Microscopic derivation of particle-based coarse-grained dynamics: Exact expression for memory function

Maximum Entropy Theory of Multiscale Coarse-Graining via Matching Thermodynamic Forces: Application to a Molecular Crystal (TATB)

The MSCG/FM (multiscale coarse-graining via force-matching) approach is an efficient supervised machine learning method to develop microscopically informed coarse-grained (CG) models. We present a theory based on the principle of maximum entropy (PME) enveloping the existing MSCG/FM approaches. This theory views the MSCG/FM method as a special case of matching the thermodynamic forces from the extended ensemble described by the set of thermodynamic (relevant) system coordinates. This set may include CG coordinates, the stress tensor, applied external fields, and so forth, and may be characterized by nonequilibrium conditions. Following the presentation of the theory, we discuss the consistent matching of both bonded and nonbonded interactions. The proposed PME formulation is used as a starting point to extend the MSCG/FM method to the constant strain ensemble, which together with the explicit matching of the bonded forces is better suited for coarse-graining anisotropic media at a submolecular resolution. The theory is demonstrated by performing the fine coarse-graining of crystalline 1,3,5-triamino-2,4,6-trinitrobenzene (TATB), a well-known insensitive molecular energetic material, which exhibits highly anisotropic mechanical properties.

Understanding dynamics in coarse-grained models. III. Roles of rotational motion and translation-rotation coupling in coarse-grained dynamics

This paper series aims to establish a complete correspondence between fine-grained (FG) and coarse-grained (CG) dynamics by way of excess entropy scaling (introduced in Paper I). While Paper II successfully captured translational motions in CG systems using a hard sphere mapping, the absence of rotational motions in single-site CG models introduces differences between FG and CG dynamics. In this third paper, our objective is to faithfully recover atomistic diffusion coefficients from CG dynamics by incorporating rotational dynamics. By extracting FG rotational diffusion, we unravel, for the first time reported to our knowledge, a universality in excess entropy scaling between the rotational and translational diffusion. Once the missing rotational dynamics are integrated into the CG translational dynamics, an effective translation-rotation coupling becomes essential. We propose two different approaches for estimating this coupling parameter: the rough hard sphere theory with acentric factor (temperature-independent) or the rough Lennard-Jones model with CG attractions (temperature-dependent). Altogether, we demonstrate that FG diffusion coefficients can be recovered from CG diffusion coefficients by (1) incorporating "entropy-free" rotational diffusion with translation-rotation coupling and (2) recapturing the missing entropy. Our findings shed light on the fundamental relationship between FG and CG dynamics in molecular fluids.

Comparison of Friction Parametrization from Dynamics and Material Properties for a Coarse-Grained Polymer Melt

In this work, we extend an approach to coarse-grained (CG) modeling for polymer melts in which the conservative potential is parametrized using the iterative Boltzmann inversion (IBI) method and the accelerated dynamics inherent to IBI are corrected using the dissipative Langevin thermostat with a single tunable friction parameter (J. Chem. Phys. 2021, 154, 084114). Diffusive measures from picoseconds to nanoseconds are used to determine the Langevin friction factor to apply to the CG model to recover all-atom (AA) dynamics; the resulting friction factors are then compared for consistency. Here, we additionally parametrize the CG dynamics using a material property, the zero-shear viscosity, which we measure using the Green-Kubo (GK) method. Two materials are studied, squalane as a function of temperature and the same polystyrene oligomers previously studied as a function of chain length. For squalane, the friction derived from the long-time diffusive measures and the viscosity all strongly increase with decreasing temperature, showing an Arrhenius-like dependence, and remain consistent with each other over the entire temperature range. In contrast, the friction required for the picosecond diffusive measurement, the Debye-Waller factor, is somewhat lower than the friction from long-time measures and relatively insensitive to temperature. A time-dependent friction would be required to exactly reproduce the AA measurements during the caging transition connecting these two extremes over the entire timespan at this level of coarse-graining. For the polystyrene oligomers for which we previously characterized the diffusive friction, the viscosity-parametrized frictions are consistent with the diffusive measures for the smallest chain length. However, for the longer chains, we find different trends based on measurement method with friction derived from rotational diffusion remaining nearly constant, friction derived from translational diffusion showing a modestly increasing trend, and viscosity-derived friction showing a modest decreasing trend. This seems to indicate that there is some sensitivity of the friction measurement method for systems with increased relaxation times and that in particular, the unsteady dynamics of the individual parametrization schemes plays a role in this. Increased difficulty in applying the GK method with increasing relaxation time of the longer chain systems is also discussed. Overall, we find that when the material is in a high-temperature melt state and the viscosity measurement is reliable, the friction parametrization from the diffusive friction measures is consistent and the lower cost diffusive parametrization is a reliable means for modeling viscosity. Our data give insight into the time-dependent friction one might compute using a non-Markovian approach to enable the recovery of AA dynamics over a wider range of time scales than can be computed using a single friction.

Hierarchical Machine Learning of Low-Resolution Coarse-Grained Free Energy Potentials

A force-matching-based method for supervised machine learning (ML) of coarse-grained (CG) free energy (FE) potentials─known as multiscale coarse-graining via force-matching (MSCG/FM)─is an efficient method to develop microscopically informed CG models that are thermodynamically and statistically equivalent to the reference microscopic models. For low-resolution models, when the coarse-graining is at supramolecular scales, objective-oriented clustering of nonbonded particles is required and the reduced description becomes a function of the clustering algorithm. In the present work, we explore the dependence of the ML of the CG Helmholtz FE potential on the clustering algorithm. We consider coarse-graining based on partitional (k-means, leading to Voronoi diagram) and hierarchical agglomerative (bottom-up) clustering algorithms common in unsupervised ML and develop theory connecting the MSCG/FM learned CG Helmholtz potential and the clustering statistics. By combining the agglomerative clustering and the MSCG/FM learning in a recursive manner, we propose an efficient ML methodology to develop the fine-to-low resolution hierarchies of the CG models. The methodology does not suffer from degrading accuracy or increased computational cost to construct larger hierarchies and as such does not impose an upper size limitation of the CG particles resulting from the extended hierarchies. The utility of the methodology is demonstrated by obtaining the bottom-up agglomerative hierarchy for liquid nitromethane from all-atom molecular dynamics (MD) simulations. For agglomerative hierarchies, we prove the existence of renormalization group transformations that indicate self-similarity and allow for learning the low-resolution MSCG/FM potentials at low computational cost by rescaling and renormalizing the certain finer-resolution members of the hierarchy. The hierarchies of the CG models can be used to carry out simulations under constant-pressure conditions.

A Mori-Zwanzig Dissipative Particle Dynamics Approach for Anisotropic Coarse Grained Molecular Dynamics

Coarse grained (CG) molecular dynamics simulations are widely used to accelerate atomistic simulations but generally lack a formalism to preserve the dynamics of the system. For spherical particles, the Mori-Zwanzig approach, while computationally complex, has ameliorated this problem. Here we present an anisotropic dissipative particle dynamics (ADPD) model as an extension of this approach, which accounts for the anisotropy for both conservative and nonconservative interactions. For a simple anisotropic system we parametrize the coarse grained force field representing ellipsoidal CG particles from the full-atomistic simulation. To represent the anisotropy of the system, both the conservative and dissipative terms are approximated using the Gay-Berne (GB) functional forms via a force-matching approach. We compare our model with other CG models and demonstrate that it yields better results in both static and dynamical properties. The inclusion of the anisotropic nonconservative force preserves the microscopic dynamical details, and hence the dynamical properties, such as diffusivity, can be better reproduced by the aspherical model. By generalizing the isotropic DPD model, this framework is effective and promising for the development of the CG model for polymers, macromolecules, and biological systems.

Understanding dynamics in coarse-grained models. II. Coarse-grained diffusion modeled using hard sphere theory

The first paper of this series [J. Chem. Phys. 158, 034103 (2023)] demonstrated that excess entropy scaling holds for both fine-grained and corresponding coarse-grained (CG) systems. Despite its universality, a more exact determination of the scaling relationship was not possible due to the semi-empirical nature. In this second paper, an analytical excess entropy scaling relation is derived for bottom-up CG systems. At the single-site CG resolution, effective hard sphere systems are constructed that yield near-identical dynamical properties as the target CG systems by taking advantage of how hard sphere dynamics and excess entropy can be analytically expressed in terms of the liquid packing fraction. Inspired by classical equilibrium perturbation theories and recent advances in constructing hard sphere models for predicting activated dynamics of supercooled liquids, we propose a new approach for understanding the diffusion of molecular liquids in the normal regime using hard sphere reference fluids. The proposed "fluctuation matching" is designed to have the same amplitude of long wavelength density fluctuations (dimensionless compressibility) as the CG system. Utilizing the Enskog theory to derive an expression for hard sphere diffusion coefficients, a bridge between the CG dynamics and excess entropy is then established. The CG diffusion coefficient can be roughly estimated using various equations of the state, and an accurate prediction of accelerated CG dynamics at different temperatures is also possible in advance of running any CG simulation. By introducing another layer of coarsening, these findings provide a more rigorous method to assess excess entropy scaling and understand the accelerated CG dynamics of molecular fluids.

Understanding dynamics in coarse-grained models. I. Universal excess entropy scaling relationship

Coarse-grained (CG) models facilitate an efficient exploration of complex systems by reducing the unnecessary degrees of freedom of the fine-grained (FG) system while recapitulating major structural correlations. Unlike structural properties, assessing dynamic properties in CG modeling is often unfeasible due to the accelerated dynamics of the CG models, which allows for more efficient structural sampling. Therefore, the ultimate goal of the present series of articles is to establish a better correspondence between the FG and CG dynamics. To assess and compare dynamical properties in the FG and the corresponding CG models, we utilize the excess entropy scaling relationship. For Paper I of this series, we provide evidence that the FG and the corresponding CG counterpart follow the same universal scaling relationship. By carefully reviewing and examining the literature, we develop a new theory to calculate excess entropies for the FG and CG systems while accounting for entropy representability. We demonstrate that the excess entropy scaling idea can be readily applied to liquid water and methanol systems at both the FG and CG resolutions. For both liquids, we reveal that the scaling exponents remain unchanged from the coarse-graining process, indicating that the scaling behavior is universal for the same underlying molecular systems. Combining this finding with the concept of mapping entropy in CG models, we show that the missing entropy plays an important role in accelerating the CG dynamics.

Bottom-up Coarse-Graining: Principles and Perspectives

Large-scale computational molecular models provide scientists a means to investigate the effect of microscopic details on emergent mesoscopic behavior. Elucidating the relationship between variations on the molecular scale and macroscopic observable properties facilitates an understanding of the molecular interactions driving the properties of real world materials and complex systems (e.g., those found in biology, chemistry, and materials science). As a result, discovering an explicit, systematic connection between microscopic nature and emergent mesoscopic behavior is a fundamental goal for this type of investigation. The molecular forces critical to driving the behavior of complex heterogeneous systems are often unclear. More problematically, simulations of representative model systems are often prohibitively expensive from both spatial and temporal perspectives, impeding straightforward investigations over possible hypotheses characterizing molecular behavior. While the reduction in resolution of a study, such as moving from an atomistic simulation to that of the resolution of large coarse-grained (CG) groups of atoms, can partially ameliorate the cost of individual simulations, the relationship between the proposed microscopic details and this intermediate resolution is nontrivial and presents new obstacles to study. Small portions of these complex systems can be realistically simulated. Alone, these smaller simulations likely do not provide insight into collectively emergent behavior. However, by proposing that the driving forces in both smaller and larger systems (containing many related copies of the smaller system) have an explicit connection, systematic bottom-up CG techniques can be used to transfer CG hypotheses discovered using a smaller scale system to a larger system of primary interest. The proposed connection between different CG systems is prescribed by (i) the CG representation (mapping) and (ii) the functional form and parameters used to represent the CG energetics, which approximate potentials of mean force (PMFs). As a result, the design of CG methods that facilitate a variety of physically relevant representations, approximations, and force fields is critical to moving the frontier of systematic CG forward. Crucially, the proposed connection between the system used for parametrization and the system of interest is orthogonal to the optimization used to approximate the potential of mean force present in all systematic CG methods. The empirical efficacy of machine learning techniques on a variety of tasks provides strong motivation to consider these approaches for approximating the PMF and analyzing these approximations.

Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium

We present a generalized Langevin equation (GLE) of motion that governs exactly the time evolution of phase-space observables in finite open systems described by classical Hamiltonians with explicitly time-dependent potentials. This formalism is based on the Mori-Zwanzig projection operator (PO) method with a time-independent Zwanzig PO within a Heisenberg (Lagrangian) picture and reduced description of Hamiltonian systems in terms of canonical relevant and irrelevant coordinates. We demonstrate that, similarly to closed systems, GLE dynamics in Hamiltonian systems in the presence of time-dependent potentials is determined by conservative, dissipative memory, and projected force fields, and that the memory functions relate to the projected force, which is a two-time process, in a way that is reminiscent of the equilibrium second fluctuation-dissipation relation. We further show that, in the most general case, the memory kernel depends on the relevant momentum gradients of the (Boltzmann) entropy of the irrelevant subsystem. Using two Zwanzig operators which are, respectively, functionals of the canonical and generalized canonical probability densities, we then derive what we call canonical and generalized canonical GLEs. Further, we can formulate the particle-based, coarse-grained (CG) GLE dynamics by transitioning to Jacobi coordinates which corresponds to a particle set partitioning of the Hamiltonian system. The obtained canonical CG GLE of motion for the relevant momenta is a generalization of the CG equation of motion known for closed systems. Also, using a Markovian approximation of the canonical CG GLE, we can extend the dissipative particle dynamics equation to open systems. A distinctive feature of our extension is a use of explicitly time-dependent frictions, which reflect the changes in the dissipation rate caused by time-dependent coupling to an external bath. Our GLE formalism and workflow constitute a general and viable framework that can be readily used as a starting point to rigorously formulate microscopically informed CG treatments for a variety of phenomena in externally forced systems far from equilibrium.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Constructing many-body dissipative particle dynamics models of fluids from bottom-up coarse-graining

Since their emergence in the 1990s, mesoscopic models of fluids have been widely used to study complex organization and transport phenomena beyond the molecular scale. Even though these models are designed based on results from physics at the meso- and macroscale, such as fluid mechanics and statistical field theory, the underlying microscopic foundation of these models is not as well defined. This paper aims to build such a systematic connection using bottom-up coarse-graining methods. From the recently developed dynamic coarse-graining scheme, we introduce a statistical inference framework of explicit many-body conservative interaction that quantitatively recapitulates the mesoscopic structure of the underlying fluid. To further consider the dissipative and fluctuation forces, we design a novel algorithm that parameterizes these forces. By utilizing this algorithm, we derive pairwise decomposable friction kernels under both non-Markovian and Markovian limits where both short- and long-time features of the coarse-grained dynamics are reproduced. Finally, through these new developments, the many-body dissipative particle dynamics type of equations of motion are successfully derived. The methodologies developed in this work thus open a new avenue for the construction of direct bottom-up mesoscopic models that naturally bridge the meso- and macroscopic physics.

Dynamically consistent coarse-grain simulation model of chemically specific polymer melts via friction parameterization

Coarse-grained (CG) models of polymers involve grouping many atoms in an all-atom (AA) representation into single sites to reduce computational effort yet retain the hierarchy of length and time scales inherent to macromolecules. Parameterization of such models is often via "bottom-up" methods, which preserve chemical specificity but suffer from artificially accelerated dynamics with respect to the AA model from which they were derived. Here, we study the combination of a bottom-up CG model with a dissipative potential as a means to obtain a chemically specific and dynamically correct model. We generate the conservative part of the force-field using the iterative Boltzmann inversion (IBI) method, which seeks to recover the AA structure. This is augmented with the dissipative Langevin thermostat, which introduces a single parameterizable friction factor to correct the unphysically fast dynamics of the IBI-generated force-field. We study this approach for linear polystyrene oligomer melts for three separate systems with 11, 21, and 41 monomers per chain and a mapping of one monomer per CG site. To parameterize the friction factor, target values are extracted from the AA dynamics using translational monomer diffusion, translational chain diffusion, and rotational chain motion to test the consistency of the parameterization across different modes of motion. We find that the value of the friction parameter needed to bring the CG dynamics in line with AA target values varies based on the mode of parameterization with short-time monomer translational dynamics requiring the highest values, long-time chain translational dynamics requiring the lowest values, and rotational dynamics falling in between. The friction ranges most widely for the shortest chains, and the span narrows with increasing chain length. For longer chains, a practical working value of the friction parameter may be derived from the rotational dynamics, owing to the contribution of multiple relaxation modes to chain rotation and a lack of sensitivity of the translational dynamics at these intermediate levels of friction. A study of equilibrium chain structure reveals that all chains studied are non-Gaussian. However, longer chains better approximate ideal chain dimensions than more rod-like shorter chains and thus are most closely described by a single friction parameter. We also find that the separability of the conservative and dissipative potentials is preserved.

Microscopic derivation of coarse-grained, energy-conserving generalized Langevin dynamics

Properly simulating nonequilibrium phenomena such as thermal transport and shock wave propagation in complex condensed matter systems require the conservation of system's internal energy. This precludes the application of the coarse-grained (CG) generalized Langevin equation (GLE) dynamics due to the presence of dissipative interactions. Attempts to address this issue have been pursued both phenomenologically and from entropy-based first principles for dissipative particle dynamics (DPD, a Markovian variant of the CG GLE dynamics) by introducing an energy conserving extension of DPD (DPD-E). We present here a rigorous microscopic derivation of two energy conserving variants of the CG GLE dynamics by extending the CG equations of motion to include the GLE for certain internal energy observables of the microscopic system. We consider two choices of such observables: the total internal energy and a set of internal energies of the CG particles. The derivation is performed using the Mori-Zwanzig projection operator method in the Heisenberg picture for time evolution of thermodynamic expectations and the recently introduced interpretation of the Zwanzig projection operator [S. Izvekov, J. Chem. Phys. 146(12), 124109 (2017)] which allows an exact calculation of the memory and projected terms. We begin with equilibrium conditions and show that the GLE dynamics for the internal energy observables is purely dissipative. Our extension of the GLE dynamics to quasiequilibrium conditions (necessary to observe heat transport) is based on the generalized canonical ensemble approach and transport equation using the nonequilibrium statistical operator (NSO) method. We derive closed microscopic expressions for conductive heat transfer coefficients in the limit of neglecting dissipation in heat transfer and in the lowest order of deviation from equilibrium. After employing the Markov approximation, we compare the equations of motion to the published DPD-E equations. Our equations contain additional energy transfer terms not reported in the previous works. Additionally, we show that, despite neglecting dissipative processes in heat transport, the heat transfer coefficients and random force are related in a way reminiscent of the fluctuation-dissipation relation. The formalism presented here is sufficiently general for the rigorous formulation of the GLE dynamics for arbitrary microscopic phase space observables as well as sampling different microscopic ensembles in CG simulations.

Recent Progress towards Chemically-Specific Coarse-Grained Simulation Models with Consistent Dynamical Properties

Coarse-grained (CG) models can provide computationally efficient and conceptually simple characterizations of soft matter systems. While generic models probe the underlying physics governing an entire family of free-energy landscapes, bottom-up CG models are systematically constructed from a higher-resolution model to retain a high level of chemical specificity. The removal of degrees of freedom from the system modifies the relationship between the relative time scales of distinct dynamical processes through both a loss of friction and a “smoothing” of the free-energy landscape. While these effects typically result in faster dynamics, decreasing the computational expense of the model, they also obscure the connection to the true dynamics of the system. The lack of consistent dynamics is a serious limitation for CG models, which not only prevents quantitatively accurate predictions of dynamical observables but can also lead to qualitatively incorrect descriptions of the characteristic dynamical processes. With many methods available for optimizing the structural and thermodynamic properties of chemically-specific CG models, recent years have seen a stark increase in investigations addressing the accurate description of dynamical properties generated from CG simulations. In this review, we present an overview of these efforts, ranging from bottom-up parameterizations of generalized Langevin equations to refinements of the CG force field based on a Markov state modeling framework. We aim to make connections between seemingly disparate approaches, while laying out some of the major challenges as well as potential directions for future efforts.

Bottom-up approach to represent dynamic properties in coarse-grained molecular simulations

Several molecular coarse-graining methods have been proposed in recent years to derive chemical- and state-point transferable force fields. While these force fields describe structural and thermodynamic properties in good agreement with fine-grained models and experiments, dynamic properties are usually overestimated. Herein, we examine if the long-time dynamic properties of molecular coarse-grained (CG) systems can be correctly represented by employing a dissipative particle dynamics (DPD) thermostat, which is "bottom-up informed" by means of a variant of the Markovian Mori-Zwanzig (MZ) DPD coarse-graining method. We report single-site and multiple-site CG models for a monomer, dimer, and 24mer based on 2,2-dimethyl propane as a chemical repeat unit and report data obtained from MZ-DPD simulations of liquids, polymer solutions, and polymer melts. We find that despite incomplete time scale separation of the molecular CG model, MZ-DPD achieves quantitative accuracy in predicting diffusive dynamics in single-component liquids and polymer solutions (24mers in a dimer solvent). We also find that MZ-DPD simulations of molecular penetrant diffusion in polymer networks do not reach quantitative agreement with the fine-grained model. Modeling diffusion governed by the activated barrier crossing of small molecular penetrants in these dense systems requires an accurate description of energy barriers, presumably combined with the treatment of memory effects. The use of a MZ-DPD thermostat extends the scope and applicability of molecular CG models for multicomponent systems where a correct description of the relative diffusion rates of the different components is important.

Adaptive resolution simulations coupling atomistic water to dissipative particle dynamics

Multiscale methods are the most efficient way to address the interlinked spatiotemporal scales encountered in soft matter and molecular liquids. In the literature reported hybrid approaches span from quantum to atomistic, coarse-grained, and continuum length scales. In this article, we present the hybrid coupling of the molecular dynamics (MD) and dissipative particle dynamics (DPD) methods, bridging the micro- and mesoscopic descriptions. The interfacing is performed within the adaptive resolution scheme (AdResS), which is a linear momentum conserving coupling technique. Our methodology is hence suitable to simulate fluids on the micro/mesoscopic scale, where hydrodynamics plays an important role. The presented approach is showcased for water at ambient conditions. The supramolecular coupling is enabled by a recently developed clustering algorithm SWINGER that assembles, disassembles, and reassembles clusters as needed during the course of the simulation. This allows for a seamless coupling between standard atomistic MD and DPD models. The developed framework can be readily applied to various applications in the fields of materials and life sciences, e.g., simulations of phospholipids and polymer melts, or to study the red blood cells behavior in normal and disease states.

Single molecule translocation in smectics illustrates the challenge for time-mapping in simulations on multiple scales

Understanding the connections between the characteristic dynamical time scales associated with a coarse-grained (CG) and a detailed representation is central to the applicability of the coarse-graining methods to understand molecular processes. The process of coarse graining leads to an accelerated dynamics, owing to the smoothening of the underlying free-energy landscapes. Often a single time-mapping factor is used to relate the time scales associated with the two representations. We critically examine this idea using a model system ideally suited for this purpose. Single molecular transport properties are studied via molecular dynamics simulations of the CG and atomistic representations of a liquid crystalline, azobenzene containing mesogen, simulated in the smectic and the isotropic phases. The out-of-plane dynamics in the smectic phase occurs via molecular hops from one smectic layer to the next. Hopping can occur via two mechanisms, with and without significant reorientation. The out-of-plane transport can be understood as a superposition of two (one associated with each mode of transport) independent continuous time random walks for which a single time-mapping factor would be rather inadequate. A comparison of the free-energy surfaces, relevant to the out-of-plane transport, qualitatively supports the above observations. Thus, this work underlines the need for building CG models that exhibit both structural and dynamical consistency to the underlying atomistic model.