1
|
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.
Collapse
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
| |
Collapse
|
2
|
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.
Collapse
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
| |
Collapse
|
3
|
Uneyama T. Application of projection operator method to coarse-grained dynamics with transient potential. Phys Rev E 2022; 105:044117. [PMID: 35590667 DOI: 10.1103/physreve.105.044117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2021] [Accepted: 03/22/2022] [Indexed: 06/15/2023]
Abstract
We show that the coarse-grained dynamics model with the time-dependent and fluctuating potential (transient potential) can be derived from the microscopic Hamiltonian dynamics. The concept of the transient potential was first introduced rather phenomenologically, and its relation to the underlying microscopic dynamics has not been clarified yet. This is in contrast to the generalized Langevin equation, the relation of which to the microscopic dynamics is well-established. In this work, we show that the dynamic equations with the transient potential can be derived for the coupled oscillator model, without any approximations. It is known that the dynamics of the coupled oscillator model can be exactly described by the generalized Langevin-type equations. This fact implies that the dynamic equations with the transient potential can be utilized as a coarse-grained dynamics model in a similar way to the generalized Langevin equation. Then we show that the dynamic equations for the transient potential can also be formally derived for the microscopic Hamiltonian dynamics, without any approximations. We use the projection operator method for the coarse-grained variables and transient potential. The dynamic equations for the coarse-grained positions and momenta are similar to those in the Hamiltonian dynamics, but the interaction potential is replaced by the transient potential. The dynamic equation for the transient potential is the generalized Langevin equation with the memory effect. Our result justifies the use of the transient potential to describe the coarse-grained dynamics. We propose several approximations to obtain the simplified dynamics model. We show that, under several approximations, the dynamic equation for the transient potential reduces to the relatively simple Markovian dynamic equation for the potential parameters. We also show that with several additional approximations, the approximate dynamics model further reduces to the Markovian Langevin-type equations with the transient potential.
Collapse
Affiliation(s)
- Takashi Uneyama
- JST-PRESTO, and Department of Materials Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
| |
Collapse
|
4
|
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.
Collapse
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
| |
Collapse
|
5
|
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.
Collapse
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.
| |
Collapse
|
6
|
Cugliandolo LF, Déjardin PM, Lozano GS, van Wijland F. Stochastic dynamics of collective modes for Brownian dipoles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032139. [PMID: 25871086 DOI: 10.1103/physreve.91.032139] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2014] [Indexed: 06/04/2023]
Abstract
The individual motion of a colloidal particle is described by an overdamped Langevin equation. When rotational degrees of freedom are relevant, these are described by a corresponding Langevin process. Our purpose is to show that the microscopic local density of colloids, in terms of a space and rotation state, also evolves according to a Langevin equation. The latter can then be used as the starting point of a variety of approaches, ranging from dynamical density functional theory to mode-coupling approximations.
Collapse
Affiliation(s)
- Leticia F Cugliandolo
- Laboratoire de Physique Théorique et Hautes Énergies, Sorbonne Universités, Université Pierre et Marie Curie-Paris 6, and UMR 7589 CNRS/P6, 4 place Jussieu, 75252 Paris cedex 05, France
| | - Pierre-Michel Déjardin
- Laboratoire de Mathématiques et de Physique, Université de Perpignan Via Domitia, 52 avenue Paul Alduy, 66860 Perpignan cedex, France
| | - Gustavo S Lozano
- Departmento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Ciudad Universitaria, Pabellon I, 1428 Buenos Aires, Argentina
| | - Frédéric van Wijland
- Laboratoire Matière et Systèmes Complexes, UMR 7057 CNRS/P7, Université Paris Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris cedex 13, France
- Department of Chemistry, University of California, Berkeley, California 94720, USA
| |
Collapse
|
7
|
|
8
|
|
9
|
Baskaran A, Baskaran A, Lowengrub J. Kinetic density functional theory of freezing. J Chem Phys 2014; 141:174506. [DOI: 10.1063/1.4900499] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023] Open
Affiliation(s)
- Arvind Baskaran
- Department of Mathematics, University of California Irvine, Irvine, California 92697-3875, USA
| | - Aparna Baskaran
- Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454, USA
| | - John Lowengrub
- Department of Mathematics, University of California Irvine, Irvine, California 92697-3875, USA
| |
Collapse
|
10
|
Das SP, Yoshimori A. Coarse-grained forms for equations describing the microscopic motion of particles in a fluid. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:043008. [PMID: 24229277 DOI: 10.1103/physreve.88.043008] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2013] [Revised: 06/12/2013] [Indexed: 06/02/2023]
Abstract
Exact equations of motion for the microscopically defined collective density ρ(x,t) and the momentum density ĝ(x,t) of a fluid have been obtained in the past starting from the corresponding Langevin equations representing the dynamics of the fluid particles. In the present work we average these exact equations of microscopic dynamics over the local equilibrium distribution to obtain stochastic partial differential equations for the coarse-grained densities with smooth spatial and temporal dependence. In particular, we consider Dean's exact balance equation for the microscopic density of a system of interacting Brownian particles to obtain the basic equation of the dynamic density functional theory with noise. Our analysis demonstrates that on thermal averaging the dependence of the exact equations on the bare interaction potential is converted to dependence on the corresponding thermodynamic direct correlation functions in the coarse-grained equations.
Collapse
Affiliation(s)
- Shankar P Das
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | | |
Collapse
|
11
|
Koide T, Kodama T. Transport coefficients of non-Newtonian fluid and causal dissipative hydrodynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:051107. [PMID: 19113095 DOI: 10.1103/physreve.78.051107] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2008] [Indexed: 05/27/2023]
Abstract
A formula to calculate the transport coefficients of the causal dissipative hydrodynamics is derived by using the projection operator method (Mori-Zwanzig formalism) [T. Koide, Phys. Rev. E 75, 060103(R) (2007)]. This is an extension of the Green-Kubo-Nakano (GKN) formula to the case of non-Newtonian fluids, which is the essential factor to preserve the relativistic causality in relativistic dissipative hydrodynamics. This formula is the generalization of the GKN formula in the sense that it can reproduce the GKN formula in a certain limit. In this work, we extend the previous work so as to apply to more general situations.
Collapse
Affiliation(s)
- T Koide
- Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postale 68528, 21945-970, Rio de Janeiro, Brazil
| | | |
Collapse
|
12
|
Abstract
Density-functional theory (DFT) and its variations provide a fruitful approach to the computational modeling of the microscopic structures and phase behavior of soft-condensed matter. The methodology takes deep root in quantum mechanics but shares a mathematical similarity with a number of classical approaches in statistical mechanics. This review discusses different strategies commonly used to formulate the free-energy functional of complex fluids for either phenomena-oriented applications or as a generic description of the thermodynamic nonideality owing to various components of intermolecular forces. We emphasize the connections among different schemes of DFT approximations, their underlying assumptions, and inherent limitations. We also address extensions of equilibrium DFT to phenomenological theories for the dynamic properties of complex fluids and for the kinetics of phase transitions. In addition, we highlight the recent literature concerning applications of DFT to diverse static and time-dependent phenomena in complex fluids.
Collapse
Affiliation(s)
- Jianzhong Wu
- Department of Chemical and Environmental Engineering, University of California, Riverside, CA 92521, USA.
| | | |
Collapse
|
13
|
Marconi UMB, Melchionna S. Phase-space approach to dynamical density functional theory. J Chem Phys 2007; 126:184109. [PMID: 17508794 DOI: 10.1063/1.2724823] [Citation(s) in RCA: 43] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The authors consider a system of interacting particles subjected to Langevin inertial dynamics and derive the governing time-dependent equation for the one-body density. They show that, after suitable truncations of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and a multiple time scale analysis, they obtain a self-consistent equation involving only the one-body density. This study extends to arbitrary dimensions previous work on a one-dimensional fluid and highlights the subtleties of kinetic theory in the derivation of dynamical density functional theory.
Collapse
|
14
|
Murata S, Yoshimori A. Nonlinear effects on solvation dynamics in simple mixtures. J Chem Phys 2006; 125:244501. [PMID: 17199349 DOI: 10.1063/1.2409713] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023] Open
Abstract
The authors applied the time dependent density functional method (TDDFM) and a linear model to solvation dynamics in simple binary solvents. Changing the solute-solvent interactions at t=0, the authors calculated the time evolution of density fields for solvent particles after the change (t>0) by the TDDFM and linear model. First, the authors changed the interaction of only one component of solvents. In this case, the TDDFM showed that the solvation time decreased monotonically with a mole fraction of the solvent strongly interacting with the solute. The monotonical decreases agreed with experimental results, while the linear model did not reproduce these results. The authors also calculated the solvation time by changing the interaction of both components. The calculation showed that the mole fraction dependence had the peak. The TDDFM presented a much higher peak than the linear model. The difference between the TDDFM and the linear model was caused by a nonlinear effect on an exchange process of solvent particles.
Collapse
Affiliation(s)
- Shuhei Murata
- Department of Physics, Kyushu University, Fukuoka 812-8581, Japan
| | | |
Collapse
|
15
|
Burghardt I, Bagchi B. On the non-adiabatic dynamics of solvation: A molecular hydrodynamic formulation. Chem Phys 2006. [DOI: 10.1016/j.chemphys.2006.07.014] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
|
16
|
Rex M, Likos CN, LÖwen H, Dzubiella J. Ultrasoft colloids in cavities of oscillating size or sharpness. Mol Phys 2006. [DOI: 10.1080/00268970500460382] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
|