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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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Netz RR. Derivation of the nonequilibrium generalized Langevin equation from a time-dependent many-body Hamiltonian. Phys Rev E 2024; 110:014123. [PMID: 39160956 DOI: 10.1103/physreve.110.014123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2023] [Accepted: 06/20/2024] [Indexed: 08/21/2024]
Abstract
It has become standard practice to describe systems that remain far from equilibrium even in their steady state by Langevin equations with colored noise which is chosen independently from the friction contribution. Since these Langevin equations are typically not derived from first-principle Hamiltonian dynamics, it is not clear whether they correspond to physically realizable scenarios. By exact Mori projection in phase space we derive the nonequilibrium generalized Langevin equation (GLE) for an arbitrary phase-space dependent observable A from a generic many-body Hamiltonian with a time-dependent external force h(t) acting on the same observable A. This is the same Hamiltonian from which the standard fluctuation-dissipation theorem is derived, which reflects the generality of our approach. The observable A could, for example, be the position of an atom, of a molecule or of a macroscopic object, the distance between two such entities or a more complex phase-space function such as the reaction coordinate of a chemical reaction or of the folding of a protein. The Hamiltonian could, for example, describe a fluid, a solid, a viscoelastic medium, or even a turbulent inhomogeneous environment. The GLE, which is a closed-form equation of motion for the observable A, is obtained in explicit form to all orders in h(t) and without restrictions on the type of many-body Hamiltonian or the observable A. If the dynamics of the observable A corresponds to a Gaussian process, the resultant GLE has a similar form as the equilibrium Mori GLE, and in particular the friction memory kernel is given by the two-point autocorrelation function of the sum of the complementary and the external force h(t). This is a nontrivial and useful result, as many observables that characterize nonequilibrium systems display Gaussian statistics. For non-Gaussian nonequilibrium observables correction terms appear in the GLE and in the relation between the force autocorrelation and the friction memory kernel, which are explicitly given in terms of cubic correlation functions of A. Interpreting the external force h(t) as a stochastic process, we derive nonequilibrium corrections to the fluctuation-dissipation theorem and present methods to extract all GLE parameters from experimental or simulation time-series data, thus making our nonequilibrium GLE a practical tool to study and model general nonequilibrium systems.
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Schmitz GJ, te Vrugt M, Haug-Warberg T, Ellingsen L, Needham P, Wittkowski R. Thermodynamics of an Empty Box. ENTROPY (BASEL, SWITZERLAND) 2023; 25:315. [PMID: 36832681 PMCID: PMC9955345 DOI: 10.3390/e25020315] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/13/2022] [Revised: 01/15/2023] [Accepted: 02/03/2023] [Indexed: 06/18/2023]
Abstract
A gas in a box is perhaps the most important model system studied in thermodynamics and statistical mechanics. Usually, studies focus on the gas, whereas the box merely serves as an idealized confinement. The present article focuses on the box as the central object and develops a thermodynamic theory by treating the geometric degrees of freedom of the box as the degrees of freedom of a thermodynamic system. Applying standard mathematical methods to the thermodynamics of an empty box allows equations with the same structure as those of cosmology and classical and quantum mechanics to be derived. The simple model system of an empty box is shown to have interesting connections to classical mechanics, special relativity, and quantum field theory.
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Affiliation(s)
- Georg J. Schmitz
- MICRESS Group, ACCESS e.V., Intzestr. 5, D-52072 Aachen, Germany
| | - Michael te Vrugt
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany
| | - Tore Haug-Warberg
- Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
| | - Lodin Ellingsen
- Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
| | - Paul Needham
- Department of Philosophy, University of Stockholm, SE-106 91 Stockholm, Sweden
| | - Raphael Wittkowski
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany
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Te Vrugt M, Wittkowski R. Perspective: New directions in dynamical density functional theory. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 35:041501. [PMID: 35917827 DOI: 10.1088/1361-648x/ac8633] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2022] [Accepted: 08/01/2022] [Indexed: 06/15/2023]
Abstract
Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.
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Affiliation(s)
- Michael Te Vrugt
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
| | - Raphael Wittkowski
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
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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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Speck T. Modeling of biomolecular machines in non-equilibrium steady states. J Chem Phys 2021; 155:230901. [PMID: 34937348 DOI: 10.1063/5.0070922] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023] Open
Abstract
Numerical computations have become a pillar of all modern quantitative sciences. Any computation involves modeling-even if often this step is not made explicit-and any model has to neglect details while still being physically accurate. Equilibrium statistical mechanics guides both the development of models and numerical methods for dynamics obeying detailed balance. For systems driven away from thermal equilibrium, such a universal theoretical framework is missing. For a restricted class of driven systems governed by Markov dynamics and local detailed balance, stochastic thermodynamics has evolved to fill this gap and to provide fundamental constraints and guiding principles. The next step is to advance stochastic thermodynamics from simple model systems to complex systems with tens of thousands or even millions of degrees of freedom. Biomolecules operating in the presence of chemical gradients and mechanical forces are a prime example for this challenge. In this Perspective, we give an introduction to isothermal stochastic thermodynamics geared toward the systematic multiscale modeling of the conformational dynamics of biomolecular and synthetic machines, and we outline some of the open challenges.
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Affiliation(s)
- Thomas Speck
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
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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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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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Te Vrugt M. The five problems of irreversibility. STUDIES IN HISTORY AND PHILOSOPHY OF SCIENCE 2021; 87:136-146. [PMID: 34111817 DOI: 10.1016/j.shpsa.2021.04.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2020] [Revised: 04/12/2021] [Accepted: 04/13/2021] [Indexed: 06/12/2023]
Abstract
Thermodynamics has a clear arrow of time, characterized by the irreversible approach to equilibrium. This stands in contrast to the laws of microscopic theories, which are invariant under time-reversal. Foundational discussions of this "problem of irreversibility" often focus on historical considerations, and do therefore not take results of modern physical research on this topic into account. In this article, I will close this gap by studying the implications of dynamical density functional theory (DDFT), a central method of modern nonequilibrium statistical mechanics not previously considered in philosophy of physics, for this debate. For this purpose, the philosophical discussion of irreversibility is structured into five problems, concerned with the source of irreversibility in thermodynamics, the definition of equilibrium and entropy, the justification of coarse-graining, the approach to equilibrium and the arrow of time. For each of these problems, it is shown that DDFT provides novel insights that are of importance for both physicists and philosophers of physics.
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Affiliation(s)
- Michael Te Vrugt
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany; Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany; Philosophisches Seminar, Westfälische Wilhelms-Universität Münster, D-48143 Münster, Germany.
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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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Melnyk SS, Usatenko OV, Yampol'skii VA. Memory-dependent noise-induced resonance and diffusion in non-Markovian systems. Phys Rev E 2021; 103:032139. [PMID: 33862761 DOI: 10.1103/physreve.103.032139] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2021] [Accepted: 03/02/2021] [Indexed: 11/07/2022]
Abstract
We study random processes with nonlocal memory and obtain solutions of the Mori-Zwanzig equation describing non-Markovian systems. We analyze the system dynamics depending on the amplitudes ν and μ_{0} of the local and nonlocal memory and pay attention to the line in the (ν, μ_{0}) plane separating the regions with asymptotically stationary and nonstationary behavior. We obtain general equations for such boundaries and consider them for three examples of nonlocal memory functions. We show that there exist two types of boundaries with fundamentally different system dynamics. On the boundaries of the first type, diffusion with memory takes place, whereas on borderlines of the second type the phenomenon of noise-induced resonance can be observed. A distinctive feature of noise-induced resonance in the systems under consideration is that it occurs in the absence of an external regular periodic force. It takes place due to the presence of frequencies in the noise spectrum, which are close to the self-frequency of the system. We analyze also the variance of the process and compare its behavior for regions of asymptotic stationarity and nonstationarity, as well as for diffusive and noise-induced-resonance borderlines between them.
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Affiliation(s)
- S S Melnyk
- O. Ya. Usikov Institute for Radiophysics and Electronics NASU, 61085 Kharkiv, Ukraine
| | - O V Usatenko
- O. Ya. Usikov Institute for Radiophysics and Electronics NASU, 61085 Kharkiv, Ukraine
| | - V A Yampol'skii
- O. Ya. Usikov Institute for Radiophysics and Electronics NASU, 61085 Kharkov, Ukraine and V. N. Karazin Kharkov National University, 61077 Kharkov, Ukraine
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Te Vrugt M, Bickmann J, Wittkowski R. Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory. Nat Commun 2020; 11:5576. [PMID: 33149128 PMCID: PMC7643184 DOI: 10.1038/s41467-020-19024-0] [Citation(s) in RCA: 44] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2020] [Accepted: 09/25/2020] [Indexed: 12/13/2022] Open
Abstract
For preventing the spread of epidemics such as the coronavirus disease COVID-19, social distancing and the isolation of infected persons are crucial. However, existing reaction-diffusion equations for epidemic spreading are incapable of describing these effects. In this work, we present an extended model for disease spread based on combining a susceptible-infected-recovered model with a dynamical density functional theory where social distancing and isolation of infected persons are explicitly taken into account. We show that the model exhibits interesting transient phase separation associated with a reduction of the number of infections, and allows for new insights into the control of pandemics.
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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
| | - Jens Bickmann
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149, Münster, Germany
| | - Raphael Wittkowski
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, D-48149, Münster, Germany.
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Meyer H, Wolf S, Stock G, Schilling T. A Numerical Procedure to Evaluate Memory Effects in Non‐Equilibrium Coarse‐Grained Models. ADVANCED THEORY AND SIMULATIONS 2020. [DOI: 10.1002/adts.202000197] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/28/2022]
Affiliation(s)
- Hugues Meyer
- Research Unit in Engineering Science Université du Luxembourg 6 Avenue de la Fonte Esch‐sur‐Alzette L‐4364 Luxembourg
| | - Steffen Wolf
- Physikalisches Institut Albert‐Ludwigs Universität Hermann‐Herder‐Straße 3a Freiburg im Breisgau 79104 Germany
| | - Gerhard Stock
- Physikalisches Institut Albert‐Ludwigs Universität Hermann‐Herder‐Straße 3a Freiburg im Breisgau 79104 Germany
| | - Tanja Schilling
- Physikalisches Institut Albert‐Ludwigs Universität Hermann‐Herder‐Straße 3a Freiburg im Breisgau 79104 Germany
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Meyer H, Pelagejcev P, Schilling T. Non-Markovian out-of-equilibrium dynamics: A general numerical procedure to construct time-dependent memory kernels for coarse-grained observables. ACTA ACUST UNITED AC 2020. [DOI: 10.1209/0295-5075/128/40001] [Citation(s) in RCA: 22] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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