Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism

Modulating Coarse-Grained Dynamics by Perturbing Free Energy Landscapes

We introduce an approach to describe the long-time dynamics of multiatomic molecules by modulating the free energy landscape (FEL) to capture dominant features of the energy-barrier crossing dynamics of the all-atom (AA) system. Notably, we establish that the self-diffusion coefficient of coarse-grained (CG) systems can be accurately delineated by enhancing conservative force fields with high-frequency perturbations. Using theoretical arguments, we show that these perturbations do not alter the lower-order distribution functions, thereby preserving the structure of the AA system after coarse-graining. We demonstrate the utility of this approach using molecular dynamics simulations of simple molecules in bulk with distinct dynamical characteristics with and without time scale separations as well as for inhomogeneous systems where a fluid is confined in a slit-like nanochannel. Additionally, we also apply our approach to more powerful many-body potentials optimized by using machine learning (ML).

Data-Driven Learning of the Generalized Langevin Equation with State-Dependent Memory

We present a data-driven method to learn stochastic reduced models of complex systems that retain a state-dependent memory beyond the standard generalized Langevin equation with a homogeneous kernel. The constructed model naturally encodes the heterogeneous energy dissipation by jointly learning a set of state features and the non-Markovian coupling among the features. Numerical results demonstrate the limitation of the standard generalized Langevin equation and the essential role of the broadly overlooked state-dependency nature in predicting molecule kinetics related to conformation relaxation and transition.

Understanding dynamics in coarse-grained models. IV. Connection of fine-grained and coarse-grained dynamics with the Stokes-Einstein and Stokes-Einstein-Debye relations

Applying an excess entropy scaling formalism to the coarse-grained (CG) dynamics of liquids, we discovered that missing rotational motions during the CG process are responsible for artificially accelerated CG dynamics. In the context of the dynamic representability between the fine-grained (FG) and CG dynamics, this work introduces the well-known Stokes-Einstein and Stokes-Einstein-Debye relations to unravel the rotational dynamics underlying FG trajectories, thereby allowing for an indirect evaluation of the effective rotations based only on the translational information at the reduced CG resolution. Since the representability issue in CG modeling limits a direct evaluation of the shear stress appearing in the Stokes-Einstein and Stokes-Einstein-Debye relations, we introduce a translational relaxation time as a proxy to employ these relations, and we demonstrate that these relations hold for the ambient conditions studied in our series of work. Additional theoretical links to our previous work are also established. First, we demonstrate that the effective hard sphere radius determined by the classical perturbation theory can approximate the complex hydrodynamic radius value reasonably well. Furthermore, we present a simple derivation of an excess entropy scaling relationship for viscosity by estimating the elliptical integral of molecules. In turn, since the translational and rotational motions at the FG level are correlated to each other, we conclude that the "entropy-free" CG diffusion only depends on the shape of the reference molecule. Our results and analyses impart an alternative way of recovering the FG diffusion from the CG description by coupling the translational and rotational motions at the hydrodynamic level.

A Gauss-Newton method for iterative optimization of memory kernels for generalized Langevin thermostats in coarse-grained molecular dynamics simulations

In molecular dynamics simulations, dynamically consistent coarse-grained (CG) models commonly use stochastic thermostats to model friction and fluctuations that are lost in a CG description. While Markovian, i.e., time-local, formulations of such thermostats allow for an accurate representation of diffusivities/long-time dynamics, a correct description of the dynamics on all time scales generally requires non-Markovian, i.e., non-time-local, thermostats. These thermostats typically take the form of a Generalized Langevin Equation (GLE) determined by a memory kernel. In this work, we use a Markovian embedded formulation of a position-independent GLE thermostat acting independently on each CG degree of freedom. Extracting the memory kernel of this CG model from atomistic reference data requires several approximations. Therefore, this task is best understood as an inverse problem. While our recently proposed approximate Newton scheme allows for the iterative optimization of memory kernels (IOMK), Markovian embedding remained potentially error-prone and computationally expensive. In this work, we present an IOMK-Gauss-Newton scheme (IOMK-GN) based on IOMK that allows for the direct parameterization of a Markovian embedded model.

Score Dynamics: Scaling Molecular Dynamics with Picoseconds Time Steps via Conditional Diffusion Model

We propose score dynamics (SD), a general framework for learning accelerated evolution operators with large timesteps from molecular dynamics (MD) simulations. SD is centered around scores or derivatives of the transition log-probability with respect to the dynamical degrees of freedom. The latter play the same role as force fields in MD but are used in denoising diffusion probability models to generate discrete transitions of the dynamical variables in an SD time step, which can be orders of magnitude larger than a typical MD time step. In this work, we construct graph neural network-based SD models of realistic molecular systems that are evolved with 10 ps timesteps. We demonstrate the efficacy of SD with case studies of the alanine dipeptide and short alkanes in aqueous solution. Both equilibrium predictions derived from the stationary distributions of the conditional probability and kinetic predictions for the transition rates and transition paths are in good agreement with MD. Our current SD implementation is about 2 orders of magnitude faster than the MD counterpart for the systems studied in this work. Open challenges and possible future remedies to improve SD are also discussed.

Chemically Specific Systematic Coarse-Grained Polymer Model with Both Consistently Structural and Dynamical Properties

The coarse-grained (CG) model serves as a powerful tool for the simulation of polymer systems; its reliability depends on the accurate representation of both structural and dynamical properties. However, strong correlations between structural and dynamical properties on different scales and also a strong memory effect, enforced by chain connectivity between monomers in polymer systems, render developing a chemically specific systematic CG model a formidable task. In this study, we report a systematic CG approach that combines the iterative Boltzmann inversion (IBI) method and the generalized Langevin equation (GLE) dynamics. Structural properties are ensured by using conservative CG potentials derived from the IBI method. To retrieve the correct dynamical properties in the system, we demonstrate that using a combination of a Rouse-type delta function and a time-dependent short-time kernel in the GLE simulation is practically efficient. The former can be used to adjust the long-time diffusion dynamics, and the latter can be reconstructed from an iterative procedure according to the velocity autocorrelation function (ACF) from all-atomistic (AA) simulations. Taking the polystyrene as an example, we show that not only structural properties of radial distribution function, intramolecular bond, and angle distributions can be reproduced but also dynamical properties of mean-square displacement, velocity ACF, and force ACF resulted from our CG model have quantitative agreement with the reference AA model. In addition, reasonable agreements are observed in other collective properties between our GLE-CG model and the AA simulations as well.

Force renormalization for probes immersed in an active bath

Langevin equations or generalized Langevin equations (GLEs) are popular models for describing the motion of a particle in a fluid medium in an effective manner. Here we examine particles immersed in an inherently nonequilibrium fluid, i.e., an active bath, which are subject to an external force. Specifically, we consider two types of forces that are highly relevant for microrheological studies: A harmonic, trapping force and a constant, "drag" force. We study such systems by molecular simulations and use the simulation data to extract an effective GLE description. We find that within this description, in an active bath, the external force in the GLE is not equal to the physical external force, but rather a renormalized external force, which can be significantly smaller. The effect cannot be attributed to the mere temperature renormalization, which is also observed.

Representability and Dynamical Consistency in Coarse-Grained Models

We address the challenge of representativity and dynamical consistency when unbonded fine-grained particles are collected together into coarse-grained particles. We implement a hybrid procedure for identifying and tracking the underlying fine-grained particles─e.g., atoms or molecules─by exchanging them between the coarse-grained particles periodically at a characteristic time. The exchange involves a back-mapping of the coarse-grained particles into fine-grained particles and a subsequent reassignment to coarse-grained particles conserving total mass and momentum. We find that an appropriate choice of the characteristic exchange time can lead to the correct effective diffusion rate of the fine-grained particles when simulated in hybrid coarse-grained dynamics. In the compressed (supercritical) fluid regime, without the exchange term, fine-grained particles remain associated with a given coarse-grained particle, leading to substantially lower diffusion rates than seen in all-atom molecular dynamics of the fine-grained particles. Thus, this work confirms the need for addressing the representativity of fine-grained particles within coarse-grained particles and offers a simple exchange mechanism so as to retain dynamical consistency between the fine- and coarse-grained scales.

Einstein-Stokes relation for small bubbles at the nanoscale

As the physicochemical properties of ultrafine bubble systems are governed by their size, it is crucial to determine the size and distribution of such bubble systems. At present, the size or size distribution of nanometer-sized bubbles in suspension is often measured by either dynamic light scattering or the nanoparticle tracking analysis. Both techniques determine the bubble size via the Einstein-Stokes equation based on the theory of the Brownian motion. However, it is not yet clear to which extent the Einstein-Stokes equation is applicable for such ultrafine bubbles. In this work, using atomic molecular dynamics simulation, we evaluate the applicability of the Einstein-Stokes equation for gas nanobubbles with a diameter less than 10 nm, and for a comparative analysis, both vacuum nanobubbles and copper nanoparticles are also considered. The simulation results demonstrate that the diffusion coefficient for rigid nanoparticles in water is found to be highly consistent with the Einstein-Stokes equation, with slight deviation only found for nanoparticle with a radius less than 1 nm. For nanobubbles, including both methane and vacuum nanobubbles, however, large deviation from the Einstein-Stokes equation is found for the bubble radius larger than 3 nm. The deviation is attributed to the deformability of large nanobubbles that leads to a cushioning effect for collision-induced bubble diffusion.

Data-driven dynamical coarse-graining for condensed matter systems

Simulations of condensed matter systems often focus on the dynamics of a few distinguished components but require integrating the full system. A prime example is a molecular dynamics simulation of a (macro)molecule in a solution, where the molecule(s) and the solvent dynamics need to be integrated, rendering the simulations computationally costly and often unfeasible for physically/biologically relevant time scales. Standard coarse graining approaches can reproduce equilibrium distributions and structural features but do not properly include the dynamics. In this work, we develop a general data-driven coarse-graining methodology inspired by the Mori-Zwanzig formalism, which shows that macroscopic systems with a large number of degrees of freedom can be described by a few relevant variables and additional noise and memory terms. Our coarse-graining method consists of numerical integrators for the distinguished components, where the noise and interaction terms with other system components are substituted by a random variable sampled from a data-driven model. The model is parameterized using data from multiple short-time full-system simulations, and then, it is used to run long-time simulations. Applying our methodology to three systems-a distinguished particle under a harmonic and a bistable potential and a dimer with two metastable configurations-the resulting coarse-grained models are capable of reproducing not only the equilibrium distributions but also the dynamic behavior due to temporal correlations and memory effects. Remarkably, our method even reproduces the transition dynamics between metastable states, which is challenging to capture correctly. Our approach is not constrained to specific dynamics and can be extended to systems beyond Langevin dynamics, and, in principle, even to non-equilibrium dynamics.

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.

A deep learning approach to the measurement of long-lived memory kernels from generalized Langevin dynamics

Memory effects are ubiquitous in a wide variety of complex physical phenomena, ranging from glassy dynamics and metamaterials to climate models. The Generalized Langevin Equation (GLE) provides a rigorous way to describe memory effects via the so-called memory kernel in an integro-differential equation. However, the memory kernel is often unknown, and accurately predicting or measuring it via, e.g., a numerical inverse Laplace transform remains a herculean task. Here, we describe a novel method using deep neural networks (DNNs) to measure memory kernels from dynamical data. As a proof-of-principle, we focus on the notoriously long-lived memory effects of glass-forming systems, which have proved a major challenge to existing methods. In particular, we learn the operator mapping dynamics to memory kernels from a training set generated with the Mode-Coupling Theory (MCT) of hard spheres. Our DNNs are remarkably robust against noise, in contrast to conventional techniques. Furthermore, we demonstrate that a network trained on data generated from analytic theory (hard-sphere MCT) generalizes well to data from simulations of a different system (Brownian Weeks-Chandler-Andersen particles). Finally, we train a network on a set of phenomenological kernels and demonstrate its effectiveness in generalizing to both unseen phenomenological examples and supercooled hard-sphere MCT data. We provide a general pipeline, KernelLearner, for training networks to extract memory kernels from any non-Markovian system described by a GLE. The success of our DNN method applied to noisy glassy systems suggests that deep learning can play an important role in the study of dynamical systems with memory.

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.

Bottom-Up Informed and Iteratively Optimized Coarse-Grained Non-Markovian Water Models with Accurate Dynamics

Molecular dynamics (MD) simulations based on coarse-grained (CG) particle models of molecular liquids generally predict accelerated dynamics and misrepresent the time scales for molecular vibrations and diffusive motions. The parametrization of Generalized Langevin Equation (GLE) thermostats based on the microscopic dynamics of the fine-grained model provides a promising route to address this issue, in conjunction with the conservative interactions of the CG model obtained with standard coarse graining methods, such as iterative Boltzmann inversion, force matching, or relative entropy minimization. We report the application of a recently introduced bottom-up dynamic coarse graining method, based on the Mori-Zwanzig formalism, which provides accurate estimates of isotropic GLE memory kernels for several CG models of liquid water. We demonstrate that, with an additional iterative optimization of the memory kernels (IOMK) for the CG water models based on a practical iterative optimization technique, the velocity autocorrelation function of liquid water can be represented very accurately within a few iterations. By considering the distinct Van Hove function, we demonstrate that, with the presented methods, an accurate representation of structural relaxation can be achieved. We consider several distinct CG potentials to study how the choice of the CG potential affects the performance of bottom-up informed and iteratively optimized models.

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.

Generalized Energy-Conserving Dissipative Particle Dynamics with Mass Transfer. Part 1: Theoretical Foundation and Algorithm

An extension of the generalized energy-conserving dissipative particle dynamics method (GenDPDE) that allows mass transfer between mesoparticles via a diffusion process is presented. By considering the concept of the mesoparticles as property carriers, the complexity and flexibility of the GenDPDE framework were enhanced to allow for interparticle mass transfer under isoenergetic conditions, notated here as GenDPDE-M. In the formulation, diffusion is described via the theory of mesoscale irreversible processes based on linear relationships between the fluxes and thermodynamic forces, where their fluctuations are described by Langevin-like equations. The mass exchange between mesoparticles is such that the mass of the mesoparticle remains unchanged after the transfer process and requires additional considerations regarding the coupling with other system properties such as the particle internal energy. The proof-of-concept work presented in this article is the first part of a two-part article series. In Part 1, the development of the GenDPDE-M theoretical framework and the derivation of the algorithm are presented in detail. Part 2 of this article series is targeted for practitioners, where applications, demonstrations, and practical considerations for implementing the GenDPDE-M method are presented and discussed.

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.

Cross-Correlation Corrected Friction in Generalized Langevin Models: Application to the continuous Asakura-Oosawa Model

In a previous study we proposed a method to parameterize isotropic, configuration independent, non-Markovian generalized Langevin thermostats to achieve dynamic consistency in coarse-grained models. In the current study, by applying the same strategy, we develop coarse-grained implicit solvent models for the continuous Asakura-Oosawa model, which under certain conditions allows to develop very accurate coarse-grained potentials. By developing coarse-grained models for different reference systems with varying parameters, we test the broader applicability of the proposed procedure and demonstrate the relevance of accurate coarse-grained potentials in bottom-up derived dissipative models. We study how different system parameters affect the dynamic representability of the coarse-grained models. In particular we find that the quality of the coarse-grained potential is crucial to correctly model the backscattering effect due to collisions on the coarse-grained scale. In the dynamics of colloid suspensions the hydrodynamic interactions affect the long-time scale dynamics by solvent mediated momentum transfer. These interactions are not explicitly modeled in the presented coarse-grained models, which poses some limitations to the proposed coarse-graining scheme. The Asakura-Oosawa model allows a tuning of system parameters, to gain an improved understanding of these limitations. We also propose three new iterative optimization schemes to fine tune the generalized Langevin thermostat to exactly match the reference velocity-autocorrelation function.

Molecular Modeling in Anion Exchange Membrane Research: A Brief Review of Recent Applications

Anion Exchange Membrane (AEM) fuel cells have attracted growing interest, due to their encouraging advantages, including high power density and relatively low cost. AEM is a polymer matrix, which conducts hydroxide (OH-) ions, prevents physical contact of electrodes, and has positively charged head groups (mainly quaternary ammonium (QA) groups), covalently bound to the polymer backbone. The chemical instability of the quaternary ammonium (QA)-based head groups, at alkaline pH and elevated temperature, is a significant threshold in AEMFC technology. This review work aims to introduce recent studies on the chemical stability of various QA-based head groups and transportation of OH- ions in AEMFC, via modeling and simulation techniques, at different scales. It starts by introducing the fundamental theories behind AEM-based fuel-cell technology. In the main body of this review, we present selected computational studies that deal with the effects of various parameters on AEMs, via a variety of multi-length and multi-time-scale modeling and simulation methods. Such methods include electronic structure calculations via the quantum Density Functional Theory (DFT), ab initio, classical all-atom Molecular Dynamics (MD) simulations, and coarse-grained MD simulations. The explored processing and structural parameters include temperature, hydration levels, several QA-based head groups, various types of QA-based head groups and backbones, etc. Nowadays, many methods and software packages for molecular and materials modeling are available. Applications of such methods may help to understand the transportation mechanisms of OH- ions, the chemical stability of functional head groups, and many other relevant properties, leading to a performance-based molecular and structure design as well as, ultimately, improved AEM-based fuel cell performances. This contribution aims to introduce those molecular modeling methods and their recent applications to the AEM-based fuel cells research community.

Likelihood-based non-Markovian models from molecular dynamics

SignificanceThe analysis of complex systems with many degrees of freedom generally involves the definition of low-dimensional collective variables more amenable to physical understanding. Their dynamics can be modeled by generalized Langevin equations, whose coefficients have to be estimated from simulations of the initial high-dimensional system. These equations feature a memory kernel describing the mutual influence of the low-dimensional variables and their environment. We introduce and implement an approach where the generalized Langevin equation is designed to maximize the statistical likelihood of the observed data. This provides an efficient way to generate reduced models to study dynamical properties of complex processes such as chemical reactions in solution, conformational changes in biomolecules, or phase transitions in condensed matter systems.

Non-Markovian systems out of equilibrium: exact results for two routes of coarse graining

Generalized Langevin equations (GLEs) can be systematically derived via dimensional reduction from high-dimensional microscopic systems. For linear models the derivation can either be based on projection operator techniques such as the Mori-Zwanzig (MZ) formalism or by 'integrating out' the bath degrees of freedom. Based on exact analytical results we show that both routes can lead to fundamentally different GLEs and that the origin of these differences is based inherently on the non-equilibrium nature of the microscopic stochastic model. The most important conceptional difference between the two routes is that the MZ result intrinsically fulfills the generalized second fluctuation-dissipation theorem while the integration result can lead to its violation. We supplement our theoretical findings with numerical and simulation results for two popular non-equilibrium systems: time-delayed feedback control and the active Ornstein-Uhlenbeck process.

Optimal renormalization of multiscale systems

While model order reduction is a promising approach in dealing with multiscale time-dependent systems that are too large or too expensive to simulate for long times, the resulting reduced order models can suffer from instabilities. We have recently developed a time-dependent renormalization approach to stabilize such reduced models. In the current work, we extend this framework by introducing a parameter that controls the time decay of the memory of such models and optimally select this parameter based on limited fully resolved simulations. First, we demonstrate our framework on the inviscid Burgers equation whose solution develops a finite-time singularity. Our renormalized reduced order models are stable and accurate for long times while using for their calibration only data from a full order simulation before the occurrence of the singularity. Furthermore, we apply this framework to the three-dimensional (3D) Euler equations of incompressible fluid flow, where the problem of finite-time singularity formation is still open and where brute force simulation is only feasible for short times. Our approach allows us to obtain a perturbatively renormalizable model which is stable for long times and includes all the complex effects present in the 3D Euler dynamics. We find that, in each application, the renormalization coefficients display algebraic decay with increasing resolution and that the parameter which controls the time decay of the memory is problem-dependent.

The Middle Science: Traversing Scale In Complex Many-Body Systems

A roadmap is developed that integrates simulation methodology and data science methods to target new theories that traverse the multiple length- and time-scale features of many-body phenomena.

Data-driven coarse-grained modeling of non-equilibrium systems

Modeling a high-dimensional Hamiltonian system in reduced dimensions with respect to coarse-grained (CG) variables can greatly reduce computational cost and enable efficient bottom-up prediction of main features of the system for many applications. However, it usually experiences significantly altered dynamics due to loss of degrees of freedom upon coarse-graining. To establish CG models that can faithfully preserve dynamics, previous efforts mainly focused on equilibrium systems. In contrast, various soft matter systems are known to be out of equilibrium. Therefore, the present work concerns non-equilibrium systems and enables accurate and efficient CG modeling that preserves non-equilibrium dynamics and is generally applicable to any non-equilibrium process and any observable of interest. To this end, the dynamic equation of a CG variable is built in the form of the non-stationary generalized Langevin equation (nsGLE), where the two-time memory kernel is determined from the data of the auto-correlation function of the observable of interest. By embedding the nsGLE in an extended dynamics framework, the nsGLE can be solved efficiently to predict the non-equilibrium dynamics of the CG variable. To prove and exploit the equivalence of the nsGLE and extended dynamics, the memory kernel is parameterized in a two-time exponential expansion. A data-driven hybrid optimization process is proposed for the parameterization, which integrates the differential-evolution method with the Levenberg-Marquardt algorithm to efficiently tackle a non-convex and high-dimensional optimization problem.

Fluctuation-dissipation relations far from equilibrium: a case study

Fluctuation-dissipation relations or "theorems" (FDTs) are fundamental for statistical physics and can be rigorously derived for equilibrium systems. Their applicability to non-equilibrium systems is, however, debated. Here, we simulate an active microrheology experiment, in which a spherical colloid is pulled with a constant external force through a fluid, creating near-equilibrium and far-from-equilibrium systems. We characterize the structural and dynamical properties of these systems, and reconstruct an effective generalized Langevin equation (GLE) for the colloid dynamics. Specifically, we test the validity of two FDTs: The first FDT relates the non-equilibrium response of a system to equilibrium correlation functions, and the second FDT relates the memory friction kernel in the GLE to the stochastic force. We find that the validity of the first FDT depends strongly on the strength of the external driving: it is fulfilled close to equilibrium and breaks down far from it. In contrast, we observe that the second FDT is always fulfilled. We provide a mathematical argument why this generally holds for memory kernels reconstructed from a deterministic Volterra equation for correlation functions, even for non-stationary non-equilibrium systems. Motivated by the Mori-Zwanzig formalism, we therefore suggest to impose an orthogonality constraint on the stochastic force, which is in fact equivalent to the validity of this Volterra equation. Such GLEs automatically satisfy the second FDT and are unique, which is desirable when using GLEs for coarse-grained modeling.

Cross-correlation corrected friction in (generalized) Langevin models

We propose a route for parameterizing isotropic (generalized) Langevin [(G)LE] thermostats with the aim to correct the dynamics of coarse-grained (CG) models with pairwise conservative interactions. The approach is based on the Mori-Zwanzig formalism and derives the memory kernels from Q-projected time correlation functions. Bottom-up informed (GLE and LE) thermostats for a CG star-polymer melt are investigated, and it is demonstrated that the inclusion of memory in the CG simulation leads to predictions of polymer diffusion in quantitative agreement with fine-grained simulations. Interestingly, memory effects are observed in the diffusive regime. We demonstrate that previously neglected cross-correlations between the "irrelevant" and the CG degree of freedom are important and lie at the origin of shortcomings in previous CG simulations.

Petrov-Galerkin methods for the construction of non-Markovian dynamics preserving nonlocal statistics

A common observation in coarse-graining a molecular system is the non-Markovian behavior, primarily due to the lack of scale separations. This is reflected in the strong memory effect and the non-white noise spectrum, which must be incorporated into a coarse-grained description to correctly predict dynamic properties. To construct a stochastic model that gives rise to the correct non-Markovian dynamics, we propose a Galerkin projection approach, which transforms the exhausting effort of finding an appropriate model to choosing appropriate subspaces in terms of the derivatives of the coarse-grained variables and, at the same time, provides an accurate approximation to the generalized Langevin equation. We introduce the notion of fractional statistics that embodies nonlocal properties. More importantly, we show how to pick subspaces in the Galerkin projection so that those statistics are automatically matched.

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.

Dynamic coarse-graining of polymer systems using mobility functions

We propose a dynamic coarse-graining (CG) scheme for mapping heterogeneous polymer fluids onto extremely CG models in a dynamically consistent manner. The idea is to use as target function for the mapping a wave-vector dependent mobility function derived from the single-chain dynamic structure factor, which is calculated in the microscopic reference system. In previous work, we have shown that dynamic density functional calculations based on this mobility function can accurately reproduce the order/disorder kinetics in polymer melts, thus it is a suitable starting point for dynamic mapping. To enable the mapping over a range of relevant wave vectors, we propose to modify the CG dynamics by introducing internal friction parameters that slow down the CG monomer dynamics on local scales, without affecting the static equilibrium structure of the system. We illustrate and discuss the method using the example of infinitely long linear Rouse polymers mapped onto ultrashort CG chains. We show that our method can be used to construct dynamically consistent CG models for homopolymers with CG chain lengthN= 4, whereas for copolymers, longer CG chain lengths are necessary.

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.

Dynamic-mode decomposition and optimal prediction

Dynamic-mode decomposition (DMD) is a well-established data-driven method of finding temporally evolving linear-mode decompositions of a nonlinear time series. Traditionally, this method presumes that all relevant dimensions are sampled through measurement. To address dynamical systems in which the data may be incomplete or represent only partial observation of a more complex system, we extend the DMD algorithm by including a Mori-Zwanzig decomposition to derive memory kernels that capture the averaged dynamics of the unresolved variables as projected onto the resolved dimensions. From this, we then derive what we call the memory-dependent dynamic mode decomposition (MDDMD). Through numerical examples, the MDDMD method is shown to produce reasonable approximations of the ensemble-averaged dynamics of the full system given a single time series measurement of the resolved variables.

Heat diffusion-related damping process in a highly precise coarse-grained model for nonlinear motion of SWCNT

Second sound and heat diffusion in single-walled carbon nanotubes (SWCNT) are well-known phenomena which is related to the high thermal conductivity of this material. In this paper, we have shown that the heat diffusion along the tube axis affects the macroscopic motion of SWCNT and adapting this phenomena to coarse-grained (CG) model can improve the precision of the coarse-grained molecular dynamics (CGMD) exceptionally. The nonlinear macroscopic motion of SWCNT in the free thermal vibration condition in adiabatic environment is demonstrated in the most simplified version of CG modeling as maintaining finite temperature and total energy with suggested dissipation process derived from internal heat diffusion. The internal heat diffusion related to the cross correlated momentum from different potential energy functions is considered, and it can reproduce the nonlinear dynamic nature of SWCNTs without external thermostatting in CG model. Memory effect and thermostat with random noise distribution are not included, and the effect of heat diffusion on memory effect is quantified through Mori-Zwanzig formalism. This diffusion shows perfect syncronization of the motion between that of CGMD and MD simulation, which is started with initial conditions from the molecular dynamics (MD) simulation. The heat diffusion related to this process has shown the same dispersive characteristics to second wave in SWCNT. This replication with good precision indicates that the internal heat diffusion process is the essential cause of the nonlinearity of the tube. The nonlinear dynamic characteristics from the various scale of simple beads systems are examined with expanding its time step and node length.

Data-Driven Model Reduction for Stochastic Burgers Equations

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.

Optimal estimates of self-diffusion coefficients from molecular dynamics simulations

Translational diffusion coefficients are routinely estimated from molecular dynamics simulations. Linear fits to mean squared displacement (MSD) curves have become the de facto standard, from simple liquids to complex biomacromolecules. Nonlinearities in MSD curves at short times are handled with a wide variety of ad hoc practices, such as partial and piece-wise fitting of the data. Here, we present a rigorous framework to obtain reliable estimates of the self-diffusion coefficient and its statistical uncertainty. We also assess in a quantitative manner if the observed dynamics is, indeed, diffusive. By accounting for correlations between MSD values at different times, we reduce the statistical uncertainty of the estimator and, thereby, increase its efficiency. With a Kolmogorov-Smirnov test, we check for possible anomalous diffusion. We provide an easy-to-use Python data analysis script for the estimation of self-diffusion coefficients. As an illustration, we apply the formalism to molecular dynamics simulation data of pure TIP4P-D water and a single ubiquitin protein. In another paper [S. von Bülow, J. T. Bullerjahn, and G. Hummer, J. Chem. Phys. 153, 021101 (2020)], we demonstrate its ability to recognize deviations from regular diffusion caused by systematic errors in a common trajectory "unwrapping" scheme that is implemented in popular simulation and visualization software.

Generalised dissipative particle dynamics with energy conservation: density- and temperature-dependent potentials

We present a generalised, energy-conserving dissipative particle dynamics (DPDE) method appropriate for the non-isothermal simulation of particle interaction force fields that are both density- and temperature-dependent. A detailed derivation is formulated in a bottom-up manner by considering the thermodynamics of small systems with the appropriate consideration of the fluctuations. Connected to the local volume is a local density and corresponding local pressure, which is determined from an equation-of-state based force field that depends also on a particle temperature. Compared to the original DPDE method, the formulation of the generalised DPDE method requires a change in the independent variable from the particle internal energy to the particle entropy. As part of the re-formulation, the terms dressed particle entropy and the corresponding dressed particle temperature are introduced, which depict the many-body contributions in the local volume. The generalised DPDE method has similarities to the energy form of the smoothed dissipative particle dynamics method, yet fundamental differences exist, which are described in the manuscript. The basic dynamic equations are presented along with practical considerations for implementing the generalised DPDE method, including a numerical integration scheme based on the Shardlow-like splitting algorithm. Demonstrations and validation tests are performed using analytical equation-of-states for the van der Waals and Lennard-Jones fluids. Particle probability distributions are analysed, where excellent agreement with theoretical estimates is demonstrated. As further validation of the generalised DPDE method, both equilibrium and non-equilibrium simulation scenarios are considered, including adiabatic flash heating response and vapour-liquid phase separation.

Implicit-solvent coarse-grained modeling for polymer solutions via Mori-Zwanzig formalism

We present a bottom-up coarse-graining (CG) method to establish implicit-solvent CG modeling for polymers in solution, which conserves the dynamic properties of the reference microscopic system. In particular, tens to hundreds of bonded polymer atoms (or Lennard-Jones beads) are coarse-grained as one CG particle, and the solvent degrees of freedom are eliminated. The dynamics of the CG system is governed by the generalized Langevin equation (GLE) derived via the Mori-Zwanzig formalism, by which the CG variables can be directly and rigorously linked to the microscopic dynamics generated by molecular dynamics (MD) simulations. The solvent-mediated dynamics of polymers is modeled by the non-Markovian stochastic dynamics in GLE, where the memory kernel can be computed from the MD trajectories. To circumvent the difficulty in direct evaluation of the memory term and generation of colored noise, we exploit the equivalence between the non-Markovian dynamics and Markovian dynamics in an extended space. To this end, the CG system is supplemented with auxiliary variables that are coupled linearly to the momentum and among themselves, subject to uncorrelated Gaussian white noise. A high-order time-integration scheme is used to solve the extended dynamics to further accelerate the CG simulations. To assess, validate, and demonstrate the established implicit-solvent CG modeling, we have applied it to study four different types of polymers in solution. The dynamic properties of polymers characterized by the velocity autocorrelation function, diffusion coefficient, and mean square displacement as functions of time are evaluated in both CG and MD simulations. Results show that the extended dynamics with auxiliary variables can construct arbitrarily high-order CG models to reproduce dynamic properties of the reference microscopic system and to characterize long-time dynamics of polymers in solution.

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.

The multi-dimensional generalized Langevin equation for conformational motion of proteins

Using the generalized Langevin equation (GLE) is a promising approach to build coarse-grained (CG) models of molecular systems since the GLE model often leads to more accurate thermodynamic and kinetic predictions than Brownian dynamics or Langevin models by including a more sophisticated friction with memory. The GLE approach has been used for CG coordinates such as the center of mass of a group of atoms with pairwise decomposition and for a single CG coordinate. We present a GLE approach when CG coordinates are multiple generalized coordinates, defined, in general, as nonlinear functions of microscopic atomic coordinates. The CG model for multiple generalized coordinates is described by the multidimensional GLE from the Mori-Zwanzig formalism, which includes an exact memory matrix. We first present a method to compute the memory matrix in a multidimensional GLE using trajectories of a full system. Then, in order to reduce the computational cost of computing the multidimensional friction with memory, we introduce a method that maps the GLE to an extended Markovian system. In addition, we study the effect of using a nonconstant mass matrix in the CG model. In particular, we include mass-dependent terms in the mean force. We used the proposed CG model to describe the conformational motion of a solvated alanine dipeptide system, with two dihedral angles as the CG coordinates. We showed that the CG model can accurately reproduce two important kinetic quantities: the velocity autocorrelation function and the distribution of first passage times.

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.

Systematic derivation of hybrid coarse-grained models

Molecular dynamics represents a key enabling technology for applications ranging from biology to the development of new materials. However, many real-world applications remain inaccessible to fully resolved simulations due to their unsustainable computational costs and must therefore rely on semiempirical coarse-grained models. Significant efforts have been devoted in the last decade towards improving the predictivity of these coarse-grained models and providing a rigorous justification of their use, through a combination of theoretical studies and data-driven approaches. One of the most promising research efforts is the (re)discovery of the Mori-Zwanzig projection as a generic, yet systematic, theoretical tool for deriving coarse-grained models. Despite its clean mathematical formulation and generality, there are still many open questions about its applicability and assumptions. In this work, we propose a detailed derivation of a hybrid multiscale system, generalizing and further investigating the approach developed in Español [Europhys. Lett. 88, 40008 (2009)10.1209/0295-5075/88/40008]. Issues such as the general coexistence of atoms (fully resolved degrees of freedom) and beads (larger coarse-grained units), the role of the fine-to-coarse mapping chosen, and the approximation of effective potentials are discussed. The theoretical discussion is supported by numerical simulations of a monodimensional nonlinear periodic benchmark system with an open-source parallel Julia code, easily extensible to arbitrary potential models and fine-to-coarse mapping functions. The results presented highlight the importance of introducing, in the macroscopic model, nonconstant fluctuating and dissipative terms, given by the Mori-Zwanzig approach, to correctly reproduce the reference fine-grained results, without requiring ad hoc calibration of interaction potentials and thermostats.

Generalized Langevin dynamics: construction and numerical integration of non-Markovian particle-based models

We propose a generalized Langevin dynamics (GLD) technique to construct non-Markovian particle-based coarse-grained models from fine-grained reference simulations and to efficiently integrate them. The proposed GLD model has the form of a discretized generalized Langevin equation with distance-dependent two-particle contributions to the self- and pair-memory kernels. The memory kernels are iteratively reconstructed from the dynamical correlation functions of an underlying fine-grained system. We develop a simulation algorithm for this class of non-Markovian models that scales linearly with the number of coarse-grained particles. Our GLD method is suitable for coarse-grained studies of systems with incomplete time scale separation, as is often encountered, e.g., in soft matter systems. We apply the method to a suspension of nanocolloids with frequency-dependent hydrodynamic interactions. We show that the results from GLD simulations perfectly reproduce the dynamics of the underlying fine-grained system. The effective speedup of these simulations amounts to a factor of about 104. Additionally, the transferability of the coarse-grained model with respect to changes of the nanocolloid density is investigated. The results indicate that the model is transferable to systems with nanocolloid densities that differ by up to one order of magnitude from the density of the reference system.

Memory-induced acceleration and slowdown of barrier crossing

We study the mean first-passage time τ_MFP for the barrier crossing of a single massive particle with non-Markovian memory by Langevin simulations in one dimension. In the Markovian limit of short memory time τ_Γ, the expected Kramers turnover between the overdamped (high-friction) and the inertial (low-friction) limits is recovered. Compared to the Markovian case, we find barrier crossing to be accelerated for intermediate memory time, while for long memory time, barrier crossing is slowed down and τ_MFP increases with τ_Γ as a power law τ_MFP∼τ_Γ². Both effects are derived from an asymptotic propagator analysis: while barrier crossing acceleration at intermediate memory can be understood as an effective particle mass reduction, slowing down for long memory is caused by the slow kinetics of energy diffusion. A simple and globally accurate heuristic formula for τ_MFP in terms of all relevant time scales of the system is presented and used to establish a scaling diagram featuring the Markovian overdamped and the Markovian inertial regimes, as well as the non-Markovian intermediate memory time regime where barrier crossing is accelerated and the non-Markovian long memory time regime where barrier crossing is slowed down.

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.