Functional thermo-dynamics: A generalization of dynamic density functional theory to non-isothermal situations

Anero, Jesús; Español, Pep; Tarazona, P.

doi:10.1063/1.4811655

Cited by 28 publications

(46 citation statements)

References 43 publications

(58 reference statements)

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“…Moreover, as in the case of time-independent projection operators, we replace G(s, t) by exp(iL(t−s)). Finally, we substitute s → t − s and switch integration boundaries 16. Since we assume that the velocities vary quickly compared to the positions, we useρ(t) for the expectation value of the positions andρ(s) for the expectation value of the velocities.…”

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confidence: 99%

Projection operators in statistical mechanics: a pedagogical approach

Vrugt

Wittkowski

2020

Eur. J. Phys.

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The Mori-Zwanzig projection operator formalism is one of the central tools of nonequilibrium statistical mechanics, allowing to derive macroscopic equations of motion from the microscopic dynamics through a systematic coarse-graining procedure. It is important as a method in physical research and gives many insights into the general structure of nonequilibrium transport equations and the general procedure of microscopic derivations. Therefore, it is a valuable ingredient of basic and advanced courses in statistical mechanics. However, accessible introductions to this methodin particular in its more advanced forms -are extremely rare. In this article, we give a simple and systematic introduction to the Mori-Zwanzig formalism, which allows students to understand the methodology in the form it is used in current research. This includes both basic and modern versions of the theory. Moreover, we relate the formalism to more general aspects of statistical mechanics and quantum mechanics. Thereby, we explain how this method can be incorporated into a lecture course on statistical mechanics as a way to give a general introduction to the study of nonequilibrium systems. Applications, in particular to spin relaxation and dynamical density functional theory, are also discussed. * Corresponding author: raphael.wittkowski@uni-muenster.de 1 Active particles are particles that convert energy into directed motion. A good example are swimming microorganisms [1]. 2 This is not possible for active systems [7].

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confidence: 99%

Projection operators in statistical mechanics: a pedagogical approach

Vrugt

Wittkowski

2020

Eur. J. Phys.

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“…Entropy has been considered in the context of DDFT in recent work by Anero et al [66] and Schmidt [67]. In our work, described in the present paper, a generalized Helmholtz freeenergy functional is chosen as an appropriate thermodynamic functional, the DDFT equations in the work of Anero et al are based on an entropy functional and do not include the entropy density as a thermodynamic variable [66].…”

Section: Discussionmentioning

confidence: 99%

“…In our work, described in the present paper, a generalized Helmholtz freeenergy functional is chosen as an appropriate thermodynamic functional, the DDFT equations in the work of Anero et al are based on an entropy functional and do not include the entropy density as a thermodynamic variable [66]. As a further difference, the balance equation for the entropy density and the dissipation functional are not used in the approach of Anero et al Schmidt, on the other hand, proposed DDFT equations for a one-particle density and an internal energy density on the basis of a generalized grand-canonical potential functional that depends functionally on the one-particle density and the entropy density and is minimized by both densities in thermodynamic equilibrium [67].…”

Section: Discussionmentioning

confidence: 99%

Microscopic approach to entropy production

Wittkowski¹,

Löwen²,

Brand³

2013

J. Phys. A: Math. Theor.

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It is a great challenge of nonequilibrium statistical mechanics to calculate entropy production within a microscopic theory. In the framework of linear irreversible thermodynamics, we combine the Mori-Zwanzig-Forster projection operator technique with the first and second law of thermodynamics to obtain microscopic expressions for the entropy production as well as for the transport equations of the entropy density and its time correlation function. We further present a microscopic derivation of a dissipation functional from which the dissipative dynamics of an extended dynamical density functional theory can be obtained in a formally elegant way.

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“…This is facilitated, if one can assume -as we do for this derivation -that the system is initially prepared in the state ρ(0) =ρ(0) [3]. Following Grabert [7] and Anero et al [16], we use the microcanonical form 3ρ…”

Section: B Projection Operator and Correlatormentioning

confidence: 99%

Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

Vrugt

Wittkowski

2019

Phys. Rev. E

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The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in a lot of scenarios like, for example, driven soft matter or nuclear magnetic resonance. In this article, we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.The currently used form of the Mori-Zwanzig formalism faces the problem that it cannot be directly applied to systems with time-dependent Hamiltonians. Those, however, are relevant for a lot of scenarios including soft matter systems subject to time-dependent external driving forces [30,31] or nuclear magnetic resonance (NMR) measurements with rapidly varying electromagnetic pulses [21].If arising, time-dependent Hamiltonians can sometimes be treated as additional external perturbations [3]. This requires, however, that the perturbation is sufficiently small and couples to the macroscopic variables only. Generalizations of the projection operator method towards non-Hamiltonian dynamical systems have been developed by Chorin, Hald, and Kupferman [27,32]. Xing and Kim use mappings between dissipative and Hamiltonian systems [33] to apply projection operators in the non-Hamiltonian case [34]. The methods from Refs. [32][33][34], however, are not applicable to quantum-mechanical systems. Moreover, approximation methods commonly applied in the context of the Mori-Zwanzig formalism, such as the linearization around thermodynamic equilibrium [3], rely on the existence of a Hamiltonian.The Mori-Zwanzig formalism exists in a variety of forms. The original theory developed by Mori [1] can be applied to systems with time-independent Hamiltonians that are close to thermal equilibrium. A generalization towards systems far from equilibrium using timedependent projection operators has been presented by Robertson [35], Kawasaki and Gunton [36], and Grabert [3,7]. Recently, Bouchard has derived an extension of the Mori theory towards systems with time-dependent Hamiltonians [21] that can be applied close to equilibrium. What is still missing, however...

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Functional thermo-dynamics: A generalization of dynamic density functional theory to non-isothermal situations

Cited by 28 publications

References 43 publications

Projection operators in statistical mechanics: a pedagogical approach

Projection operators in statistical mechanics: a pedagogical approach

Microscopic approach to entropy production

Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

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