1
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Te Vrugt M, Topp L, Wittkowski R, Heuer A. Microscopic derivation of the thin film equation using the Mori-Zwanzig formalism. J Chem Phys 2024; 161:094904. [PMID: 39225531 DOI: 10.1063/5.0217535] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2024] [Accepted: 07/19/2024] [Indexed: 09/04/2024] Open
Abstract
The hydrodynamics of thin films is typically described using macroscopic models whose connection to the microscopic particle dynamics is a subject of ongoing research. Existing methods based on density functional theory provide a good description of static thin films but are not sufficient for understanding nonequilibrium dynamics. In this work, we present a microscopic derivation of the thin film equation using the Mori-Zwanzig projection operator formalism. This method allows to directly obtain the correct gradient dynamics structure along with microscopic expressions for mobility and free energy. Our results are verified against molecular dynamics simulations for both simple fluids and polymers.
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Affiliation(s)
- Michael Te Vrugt
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
| | - Leon Topp
- Institute of Physical Chemistry, Universität Münster, 48149 Münster, Germany
| | - Raphael Wittkowski
- Institute of Theoretical Physics, Center for Soft Nanoscience, Universität Münster, 48149 Münster, Germany
| | - Andreas Heuer
- Institute of Physical Chemistry, Universität Münster, 48149 Münster, Germany
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2
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Xie P, Car R, E W. Ab initio generalized Langevin equation. Proc Natl Acad Sci U S A 2024; 121:e2308668121. [PMID: 38551836 PMCID: PMC10998567 DOI: 10.1073/pnas.2308668121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2023] [Accepted: 02/22/2024] [Indexed: 04/08/2024] Open
Abstract
We introduce a machine learning-based approach called ab initio generalized Langevin equation (AIGLE) to model the dynamics of slow collective variables (CVs) in materials and molecules. In this scheme, the parameters are learned from atomistic simulations based on ab initio quantum mechanical models. Force field, memory kernel, and noise generator are constructed in the context of the Mori-Zwanzig formalism, under the constraint of the fluctuation-dissipation theorem. Combined with deep potential molecular dynamics and electronic density functional theory, this approach opens the way to multiscale modeling in a variety of situations. Here, we demonstrate this capability with a study of two mesoscale processes in crystalline lead titanate, namely the field-driven dynamics of a planar ferroelectric domain wall, and the dynamics of an extensive lattice of coarse-grained electric dipoles. In the first case, AIGLE extends the reach of ab initio simulations to a regime of noise-driven motions not accessible to molecular dynamics. In the second case, AIGLE deals with an extensive set of CVs by adopting a local approximation for the memory kernel and retaining only short-range noise correlations. The scheme is computationally more efficient than molecular dynamics by several orders of magnitude and mimics the microscopic dynamics at low frequencies where it reproduces accurately the dominant far-infrared absorption frequency.
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Affiliation(s)
- Pinchen Xie
- Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ08544
| | - Roberto Car
- Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ08544
- Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, NJ08544
- Department of Physics, Princeton University, Princeton, NJ08544
| | - Weinan E
- AI for Science Institute, Beijing100080, China
- Center for Machine Learning Research and School of Mathematical Sciences, Peking University, Beijing100084, China
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3
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Zhang XZ, Shi R, Lu ZY, Qian HJ. Chemically Specific Systematic Coarse-Grained Polymer Model with Both Consistently Structural and Dynamical Properties. JACS AU 2024; 4:1018-1030. [PMID: 38559727 PMCID: PMC10976574 DOI: 10.1021/jacsau.3c00756] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/30/2023] [Revised: 02/23/2024] [Accepted: 02/28/2024] [Indexed: 04/04/2024]
Abstract
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.
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Affiliation(s)
| | | | - Zhong-Yuan Lu
- State Key Laboratory of Supramolecular
Structure and Materials, Institute of Theoretical Chemistry, College
of Chemistry, Jilin University, Changchun 130021, China
| | - Hu-Jun Qian
- State Key Laboratory of Supramolecular
Structure and Materials, Institute of Theoretical Chemistry, College
of Chemistry, Jilin University, Changchun 130021, China
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4
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Girardier DD, Vroylandt H, Bonella S, Pietrucci F. Inferring free-energy barriers and kinetic rates from molecular dynamics via underdamped Langevin models. J Chem Phys 2023; 159:164111. [PMID: 37882336 DOI: 10.1063/5.0169050] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2023] [Accepted: 09/27/2023] [Indexed: 10/27/2023] Open
Abstract
Rare events include many of the most interesting transformation processes in condensed matter, from phase transitions to biomolecular conformational changes to chemical reactions. Access to the corresponding mechanisms, free-energy landscapes and kinetic rates can in principle be obtained by different techniques after projecting the high-dimensional atomic dynamics on one (or a few) collective variable. Even though it is well-known that the projected dynamics approximately follows - in a statistical sense - the generalized, underdamped or overdamped Langevin equations (depending on the time resolution), to date it is nontrivial to parameterize such equations starting from a limited, practically accessible amount of non-ergodic trajectories. In this work we focus on Markovian, underdamped Langevin equations, that arise naturally when considering, e.g., numerous water-solution processes at sub-picosecond resolution. After contrasting the advantages and pitfalls of different numerical approaches, we present an efficient parametrization strategy based on a limited set of molecular dynamics data, including equilibrium trajectories confined to minima and few hundreds transition path sampling-like trajectories. Employing velocity autocorrelation or memory kernel information for learning the friction and likelihood maximization for learning the free-energy landscape, we demonstrate the possibility to reconstruct accurate barriers and rates both for a benchmark system and for the interaction of carbon nanoparticles in water.
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Affiliation(s)
- David Daniel Girardier
- Sorbonne Université, Musée National d'Histoire Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Materiaux et de Cosmochimie, IMPMC, F-75005 Paris, France
| | - Hadrien Vroylandt
- Sorbonne Université, Institut des Sciences du Calcul et des données, ISCD, F-75005 Paris, France
| | - Sara Bonella
- Centre Européen de Calcul Atomique et Moléculaire (CECAM), Ecole Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
| | - Fabio Pietrucci
- Sorbonne Université, Musée National d'Histoire Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Materiaux et de Cosmochimie, IMPMC, F-75005 Paris, France
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5
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Cai W, Jäger M, Bullerjahn JT, Hugel T, Wolf S, Balzer BN. Anisotropic Friction in a Ligand-Protein Complex. NANO LETTERS 2023; 23:4111-4119. [PMID: 36948207 DOI: 10.1021/acs.nanolett.2c04632] [Citation(s) in RCA: 9] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
The effect of an externally applied directional force on molecular friction is so far poorly understood. Here, we study the force-driven dissociation of the ligand-protein complex biotin-streptavidin and identify anisotropic friction as a not yet described type of molecular friction. Using AFM-based stereographic single molecule force spectroscopy and targeted molecular dynamics simulations, we find that the rupture force and friction for biotin-streptavidin vary with the pulling angle. This observation holds true for friction extracted from Kramers' rate expression and by dissipation-corrected targeted molecular dynamics simulations based on Jarzynski's identity. We rule out ligand solvation and protein-internal friction as sources of the angle-dependent friction. Instead, we observe a heterogeneity in free energy barriers along an experimentally uncontrolled orientation parameter, which increases the rupture force variance and therefore the overall friction. We anticipate that anisotropic friction needs to be accounted for in a complete understanding of friction in biomolecular dynamics and anisotropic mechanical environments.
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Affiliation(s)
- Wanhao Cai
- Institute of Physical Chemistry, University of Freiburg, Albertstr. 21, 79104 Freiburg, Germany
| | - Miriam Jäger
- Biomolecular Dynamics, Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
| | - Jakob T Bullerjahn
- Department of Theoretical Biophysics, Max Planck Institute of Biophysics, Max-von-Laue-Str. 3, 60438 Frankfurt am Main, Germany
| | - Thorsten Hugel
- Institute of Physical Chemistry, University of Freiburg, Albertstr. 21, 79104 Freiburg, Germany
- Cluster of Excellence livMatS @ FIT - Freiburg Center for Interactive Materials and Bioinspired Technologies, University of Freiburg, Georges-Köhler-Allee 105, 79110 Freiburg, Germany
| | - Steffen Wolf
- Biomolecular Dynamics, Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
| | - Bizan N Balzer
- Institute of Physical Chemistry, University of Freiburg, Albertstr. 21, 79104 Freiburg, Germany
- Cluster of Excellence livMatS @ FIT - Freiburg Center for Interactive Materials and Bioinspired Technologies, University of Freiburg, Georges-Köhler-Allee 105, 79110 Freiburg, Germany
- Freiburg Materials Research Center (FMF), University of Freiburg, Stefan-Meier-Str. 21, 79104 Freiburg, Germany
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6
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Brünig F, Daldrop JO, Netz RR. Pair-Reaction Dynamics in Water: Competition of Memory, Potential Shape, and Inertial Effects. J Phys Chem B 2022; 126:10295-10304. [PMID: 36473702 PMCID: PMC9761671 DOI: 10.1021/acs.jpcb.2c05923] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2022] [Revised: 11/11/2022] [Indexed: 12/12/2022]
Abstract
When described by a one-dimensional reaction coordinate, pair-reaction rates in a solvent depend, in addition to the potential barrier height and the friction coefficient, on the potential shape, the effective mass, and the friction relaxation spectrum, but a rate theory that accurately accounts for all of these effects does not exist. After a review of classical reaction-rate theories, we show how to extract all parameters of the generalized Langevin equation (GLE) and, in particular, the friction memory function from molecular dynamics (MD) simulations of two prototypical pair reactions in water, the dissociation of NaCl and of two methane molecules. The memory exhibits multiple time scales and, for NaCl, pronounced oscillatory components. Simulations of the GLE by Markovian embedding techniques accurately reproduce the pair-reaction kinetics from MD simulations without any fitting parameters, which confirms the accuracy of the approximative form of the GLE and of the parameter extraction techniques. By modification of the GLE parameters, we investigate the relative importance of memory, mass, and potential shape effects. Neglect of memory slows down NaCl and methane dissociation by roughly a factor of 2; neglect of mass accelerates reactions by a similar factor, and the harmonic approximation of the potential shape gives rise to slight acceleration. This partial error cancellation explains why Kramers' theory, which neglects memory effects and treats the potential shape in harmonic approximation, describes reaction rates better than more sophisticated theories. In essence, all three effects, friction memory, inertia, and the potential shape nonharmonicity, are important to quantitatively describe pair-reaction kinetics in water.
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Affiliation(s)
- Florian
N. Brünig
- Fachbereich Physik, Freie Universität
Berlin, Arnimallee 14, 14195Berlin, Germany
| | - Jan O. Daldrop
- Fachbereich Physik, Freie Universität
Berlin, Arnimallee 14, 14195Berlin, Germany
| | - Roland R. Netz
- Fachbereich Physik, Freie Universität
Berlin, Arnimallee 14, 14195Berlin, Germany
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7
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Widder C, Koch F, Schilling T. Generalized Langevin dynamics simulation with non-stationary memory kernels: How to make noise. J Chem Phys 2022; 157:194107. [DOI: 10.1063/5.0127557] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
We present a numerical method to produce stochastic dynamics according to the generalized Langevin equation with a non-stationary memory kernel. This type of dynamics occurs when a microscopic system with an explicitly time-dependent Liouvillian is coarse-grained by means of a projection operator formalism. We show how to replace the deterministic fluctuating force in the generalized Langevin equation by a stochastic process, such that the distributions of the observables are reproduced up to moments of a given order. Thus, in combination with a method to extract the memory kernel from simulation data of the underlying microscopic model, the method introduced here allows us to construct and simulate a coarse-grained model for a driven process.
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Affiliation(s)
- Christoph Widder
- Institut für Physik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg im Breisgau, Germany
| | - Fabian Koch
- Institut für Physik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg im Breisgau, Germany
| | - Tanja Schilling
- Institut für Physik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg im Breisgau, Germany
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8
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Schmid F. Understanding and Modeling Polymers: The Challenge of Multiple Scales. ACS POLYMERS AU 2022. [DOI: 10.1021/acspolymersau.2c00049] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Affiliation(s)
- Friederike Schmid
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128Mainz, Germany
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9
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Brünig FN, Netz RR, Kappler J. Barrier-crossing times for different non-Markovian friction in well and barrier: A numerical study. Phys Rev E 2022; 106:044133. [PMID: 36397504 DOI: 10.1103/physreve.106.044133] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Accepted: 07/21/2022] [Indexed: 06/16/2023]
Abstract
We introduce a generalized Langevin model system for different non-Markovian effects in the well and barrier regions of a potential, and use it to numerically study the barrier-crossing time. In the appropriate limits, our model interpolates between the theoretical barrier-crossing-time predictions by Grote and Hynes (GH), as well as by Pollak et al., which for a single barrier memory time can differ by several orders of magnitude. Our model furthermore allows one to test an analytic rate theory for space-inhomogeneous memory, which disagrees with our numerical results in the long well-memory regime. In this regime, we find that short barrier memory decreases the barrier-crossing time as compared to long barrier memory. This is in contrast with the short well-memory regime, where both our numerical results and the GH theory predict an acceleration of the barrier crossing time with increasing barrier memory time. Both effects, the "Markovian-barrier acceleration" and GH "non-Markovian-barrier acceleration," can be understood from a committor analysis. Our model combines finite relaxation times of orthogonal degrees of freedom with a space-inhomogeneous coupling to such degrees and represents a step towards more realistic modeling of reaction coordinates.
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Affiliation(s)
- Florian N Brünig
- Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
| | - Roland R Netz
- Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
| | - Julian Kappler
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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10
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Post M, Wolf S, Stock G. Molecular Origin of Driving-Dependent Friction in Fluids. J Chem Theory Comput 2022; 18:2816-2825. [PMID: 35442659 DOI: 10.1021/acs.jctc.2c00190] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The friction coefficient of fluids may become a function of the velocity at increased external driving. This non-Newtonian behavior is of general theoretical interest and of great practical importance, for example, for the design of lubricants. Although the effect has been observed in large-scale atomistic simulations of bulk liquids, its theoretical formulation and microscopic origin are not well understood. Here, we use dissipation-corrected targeted molecular dynamics, which pulls apart two tagged liquid molecules in the presence of surrounding molecules, and analyze this nonequilibrium process via a generalized Langevin equation. The approach is based on a second-order cumulant expansion of Jarzynski's identity, which is shown to be valid for fluids and therefore allows for an exact computation of the friction profile as well of the underlying memory kernel. We show that velocity-dependent friction in fluids results from an intricate interplay of near-order structural effects and the non-Markovian behavior of the friction memory kernel. For complex fluids such as the model lubricant C40H82, the memory kernel exhibits a stretched-exponential long-time decay, which reflects the multitude of timescales of the system.
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Affiliation(s)
- Matthias Post
- Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, Freiburg 79104, Germany
| | - Steffen Wolf
- Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, Freiburg 79104, Germany
| | - Gerhard Stock
- Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, Freiburg 79104, Germany
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11
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Nagahata Y, Hernandez R, Komatsuzaki T. Phase space geometry of isolated to condensed chemical reactions. J Chem Phys 2021; 155:210901. [PMID: 34879678 DOI: 10.1063/5.0059618] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The complexity of gas and condensed phase chemical reactions has generally been uncovered either approximately through transition state theories or exactly through (analytic or computational) integration of trajectories. These approaches can be improved by recognizing that the dynamics and associated geometric structures exist in phase space, ensuring that the propagator is symplectic as in velocity-Verlet integrators and by extending the space of dividing surfaces to optimize the rate variationally, respectively. The dividing surface can be analytically or variationally optimized in phase space, not just over configuration space, to obtain more accurate rates. Thus, a phase space perspective is of primary importance in creating a deeper understanding of the geometric structure of chemical reactions. A key contribution from dynamical systems theory is the generalization of the transition state (TS) in terms of the normally hyperbolic invariant manifold (NHIM) whose geometric phase-space structure persists under perturbation. The NHIM can be regarded as an anchor of a dividing surface in phase space and it gives rise to an exact non-recrossing TS theory rate in reactions that are dominated by a single bottleneck. Here, we review recent advances of phase space geometrical structures of particular relevance to chemical reactions in the condensed phase. We also provide conjectures on the promise of these techniques toward the design and control of chemical reactions.
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Affiliation(s)
- Yutaka Nagahata
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Tamiki Komatsuzaki
- Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0 020, Japan
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12
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Te Vrugt M, Hossenfelder S, Wittkowski R. Mori-Zwanzig Formalism for General Relativity: A New Approach to the Averaging Problem. PHYSICAL REVIEW LETTERS 2021; 127:231101. [PMID: 34936793 DOI: 10.1103/physrevlett.127.231101] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2021] [Accepted: 10/20/2021] [Indexed: 06/14/2023]
Abstract
Cosmology relies on a coarse-grained description of the universe, assumed to be valid on large length scales. However, the nonlinearity of general relativity makes coarse graining extremely difficult. We here address this problem by extending the Mori-Zwanzig projection operator formalism, a highly successful coarse-graining method from statistical mechanics, towards general relativity. Using the Buchert equations, we derive a new dynamic equation for the Hubble parameter which captures the effects of averaging through a memory function. This gives an empirical prediction for the cosmic jerk.
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Affiliation(s)
- Michael Te Vrugt
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany
| | - Sabine Hossenfelder
- Frankfurt Institute for Advanced Studies, D-60438 Frankfurt am Main, Germany
| | - Raphael Wittkowski
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany
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13
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Lickert B, Wolf S, Stock G. Data-Driven Langevin Modeling of Nonequilibrium Processes. J Phys Chem B 2021; 125:8125-8136. [PMID: 34270245 DOI: 10.1021/acs.jpcb.1c03828] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Given nonstationary data from molecular dynamics simulations, a Markovian Langevin model is constructed that aims to reproduce the time evolution of the underlying process. While at equilibrium the free energy landscape is sampled, nonequilibrium processes can be associated with a biased energy landscape, which accounts for finite sampling effects and external driving. When the data-driven Langevin equation (dLE) approach [Phys. Rev. Lett. 2015, 115, 050602] is extended to the modeling of nonequilibrium processes, an efficient way to calculate multidimensional Langevin fields is outlined. The dLE is shown to correctly account for various nonequilibrium processes, including the enforced dissociation of sodium chloride in water, the pressure-jump induced nucleation of a liquid of hard spheres, and the conformational dynamics of a helical peptide sampled from nonstationary short trajectories.
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Affiliation(s)
- Benjamin Lickert
- Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, 79104 Freiburg, Germany
| | - Steffen Wolf
- Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, 79104 Freiburg, Germany
| | - Gerhard Stock
- Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, 79104 Freiburg, Germany
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14
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Wang S, Ma Z, Pan W. Data-driven coarse-grained modeling of non-equilibrium systems. SOFT MATTER 2021; 17:6404-6412. [PMID: 34132317 DOI: 10.1039/d1sm00413a] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
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.
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Affiliation(s)
- Shu Wang
- Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA.
| | - Zhan Ma
- Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA.
| | - Wenxiao Pan
- Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA.
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15
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Klippenstein V, Tripathy M, Jung G, Schmid F, van der Vegt NFA. Introducing Memory in Coarse-Grained Molecular Simulations. J Phys Chem B 2021; 125:4931-4954. [PMID: 33982567 PMCID: PMC8154603 DOI: 10.1021/acs.jpcb.1c01120] [Citation(s) in RCA: 40] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
Abstract
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.
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Affiliation(s)
- Viktor Klippenstein
- Eduard-Zintl-Institut für Anorganische und Physikalische Chemie, Technische Universität Darmstadt, 64287 Darmstadt, Germany
| | - Madhusmita Tripathy
- Eduard-Zintl-Institut für Anorganische und Physikalische Chemie, Technische Universität Darmstadt, 64287 Darmstadt, Germany
| | - Gerhard Jung
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21 A, A-6020 Innsbruck, Austria
| | - Friederike Schmid
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany
| | - Nico F A van der Vegt
- Eduard-Zintl-Institut für Anorganische und Physikalische Chemie, Technische Universität Darmstadt, 64287 Darmstadt, Germany
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16
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Glatzel F, Schilling T. Comments on the validity of the non-stationary generalized Langevin equation as a coarse-grained evolution equation for microscopic stochastic dynamics. J Chem Phys 2021; 154:174107. [PMID: 34241070 DOI: 10.1063/5.0049693] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We recently showed that the dynamics of coarse-grained observables in systems out of thermal equilibrium are governed by the non-stationary generalized Langevin equation [H. Meyer, T. Voigtmann, and T. Schilling, J. Chem. Phys. 147, 214110 (2017); 150, 174118 (2019)]. The derivation we presented in these two articles was based on the assumption that the dynamics of the microscopic degrees of freedom were deterministic. Here, we extend the discussion to stochastic microscopic dynamics. The fact that the same form of the non-stationary generalized Langevin equation as derived for the deterministic case also holds for stochastic processes implies that methods designed to estimate the memory kernel, drift term, and fluctuating force term of this equation, as well as methods designed to propagate it numerically, can be applied to data obtained in molecular dynamics simulations that employ a stochastic thermostat or barostat.
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Affiliation(s)
- Fabian Glatzel
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
| | - Tanja Schilling
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
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Meyer H, Glatzel F, Wöhler W, Schilling T. Evaluation of memory effects at phase transitions and during relaxation processes. Phys Rev E 2021; 103:022102. [PMID: 33735956 DOI: 10.1103/physreve.103.022102] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2020] [Accepted: 01/12/2021] [Indexed: 11/07/2022]
Abstract
We propose to describe the dynamics of phase transitions in terms of a nonstationary generalized Langevin equation for the order parameter. By construction, this equation is nonlocal in time, i.e., it involves memory effects whose intensity is governed by a memory kernel. In general, it is a hard task to determine the physical origin and the extent of the memory effects based on the underlying microscopic equations of motion. Therefore we propose to relate the extent of the memory kernel to quantities that are experimentally observed such as the induction time and the duration of the phase transformation process. Using a simple kinematic model, we show that the extent of the memory kernel is positively correlated with the duration of the transition, and that it is of the same order of magnitude, while the distribution of induction times does not have an effect on the memory kernel. This observation is tested at the example of several model systems, for which we have run computer simulations: a modified Potts model, a dipole gas, an anharmonic spring in a bath, and a nucleation problem. All these cases are shown to be consistent with the simple theoretical model.
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Affiliation(s)
- Hugues Meyer
- Department of Theoretical Physics and Center for Biophysics, Saarland University, 66123 Saarbrücken, Germany
| | - Fabian Glatzel
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
| | - Wilkin Wöhler
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
| | - Tanja Schilling
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
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