric potential that is able to capture some basic features of hydrophobic association. Such a model may be useful for theoretical studies of the HI over the broad range of scales observed in SFA experiments.

We show that the relative entropy, Srel, 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 Srel 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 Srel has important ramifications for evaluating and designing coarse-grained models.

We show here that molecular resolution is inherently hybrid in terms of relative separation. While nearest neighbors are characterized by a fine-grained (geometrically detailed) model, other neighbors are characterized by a coarse-grained (isotropically simplified) model. We notably present an analytical expression for relating the two models via energy conservation. This hybrid framework is correspondingly capable of retrieving the structural and thermal behavior of various multi-component and multi-phase fluids across state space.