Geometry of Multiscale Nonequilibrium Thermodynamics

Grmela, Miroslav

doi:10.3390/e17095938

Cited by 15 publications

(16 citation statements)

References 73 publications

(214 reference statements)

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“…This connection extends the existing connection for reversible Markov processes and gradient flows, and gives a microscopic basis to the non-quadratic version of GENERIC that we use and that has been identified before (see e.g. [6,7]).…”

Section: Resultssupporting

confidence: 54%

Fluctuation symmetry leads to GENERIC equations with non-quadratic dissipation

Kraaij

Lazarescu

Maes

et al. 2020

Stochastic Processes and their Applications

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We develop a formalism to discuss the properties of GENERIC systems in terms of corresponding Hamiltonians that appear in the characterization of large-deviation limits. We demonstrate how the GENERIC structure naturally arises from a certain symmetry in the Hamiltonian, which extends earlier work that has connected the large-deviation behaviour of reversible stochastic processes to the gradient-flow structure of their deterministic limit. Natural examples of application include particle systems with inertia.

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Section: Resultssupporting

confidence: 54%

Fluctuation symmetry leads to GENERIC equations with non-quadratic dissipation

Kraaij

Lazarescu

Maes

et al. 2020

Stochastic Processes and their Applications

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“…Another approach to nonequilibrium thermodynamics, which combines reversible and irreversible dynamics, is the General Equation for NonEquilibrium Reversible-Irreversible Coupling (GENERIC) [15,16]. The structure of this formalism can be derived using contact forms in the setting of the Gibbs-Legendre manifold [17,18]; it can be cast variationally; and it allows for a systematic multiscale approach [19] as well as a treatment of fluctuations [17,20].…”

Section: Deterministic Gradient Flow Stochastic Gradient Flowmentioning

confidence: 99%

Entropy production and the geometry of dissipative evolution equations

Reina¹,

Zimmer²

2015

Phys. Rev. E

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Purely dissipative evolution equations are often cast as gradient flow structures, $\dot{\mathbf{z}}=K(\mathbf{z})DS(\mathbf{z})$, where the variable $\mathbf{z}$ of interest evolves towards the maximum of a functional $S$ according to a metric defined by an operator $K$. While the functional often follows immediately from physical considerations (e.g., the thermodynamic entropy), the operator $K$ and the associated geometry does not necessarily so (e.g., Wasserstein geometry for diffusion). In this paper, we present a variational statement in the sense of maximum entropy production that directly delivers a relationship between the operator $K$ and the constraints of the system. In particular, the Wasserstein metric naturally arises here from the conservation of mass or energy, and depends on the Onsager resistivity tensor, which, itself, may be understood as another metric, as in the Steepest Entropy Ascent formalism. This new variational principle is exemplified here for the simultaneous evolution of conserved and non-conserved quantities in open systems. It thus extends the classical Onsager flux-force relationships and the associated variational statement to variables that do not have a flux associated to them. We further show that the metric structure $K$ is intimately linked to the celebrated Freidlin-Wentzell theory of stochastically perturbed gradient flows, and that the proposed variational principle encloses an infinite-dimensional fluctuation-dissipation statement

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“…This question has been answered by Hermann in [39]. The chain of arguments leading to Hermann's answer can be formulated as follows (see [40]). MaxEnt with constraints is, from the mathematical point of view, a Legendre transformation.…”

Section: Contact Geometrymentioning

confidence: 99%

“…Following [40], we extend now the above contact formulation of classical thermodynamics to the mesoscopic thermodynamics expressed in the fundamental thermodynamic relation (6). We begin by introducing a space…”

Section: Contact Geometrymentioning

confidence: 99%

“…Among the interesting open problems we mention a few: (i) to clarify the physical meaning of the time evolution governed by (14) outside the Gibbs-Legendre manifold  ( ) ( ) Meq GL , (ii) to lift the GENERIC equations discussed in section 3 to contact dynamics, (iii) to use the contact formulation for developing physically meaningful discretizations (an attempt in this direction is made in section 4.3.4 in [40]).…”

Section: Contact Geometrymentioning

confidence: 99%

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GENERIC guide to the multiscale dynamics and thermodynamics

Grmela

2018

J. Phys. Commun.

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GENERIC is an abstract equation collecting the mathematical structure guaranteeing agreement of its solutions with results of certain basic experimental observations (conservations of mass and energy and the approach to equilibrium at which the classical equilibrium thermodynamics applies). In the unifying framework provided by GENERIC we present recent developments and open challenges in the fundamental aspects of the multiscale dynamics and thermodynamics.

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Geometry of Multiscale Nonequilibrium Thermodynamics

Cited by 15 publications

References 73 publications

Fluctuation symmetry leads to GENERIC equations with non-quadratic dissipation

Fluctuation symmetry leads to GENERIC equations with non-quadratic dissipation

Entropy production and the geometry of dissipative evolution equations

GENERIC guide to the multiscale dynamics and thermodynamics

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