Dissipative hamiltonian systems: A unifying principle

Kaufman, Allan N.

doi:10.1016/0375-9601(84)90634-0

Cited by 228 publications

(190 citation statements)

References 12 publications

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“…The bracket (., . ), referred to as (semi)metric bracket, is a type of Leibniz bracket quite different from Poisson algebra [6,20]. Once a generating function Q (x) is defined so that the dynamics of the system is governed by…”

Section: Dissipative Systems and Metric Algebramentioning

confidence: 99%

“…In the context of the classical dissipative dynamical systems that will be examined in this review, entropy plays the same role of energy, when the latter wears the costume of the Hamiltonian H: entropy contributes to generating the motion of the system, even if through a mechanism algebraically different from the Hamiltonian one [6,7]. In this way entropy plays directly and vividly the role of "εν τρoπη", a Greek phrase translatable as "the inner transformer", to which the physical function owes its name.…”

Section: Introductionmentioning

confidence: 99%

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Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Materassi

2016

Entropy

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This lecture is a short review on the role entropy plays in those classical dissipative systems whose equations of motion may be expressed via a Leibniz Bracket Algebra (LBA). This means that the time derivative of any physical observable f of the system is calculated by putting this f in a "bracket" together with a "special observable" F, referred to as a Leibniz generator of the dynamics. While conservative dynamics is given an LBA formulation in the Hamiltonian framework, so that F is the Hamiltonian H of the system that generates the motion via classical Poisson brackets or quantum commutation brackets, an LBA formulation can be given to classical dissipative dynamics through the Metriplectic Bracket Algebra (MBA): the conservative component of the dynamics is still generated via Poisson algebra by the total energy H, while S, the entropy of the degrees of freedom statistically encoded in friction, generates dissipation via a metric bracket. The motivation of expressing through a bracket algebra and a motion-generating function F is to endow the theory of the system at hand with all the powerful machinery of Hamiltonian systems in terms of symmetries that become evident and readable. Here a (necessarily partial) overview of the types of systems subject to MBA formulation is presented, and the physical meaning of the quantity S involved in each is discussed. Here the aim is to review the different MBAs for isolated systems in a synoptic way. At the end of this collection of examples, the fact that dissipative dynamics may be constructed also in the absence of friction with microscopic degrees of freedom is stressed. This reasoning is a hint to introduce dissipation at a more fundamental level.

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Section: Dissipative Systems and Metric Algebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Materassi

2016

Entropy

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“…Using the terminology of Callen [28], the choice of state variables made in (2) represents the entropy representation and the choice of the fields q(r) = (s(r), q (r)) (13) as state variables represents the energy representation. In the investigation of particular realizations of the Euler structure, it turns out to be convenient and natural to use the energy representation in discussions of the nondissipative vector field (in this representation it is easier to guarantee the requirement (5)) and the entropy representation in discussions of the dissipative vector field (in this representation it is easier to guarantee the requirement of the energy conservation).…”

Section: Commentmentioning

confidence: 99%

“…In the investigation of particular realizations of the Euler structure, it turns out to be convenient and natural to use the energy representation in discussions of the nondissipative vector field (in this representation it is easier to guarantee the requirement (5)) and the entropy representation in discussions of the dissipative vector field (in this representation it is easier to guarantee the requirement of the energy conservation). In this paper we shall however use the energy representation (13) in both dissipative and nondissipative dynamics.…”

Section: Commentmentioning

confidence: 99%

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

Peshkov

Grmela

Romenski

2014

Continuum Mech. Thermodyn.

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On the example of two-phase continua experiencing stress induced solid-fluid phase transitions we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization.

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“…5 of [2]). Kaufman [89], Morrison [90], and Grmela [66] published the first attempts to include dissipation in systems described by bracket formalisms by introducing a dissipation bracket. Grmela and Carreau [1], Grmela [38,91], and Beris and Edwards [2,67,68] used the bracket formalism to formulate the equations of change of polymeric liquids in a unified framework based on introducing the conformation tensor, an expectation value or continuum variable that represents local average values of stretch and orientation of the polymer coils.…”

Section: The Generalized Bracket Approach Of Grmela Beris and Edwardsmentioning

confidence: 99%

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

Pasquali

Scriven

2004

Journal of Non-Newtonian Fluid Mechanics

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A new method is presented for accounting for microstructural features of flowing complex fluids at the level of mesoscopic, or coarse-grained, models by ensuring compatibility with macroscopic and continuum thermodynamics and classical transport phenomena. In this method, the microscopic state of the liquid is described by variables that are local expectation values of microscopic features. The hypothesis of local thermodynamic equilibrium is extended to include information on the microscopic state, i.e., the energy of the liquid is assumed to depend on the entropy, specific volume, and microscopic variables. For compatibility with classical transport phenomena, the microscopic variables are taken to be extensive variables (per unit mass or volume), which obey convection-diffusion-generation equations. Restrictions on the constitutive laws of the diffusive fluxes and generation terms are derived by separating dissipation by transport (caused by gradients in the derivatives of the energy with respect to the state variables) and by relaxation (caused by non-equilibrated microscopic processes like polymer chain stretching and orientation), and by applying isotropy. When applied to unentangled, isothermal, non-diffusing polymer solutions, the equations developed according to the new method recover those developed by the Generalized Bracket [J. Non-Newtonian Fluid Mech. 23 (1987) Rheol. 38 (1994) 769]. Minor differences with published results obtained by the Generalized Bracket are found in the equations describing flow coupled to heat and mass transfer in polymer solutions. The new method is applied to entangled polymer solutions and melts in the general case where the rate of generation of entanglements depends nonlinearly on the rate of strain. A link is drawn between the mesoscopic transport equations of entanglements and conformation and the microscopic equation describing the configurational distribution of polymer segment stretch and orientation. Constraints are derived on the generation terms in the transport equations of entanglements and conformation, and the formula for the elastic stress is generalized to account for reversible formation and destruction of entanglements. A simplified version of the transport equation of conformation is presented which includes many previously published constitutive models, separates flow-induced polymer stretching and orientation, yet is simple enough to be useful for developing large-scale computer codes for modeling coupled fluid flow and transport phenomena in two-and three-dimensional domains with complex shapes and free surfaces.

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Dissipative hamiltonian systems: A unifying principle

Cited by 228 publications

References 12 publications

Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

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