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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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2
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Dhamankar S, Webb MA. Chemically specific coarse‐graining of polymers: Methods and prospects. JOURNAL OF POLYMER SCIENCE 2021. [DOI: 10.1002/pol.20210555] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Satyen Dhamankar
- Department of Chemical and Biological Engineering Princeton University Princeton New Jersey USA
| | - Michael A. Webb
- Department of Chemical and Biological Engineering Princeton University Princeton New Jersey USA
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Izvekov S. Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium. Phys Rev E 2021; 104:024121. [PMID: 34525637 DOI: 10.1103/physreve.104.024121] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 07/20/2021] [Indexed: 11/07/2022]
Abstract
We present a generalized Langevin equation (GLE) of motion that governs exactly the time evolution of phase-space observables in finite open systems described by classical Hamiltonians with explicitly time-dependent potentials. This formalism is based on the Mori-Zwanzig projection operator (PO) method with a time-independent Zwanzig PO within a Heisenberg (Lagrangian) picture and reduced description of Hamiltonian systems in terms of canonical relevant and irrelevant coordinates. We demonstrate that, similarly to closed systems, GLE dynamics in Hamiltonian systems in the presence of time-dependent potentials is determined by conservative, dissipative memory, and projected force fields, and that the memory functions relate to the projected force, which is a two-time process, in a way that is reminiscent of the equilibrium second fluctuation-dissipation relation. We further show that, in the most general case, the memory kernel depends on the relevant momentum gradients of the (Boltzmann) entropy of the irrelevant subsystem. Using two Zwanzig operators which are, respectively, functionals of the canonical and generalized canonical probability densities, we then derive what we call canonical and generalized canonical GLEs. Further, we can formulate the particle-based, coarse-grained (CG) GLE dynamics by transitioning to Jacobi coordinates which corresponds to a particle set partitioning of the Hamiltonian system. The obtained canonical CG GLE of motion for the relevant momenta is a generalization of the CG equation of motion known for closed systems. Also, using a Markovian approximation of the canonical CG GLE, we can extend the dissipative particle dynamics equation to open systems. A distinctive feature of our extension is a use of explicitly time-dependent frictions, which reflect the changes in the dissipation rate caused by time-dependent coupling to an external bath. Our GLE formalism and workflow constitute a general and viable framework that can be readily used as a starting point to rigorously formulate microscopically informed CG treatments for a variety of phenomena in externally forced systems far from equilibrium.
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Affiliation(s)
- Sergei Izvekov
- Weapons and Materials Research Directorate, U.S. Army DEVCOM Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005, USA
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Izvekov S. Microscopic derivation of coarse-grained, energy-conserving generalized Langevin dynamics. J Chem Phys 2019; 151:104109. [PMID: 31521077 DOI: 10.1063/1.5096655] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
Properly simulating nonequilibrium phenomena such as thermal transport and shock wave propagation in complex condensed matter systems require the conservation of system's internal energy. This precludes the application of the coarse-grained (CG) generalized Langevin equation (GLE) dynamics due to the presence of dissipative interactions. Attempts to address this issue have been pursued both phenomenologically and from entropy-based first principles for dissipative particle dynamics (DPD, a Markovian variant of the CG GLE dynamics) by introducing an energy conserving extension of DPD (DPD-E). We present here a rigorous microscopic derivation of two energy conserving variants of the CG GLE dynamics by extending the CG equations of motion to include the GLE for certain internal energy observables of the microscopic system. We consider two choices of such observables: the total internal energy and a set of internal energies of the CG particles. The derivation is performed using the Mori-Zwanzig projection operator method in the Heisenberg picture for time evolution of thermodynamic expectations and the recently introduced interpretation of the Zwanzig projection operator [S. Izvekov, J. Chem. Phys. 146(12), 124109 (2017)] which allows an exact calculation of the memory and projected terms. We begin with equilibrium conditions and show that the GLE dynamics for the internal energy observables is purely dissipative. Our extension of the GLE dynamics to quasiequilibrium conditions (necessary to observe heat transport) is based on the generalized canonical ensemble approach and transport equation using the nonequilibrium statistical operator (NSO) method. We derive closed microscopic expressions for conductive heat transfer coefficients in the limit of neglecting dissipation in heat transfer and in the lowest order of deviation from equilibrium. After employing the Markov approximation, we compare the equations of motion to the published DPD-E equations. Our equations contain additional energy transfer terms not reported in the previous works. Additionally, we show that, despite neglecting dissipative processes in heat transport, the heat transfer coefficients and random force are related in a way reminiscent of the fluctuation-dissipation relation. The formalism presented here is sufficiently general for the rigorous formulation of the GLE dynamics for arbitrary microscopic phase space observables as well as sampling different microscopic ensembles in CG simulations.
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Affiliation(s)
- Sergei Izvekov
- Weapons and Materials Research Directorate, U.S. Army CCDC Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005, USA
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Recent Progress towards Chemically-Specific Coarse-Grained Simulation Models with Consistent Dynamical Properties. COMPUTATION 2019. [DOI: 10.3390/computation7030042] [Citation(s) in RCA: 34] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/26/2023]
Abstract
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.
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Yoshimoto Y, Li Z, Kinefuchi I, Karniadakis GE. Construction of non-Markovian coarse-grained models employing the Mori–Zwanzig formalism and iterative Boltzmann inversion. J Chem Phys 2017; 147:244110. [DOI: 10.1063/1.5009041] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yuta Yoshimoto
- Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - Zhen Li
- Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA
| | - Ikuya Kinefuchi
- Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - George Em Karniadakis
- Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA
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Haralson Z, Goree J. Overestimation of Viscosity by the Green-Kubo Method in a Dusty Plasma Experiment. PHYSICAL REVIEW LETTERS 2017; 118:195001. [PMID: 28548538 DOI: 10.1103/physrevlett.118.195001] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2016] [Indexed: 06/07/2023]
Abstract
The Green-Kubo (GK) method is widely used in simulations of strongly coupled plasmas to obtain the viscosity coefficient. However, the method's applicability, which is often taken for granted, has not been tested experimentally. We report an experimental test using a two-dimensional strongly coupled dusty plasma. We find that the GK viscosity is ≈60% larger than the result of a benchmark hydrodynamic method, obtained in the same experiment with the same conditions except for the presence of shear.
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Affiliation(s)
- Zach Haralson
- Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA
| | - J Goree
- Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA
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Izvekov S. Microscopic derivation of particle-based coarse-grained dynamics: Exact expression for memory function. J Chem Phys 2017; 146:124109. [DOI: 10.1063/1.4978572] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Izvekov S. Mori-Zwanzig theory for dissipative forces in coarse-grained dynamics in the Markov limit. Phys Rev E 2017; 95:013303. [PMID: 28208451 DOI: 10.1103/physreve.95.013303] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2016] [Indexed: 04/29/2023]
Abstract
We derive alternative Markov approximations for the projected (stochastic) force and memory function in the coarse-grained (CG) generalized Langevin equation, which describes the time evolution of the center-of-mass coordinates of clusters of particles in the microscopic ensemble. This is done with the aid of the Mori-Zwanzig projection operator method based on the recently introduced projection operator [S. Izvekov, J. Chem. Phys. 138, 134106 (2013)10.1063/1.4795091]. The derivation exploits the "generalized additive fluctuating force" representation to which the projected force reduces in the adopted projection operator formalism. For the projected force, we present a first-order time expansion which correctly extends the static fluctuating force ansatz with the terms necessary to maintain the required orthogonality of the projected dynamics in the Markov limit to the space of CG phase variables. The approximant of the memory function correctly accounts for the momentum dependence in the lowest (second) order and indicates that such a dependence may be important in the CG dynamics approaching the Markov limit. In the case of CG dynamics with a weak dependence of the memory effects on the particle momenta, the expression for the memory function presented in this work is applicable to non-Markov systems. The approximations are formulated in a propagator-free form allowing their efficient evaluation from the microscopic data sampled by standard molecular dynamics simulations. A numerical application is presented for a molecular liquid (nitromethane). With our formalism we do not observe the "plateau-value problem" if the friction tensors for dissipative particle dynamics (DPD) are computed using the Green-Kubo relation. Our formalism provides a consistent bottom-up route for hierarchical parametrization of DPD models from atomistic simulations.
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Affiliation(s)
- Sergei Izvekov
- Weapons and Materials Research Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, Maryland 21005, USA
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Izvekov S, Rice BM. On the importance of shear dissipative forces in coarse-grained dynamics of molecular liquids. Phys Chem Chem Phys 2015; 17:10795-804. [DOI: 10.1039/c4cp06116k] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
In this work we demonstrate from first principles that the shear frictions describing dissipative forces in the direction normal to the vector connecting the coarse-grained (CG) particles in dissipative particle dynamics (DPD) could be dominant for certain real molecular liquids at high-resolution coarse-graining.
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Affiliation(s)
- Sergei Izvekov
- Weapons and Materials Research Directorate
- U.S. Army Research Laboratory
- Aberdeen Proving Ground
- USA
| | - Betsy M. Rice
- Weapons and Materials Research Directorate
- U.S. Army Research Laboratory
- Aberdeen Proving Ground
- USA
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Izvekov S, Rice BM. Multi-scale coarse-graining of non-conservative interactions in molecular liquids. J Chem Phys 2014; 140:104104. [DOI: 10.1063/1.4866142] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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12
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Izvekov S. Microscopic derivation of particle-based coarse-grained dynamics. J Chem Phys 2013; 138:134106. [DOI: 10.1063/1.4795091] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
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Minami Y, Hidaka Y. Relativistic hydrodynamics from the projection operator method. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:023007. [PMID: 23496610 DOI: 10.1103/physreve.87.023007] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2012] [Indexed: 06/01/2023]
Abstract
We study relativistic hydrodynamics in the linear regime, based on Mori's projection operator method. In relativistic hydrodynamics, it is considered that an ambiguity about the fluid velocity occurs from the choice of a local rest frame: the Landau and Eckart frames. We find that the difference of the frames is not the choice of the local rest frame, but rather that of dynamic variables in the linear regime. We derive hydrodynamic equations in both frames by the projection operator method. We show that the natural derivation gives the linearized Landau equation. Also we find that, even for the Eckart frame, the slow dynamics is actually described by the dynamic variables for the Landau frame.
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Affiliation(s)
- Yuki Minami
- Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan
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Koide T, Nakano E, Kodama T. Shear viscosity coefficient and relaxation time of causal dissipative hydrodynamics in QCD. PHYSICAL REVIEW LETTERS 2009; 103:052301. [PMID: 19792491 DOI: 10.1103/physrevlett.103.052301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2009] [Revised: 06/21/2009] [Indexed: 05/28/2023]
Abstract
The shear viscosity coefficient and the corresponding relaxation time for causal dissipative hydrodynamics are calculated based on the microscopic formula proposed in T. Koide and T. Kodama [Phys. Rev. E 78, 051107 (2008)10.1103/PhysRevE.78.051107]. Here, the exact formula is transformed into a more compact form and applied to evaluate these transport coefficients in the chiral perturbation theory and perturbative QCD. It is shown that in the leading order calculation, the causal shear viscosity coefficient eta reduces to that of the ordinary Green-Kubo-Nakano formula, and the relaxation time tau(pi) is related to eta and pressure P by a simple relationship, tau(pi)=eta/P.
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Affiliation(s)
- T Koide
- FIAS, Johann Wolfgang Goethe-Universität, Ruth-Moufang Str. 1, 60438, Frankfurt am Main, Germany
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