Coarse-graining Scheme Research Articles

Spectroscopy with the tensor renormalization group method

We present a spectroscopy scheme for the lattice field theory by using the tensor renormalization group method combining with the transfer matrix formalism. By using the scheme, we cannot only compute the energy spectrum for the lattice theory but also determine quantum numbers of the energy eigenstates. Furthermore, the wave function of the corresponding eigenstate can also be computed. The first step of the scheme is to coarse grain the tensor network of a given lattice model by using the higher order tensor renormalization group, and then after making a matrix corresponding to a transfer matrix from the coarse-grained tensors, its eigenvalues are evaluated to extract the energy spectrum. Second, the quantum number of the eigenstates can be identified by a selection rule that requires to compute matrix elements of an associated insertion operator. The matrix elements can be represented by an impurity tensor network and computed by the coarse-graining scheme. Moreover, we can compute the wave function of the energy eigenstate by putting the impurity tensor at each point in space direction of the network. Additionally, the momentum of the eigenstate can also be identified by computing appropriate matrix elements represented by the tensor network. As a demonstration of the new scheme, we show the spectroscopy of the (1+1)d Ising model and compare it with exact results. We also present a scattering phase shift obtained from two-particle state energy using Lüscher’s formula. Published by the American Physical Society 2024

An encompassed representation of timescale hierarchies in first-order reaction network

Complex networks are pervasive in various fields such as chemistry, biology, and sociology. In chemistry, first-order reaction networks are represented by a set of first-order differential equations, which can be constructed from the underlying energy landscape. However, as the number of nodes increases, it becomes more challenging to understand complex kinetics across different timescales. Hence, how to construct an interpretable, coarse-graining scheme that preserves the underlying timescales of overall reactions is of crucial importance. Here, we develop a scheme to capture the underlying hierarchical subsets of nodes, and a series of coarse-grained (reduced-dimensional) rate equations between the subsets as a function of time resolution from the original reaction network. Each of the coarse-grained representations guarantees to preserve the underlying slow characteristic timescales in the original network. The crux is the construction of a lumping scheme incorporating a similarity measure in deciphering the underlying timescale hierarchy, which does not rely on the assumption of equilibrium. As an illustrative example, we apply the scheme to four-state Markovian models and Claisen rearrangement of allyl vinyl ether (AVE), and demonstrate that the reduced-dimensional representation accurately reproduces not only the slowest but also the faster timescales of overall reactions although other reduction schemes based on equilibrium assumption well reproduce the slowest timescale but fail to reproduce the second-to-fourth slowest timescales with the same accuracy. Our scheme can be applied not only to the reaction networks but also to networks in other fields, which helps us encompass their hierarchical structures of the complex kinetics over timescales.

Construction of Multiscale Dissipative Particle Dynamics (DPD) Models from Other Coarse-Grained Models.

We present a general scheme for converting coarse-grained models into Dissipative Particle Dynamics (DPD) models. We build the corresponding DPD models by analogy with the de novo DPD coarse-graining scheme suggested by Groot and Warren (J. Chem. Phys., 1997). Electrostatic interactions between charged DPD particles are represented though the addition of a long-range Slater Coulomb potential as suggested by González-Melchor et al. (J. Chem. Phys., 2006). The construction is illustrated by converting MARTINI models for various proteins into a DPD representation, but it not restricted to the usual potential form in the MARTINI model-viz., Lennard-Jones potentials. We further extended the DPD scheme away from the typical use of homogeneous particle sizes, therefore faithfully representing the variations in the particle sizes seen in the underlying MARTINI model. The accuracy of the resulting construction of our generalized DPD models with respect to several structural observables has been benchmarked favorably against all-atom and MARTINI models for a selected set of peptides and proteins, and variations in the scales of the coarse-graining of the water solvent.

Tensor network methods for extracting conformal field theory data from fixed-point tensors and defect coarse graining.

We present a comprehensive study on extracting conformal field theory data using tensor network methods, especially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the classical two-dimensional Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator product expansion coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of pointlike defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from the lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with pointlike defects, including graph-independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tensor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.

Modelling Binder Degradation in the Thermal Treatment of Spent Lithium-Ion Batteries by Coupling Discrete Element Method and Isoconversional Kinetics

Developing efficient recycling processes with high recycling quotas for the recovery of graphite and other critical raw materials contained in LIBs is essential and prudent. This action holds the potential to substantially diminish the supply risk of raw materials for LIBs and enhance the sustainability of their production. An essential processing step in LIB recycling involves the thermal treatment of black mass to degrade the binder. This step is crucial as it enhances the recycling efficiency in subsequent processes, such as flotation and leaching-based processing. Therefore, this paper introduces a Representative Black Mass Model (RBMM) and develops a computational framework for the simulation of the thermal degradation of polymer-based binders in black mass (BM). The models utilize the discrete element method (DEM) with a coarse-graining (CG) scheme and the isoconversional method to predict binder degradation and the required heat. Thermogravimetric analysis (TGA) of the binder polyvinylidene fluoride (PVDF) is utilized to determine the model parameters. The model simulates a specific thermal treatment case on a laboratory scale and investigates the relationship between the scale factor and heating rate. The findings reveal that, for a particular BM system, a scaling factor of 100 regarding the particle diameter is applicable within a heating rate range of 2 to 22 K/min.

A coupled MPM-DEM method for modelling soil-rock mixtures

Aiming at modelling the mechanical behaviour of soil-rock mixtures accurately and efficiently, a coupled MPM-DEM formulation combining the material point method (MPM) and the discrete element method (DEM) is proposed. It is solved concurrently via the contact force linking the two individual methods. Specifically, the soil is modelled with MPM as continuums to avoid handling the contacts between fine particles. The rocks are modelled by DEM to capture the contact characteristics of rocks. This method is validated with ball impacting and block sliding tests first for the contact between material points and DEM particles. Its capability in describing the mechanics of soil-rock mixtures is thereafter proved by comparing the simulation results with pure DEM simulations of binary mixtures and laboratory tests of soil-rock mixtures. It is demonstrated that MPM-DEM can reproduce the stress–strain response of soil-rock mixtures and capture the influence of rock contents and rock sizes. In addition, a coarse-graining modelling scheme is implemented, i.e., representing the soil particles with fewer material points, which significantly increases the efficiency compared with pure DEM. Our proposed method provides a novel way to model soil-rock mixtures with reasonable computational efforts, which sheds light on simulating large-scale soil-rock mixtures in nature or engineering.

Integrating Machine Learning in the Coarse-Grained Molecular Simulation of Polymers

Machine learning (ML) is having an increasing impact on the physical sciences, engineering, and technology and its integration into molecular simulation frameworks holds great potential to expand their scope of applicability to complex materials and facilitate fundamental knowledge and reliable property predictions, contributing to the development of efficient materials design routes. The application of ML in materials informatics in general, and polymer informatics in particular, has led to interesting results, however great untapped potential lies in the integration of ML techniques into the multiscale molecular simulation methods for the study of macromolecular systems, specifically in the context of Coarse Grained (CG) simulations. In this Perspective, we aim at presenting the pioneering recent research efforts in this direction and discussing how these new ML-based techniques can contribute to critical aspects of the development of multiscale molecular simulation methods for bulk complex chemical systems, especially polymers. Prerequisites for the implementation of such ML-integrated methods and open challenges that need to be met toward the development of general systematic ML-based coarse graining schemes for polymers are discussed.

Slicing and Dicing: Optimal Coarse-Grained Representation to Preserve Molecular Kinetics.

The aim of molecular coarse-graining approaches is to recover relevant physical properties of the molecular system via a lower-resolution model that can be more efficiently simulated. Ideally, the lower resolution still accounts for the degrees of freedom necessary to recover the correct physical behavior. The selection of these degrees of freedom has often relied on the scientist's chemical and physical intuition. In this article, we make the argument that in soft matter contexts desirable coarse-grained models accurately reproduce the long-time dynamics of a system by correctly capturing the rare-event transitions. We propose a bottom-up coarse-graining scheme that correctly preserves the relevant slow degrees of freedom, and we test this idea for three systems of increasing complexity. We show that in contrast to this method existing coarse-graining schemes such as those from information theory or structure-based approaches are not able to recapitulate the slow time scales of the system.

A data-driven optimization method for coarse-graining gene regulatory networks

One major challenge in systems biology is to understand how various genes in a gene regulatory network (GRN) collectively perform their functions and control network dynamics. This task becomes extremely hard to tackle in the case of large networks with hundreds of genes and edges, many of which have redundant regulatory roles and functions. The existing methods for model reduction usually require the detailed mathematical description of dynamical systems and their corresponding kinetic parameters, which are often not available. Here, we present a data-driven method for coarse-graining large GRNs, named SacoGraci, using ensemble-based mathematical modeling, dimensionality reduction, and gene circuit optimization by Markov Chain Monte Carlo methods. SacoGraci requires network topology as the only input and is robust against errors in GRNs. We benchmark and demonstrate its usage with synthetic, literature-based, and bioinformatics-derived GRNs. We hope SacoGraci will enhance our ability to model the gene regulation of complex biological systems.

Enhancing pressure consistency and transferability of structure-based coarse-graining.

Coarse-graining, which models molecules with coarse-grained (CG) beads, allows molecular dynamics simulations to be applied to systems with large length and time scales while preserving the essential molecular structure. However, CG models generally have insufficient representability and transferability. A commonly used method to resolve this problem is multi-state iterative Boltzmann inversion (MS-IBI) with pressure correction, which matches both the structural properties and pressures at different thermodynamic states between CG and all-atom (AA) simulations. Nevertheless, this method is usually effective only in a narrow pressure range. In this paper, we propose a modified CG scheme to overcome this limitation. We find that the fundamental reason for this limitation is that CG beads at close distances are ellipsoids rather than isotropically compressed spheres, as described in conventional CG models. Hence, we propose a method to compensate for such differences by slightly modifying the radial distribution functions (RDFs) derived from AA simulations and using the modified RDFs as references for pressure-corrected MS-IBI. We also propose a method to determine the initial non-bonded potential using both the target RDF and pressure. Using n-dodecane as a case study, we demonstrate that the CG model developed using our scheme reproduces the RDFs and pressures over a wide range of pressure states, including three reference low-pressure states and two test high-pressure states. The proposed scheme allows for accurate CG simulations of systems in which pressure or density varies with time and/or position.

Inference of time irreversibility from incomplete information: Linear systems and its pitfalls

Data from experiments and theoretical arguments are the two pillars sustaining the job of modeling physical systems through inference. In order to solve the inference problem, the data should satisfy certain conditions that depend also upon the particular questions addressed in a research. Here we focus on the characterization of systems in terms of a distance from equilibrium, typically the entropy production (time-reversal asymmetry) or the violation of the Kubo fluctuation-dissipation relation. We show how general, counterintuitive and negative for inference, is the problem of the impossibility to estimate the distance from equilibrium using a series of scalar data which have a Gaussian statistics. This impossibility occurs also when the data are correlated in time, and that is the most interesting case because it usually stems from a multi-dimensional linear Markovian system where there are many timescales associated to different variables and, possibly, thermal baths. Observing a single variable (or a linear combination of variables) results in a one-dimensional process which is always indistinguishable from an equilibrium one (unless a perturbation-response experiment is available). In a setting where only data analysis (and not new experiments) is allowed, we propose as a way out the combined use of different series of data acquired with different parameters. This strategy works when there is a sufficient knowledge of the connection between experimental parameters and model parameters. We also briefly discuss how such results emerge, similarly, in the context of Markov chains within certain coarse-graining schemes. Our conclusion is that the distance from equilibrium is related to quite a fine knowledge of the full phase space, and therefore typically hard to approximate in real experiments.

Computer-aided comprehensive explorations of RNA structural polymorphism through complementary simulation methods.

While RNA folding was originally seen as a simple problem to solve, it has been shown that the promiscuous interactions of the nucleobases result in structural polymorphism, with several competing structures generally observed for non-coding RNA. This inherent complexity limits our understanding of these molecules from experiments alone, and computational methods are commonly used to study RNA. Here, we discuss three advanced sampling schemes, namely Hamiltonian-replica exchange molecular dynamics (MD), ratchet-and-pawl MD and discrete path sampling, as well as the HiRE-RNA coarse-graining scheme, and highlight how these approaches are complementary with reference to recent case studies. While all computational methods have their shortcomings, the plurality of simulation methods leads to a better understanding of experimental findings and can inform and guide experimental work on RNA polymorphism.

Coarse-graining master equation for periodically driven systems

We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similar as for undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but invididually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at highest computational cost.

Visualization and Quantification of Geometric Diversity in Metal-Organic Frameworks.

With ever-growing numbers of metal–organic framework (MOF) materials being reported, new computational approaches are required for a quantitative understanding of structure–property correlations in MOFs. Here, we show how structural coarse-graining and embedding (“unsupervised learning”) schemes can together give new insights into the geometric diversity of MOF structures. Based on a curated data set of 1262 reported experimental structures, we automatically generate coarse-grained and rescaled representations which we couple to a kernel-based similarity metric and to widely used embedding schemes. This approach allows us to visualize the breadth of geometric diversity within individual topologies and to quantify the distributions of local and global similarities across the structural space of MOFs. The methodology is implemented in an openly available Python package and is expected to be useful in future high-throughput studies.

Coarse-graining strategies for predicting properties of closely related polymer architectures: A case study of PEEK and PEKK

The ability of atomistic and coarse-grained models to discern between two polymers of very similar architecture is examined. To this end, polyether ether ketone (PEEK) and polyether ketone ketone (PEKK) are chosen. The difference in glass transition temperature and the similarity in compressive responses of the two polymers are captured by all-atom models. A coarse-graining scheme, with 6 beads per monomer and 3 types of beads, leads to a good approximation of the structure and packing of chains of PEEK and PEKK. The CG model reproduces differences in weakly rate-dependent properties such as {T}_{mathrm{g}}. Comparison between strongly rate-dependent uniaxial stress–strain responses of these two polymers requires a knowledge of the scaling between physical strain rate in one to the effective rate in the other. The scaling can be approximately determined by comparing the variation of yield strength with strain rate, obtained from small-sized simulations.Graphic abstract

Comparative analysis of porosity coarse-graining techniques for discrete element simulations of dense particulate systems

The discrete element method (DEM) is a well-established approach to study granular materials in numerous fields of application; each granular particle is modelled individually to predict the overall behaviour. This behaviour can be then extracted by averaging, or coarse graining, the sample using a suitable method. The choice of appropriate coarse-graining method entails a compromise between accuracy and computational cost, especially in the large-scale simulations typically required by industry. A number of coarse-graining methods have been proposed in the literature, and these are reviewed and categorized in this work. Within this contribution, two novel porosity coarse-graining strategies are proposed including a voxel method where a secondary dense grid of “pixel cells” is implemented adopting a binary logic for the coarse graining and a hybrid method where both analytical formulas and pixels are utilized. The proposed methods are compared with four coarse-graining schemes that have been documented in the literature, including the particle centroid method, an analytical method, a method which solves the diffusion equation and an approach which employs averaging using kernels. The novel methods are validated for problems in both two and three dimensions through comparison with the “accurate” analytical method. It is shown that, once validated, both the proposed schemes can approximate the exact solutions quite accurately; however, there is a high computational cost associated with the voxel method. The accuracy of both methods can be adjusted allowing the user to decide between accuracy and computational time. A detailed comparison is then presented for all six schemes considering “accuracy”, “smoothness” and “computational cost”. Optimal parameters are obtained for all six methods, and recommendations for coarse-graining DEM samples are discussed.

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 length N = 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.

A Simple Model for the Calculation of Diffusion Coefficient in a Periodic Potential

Motivated by developing a simple model to calculate the diffusion coefficient in moderate friction region, a simplified model is proposed to deal with the diffusion of Brownian particles in a periodic potential. Where the internal noise is a Gaussian white noise, and the basic cell of the periodic potential is composed of a parabolic potential linked with a harmonic potential. When the particles cross the joint point of the potential, a time coarse-graining scheme is used to obtain a simple analytical expression of the probability distribution. The particles drift and diffuse from the first barrier to the second barrier, the passing probability over the second barrier corresponding to the escape rate becomes decrease serves as the long-jump probability. The theoretical result is confirmed by numerical simulation results. The approach can be extended to color noise case.

A Framework of Nonequilibrium Statistical Mechanics. II. Coarse-Graining

Abstract For a given thermodynamic system, and a given choice of coarse-grained state variables, the knowledge of a force-flux constitutive law is the basis for any nonequilibrium modeling. In the first paper of this series we established how, by a generalization of the classical fluctuation-dissipation theorem (FDT), the structure of a constitutive law is directly related to the distribution of the fluctuations of the state variables. When these fluctuations can be expressed in terms of diffusion processes, one may use Green–Kubo-type coarse-graining schemes to find the constitutive laws. In this paper we propose a coarse-graining method that is valid when the fluctuations are described by means of general Markov processes, which include diffusions as a special case. We prove the success of the method by numerically computing the constitutive law for a simple chemical reaction A ⇄ B A\rightleftarrows B . Furthermore, we show that, for such a system, one cannot find a consistent constitutive law by any Green–Kubo-like scheme.