From Variational to Bracket Formulations in Nonequilibrium Thermodynamics of Simple Systems

Gay–Balmaz, François; Yoshimura, Hidetoshi

doi:10.1007/978-3-030-26980-7_22

Cited by 5 publications

(5 citation statements)

References 21 publications

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“…The mapping H : Y → HY in (27), taking a vector field to its holonomic part, is a Lie algebra isomorphism from the space of projectable vector fields into the space of generalized vector fields of order one.…”

Section: Lemmamentioning

confidence: 99%

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Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Esen

Grmela

Gümral

et al. 2019

Entropy

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Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. This pathway is free from Poisson brackets and Hamiltonian functionals. Momentum realizations (sections on T * T * Q ) of (both compressible and incompressible) Euler’s fluid and Vlasov’s plasma are derived. Poisson mappings relating the momentum realizations with the usual field equations are constructed as duals of injective Lie algebra homomorphisms. The geometric pathway is then used to construct the evolution equations for 10-moments kinetic theory. This way the entire Grad hierarchy (including entropic fields) can be constructed in a purely geometric way. This geometric way is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets.

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Section: Lemmamentioning

confidence: 99%

“…For example, contact geometry offers a very convenient setting [13,24,25]. A variational formulation has recently been introduced in the works by the authors of [26,27]. Another variational formulation also arises in the contact geometry formulation [28].…”

Section: Introductionmentioning

confidence: 99%

Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Esen

Grmela

Gümral

et al. 2019

Entropy

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“…These formulations are shown to reduce to existing, known bracket formulations (e.g., [4,40]) for classical hydrodynamics in the case of a single component. We refer to [27] for a derivation of these bracket formalisms from the variational formulation for the case of finite dimensional thermodynamic systems.…”

Section: Introductionmentioning

confidence: 99%

Single and double generator bracket formulations of multicomponent fluids with irreversible processes

Eldred¹,

Gay–Balmaz²

2020

J. Phys. A: Math. Theor.

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The equations of reversible (inviscid, adiabatic) fluid dynamics have a well-known variational formulation based on Hamilton's principle and the Lagrangian, to which is associated a Hamiltonian formulation that involves a Poisson bracket structure. These variational and bracket structures underlie many of the most basic principles that we know about geophysical fluid flows, such as conservation laws. However, real geophysical flows also include irreversible processes, such as viscous dissipation, heat conduction, diffusion and phase changes. Recent work has demonstrated that the variational formulation can be extended to include irreversible processes and non-equilibrium thermodynamics, through the new concept of thermodynamic displacement. By design, and in accordance with fundamental physical principles, the resulting equations automatically satisfy the first and second law of thermodynamics. Irreversible processes can also be incorporated into the bracket structure through the addition of a dissipation bracket. This gives what are known as the single and double generator bracket formulations, which are the natural generalizations of the Hamiltonian formulation to include irreversible dynamics. Here the variational formulation for irreversible processes is shown to underlie these bracket formulations for fully compressible, multicomponent, multiphase geophysical fluids with a single temperature and velocity. Many previous results in the literature are demonstrated to be special cases of this approach. Finally, some limitations of the current approach (especially with regards to precipitation and nonlocal processes such as convection) are discussed, and future directions of research to overcome them are outlined.

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“…On can imagine, for instance, a nonholonomic system subject additionally to Rayleigh dissipation [6,27,28]. Another source of examples comes from thermodynamics, treated in [15][16][17] with a variational approach.…”

Section: Introductionmentioning

confidence: 99%

Contact Hamiltonian systems with nonholonomic constraints

de León,

Jiménez,

Valcázar

2019

Preprint

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In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstrained Hamiltonian vector field. Finally, we construct the nonholonomic bracket, which is an almost Jacobi bracket on the space of observables and provides the nonholonomic dynamics.

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From Variational to Bracket Formulations in Nonequilibrium Thermodynamics of Simple Systems

Cited by 5 publications

References 21 publications

Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Single and double generator bracket formulations of multicomponent fluids with irreversible processes

Contact Hamiltonian systems with nonholonomic constraints

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