Hamilton-Lagrange formalism of nonequilibrium thermodynamics

Gambár, Katalin; Márkus, Ferenc

doi:10.1103/physreve.50.1227

Cited by 58 publications

(42 citation statements)

References 25 publications

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“…The Hamiltonian formulation can be also achieved for certain differential equations involving non-selfadjoint operators like the first time derivative in the classical Fourier heat conduction. Then such potential functions are required to introduce by which the Lagrange functions can be expressed and the whole Hamiltionian theory can be constructed (Gambár & Márkus, 1994;Gambár, 2005;Márkus, 2005). The long scientific experience on this topic showed that the theories are comparable and connectable on thisLagrangian-Hamiltonian -level, thus in the further development of the theory it is useful to apply this idea and scheme.…”

Section: Lorentz Invariant Thermal Energy Propagationmentioning

confidence: 99%

“…This treating is an attempt to point out that the dynamic phase transition (Ma, 1982) between the two kinds of propagation, between a wave and a non-wave, or with another context it is better to say -between a non-dissipative and a dissipative thermal process -has a more general role and manifestation in the processes. As a starting point the Lagrange functions are given for both the Lorentz invariant heat propagation (Márkus & Gambár, 2005) and for the classical heat conduction (Fourier's heat conduction) (Gambár & Márkus, 1994). The first description is based on a Klein-Gordon type equation formulated by a negative "mass term".…”

Section: Lorentz Invariant Thermal Energy Propagationmentioning

confidence: 99%

“…Now, the Hamiltonian descriptions are written side by side -to prepare the later comparison -showing how the Lorentz invariant solution provides the classical solution in the limit of speed of light. The relevant Lagrangians, L w for the wave-like solution (Márkus & Gambár, 2005) and L c for the classical heat conduction (Gambár & Márkus, 1994) restricting our examination for the one dimensional case, are…”

Section: Lorentz Invariant Thermal Energy Propagationmentioning

confidence: 99%

“…Since, for the cases when the Lagrangian contains second order time derivatives the HamiltonianH must be expressed as follows (Gambár & Márkus, 1994;Márkus & Gambár, 1991),…”

Section: The Coupling Of the Fieldsmentioning

confidence: 99%

See 3 more Smart Citations

Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?

Márkus¹

2011

Heat Conduction - Basic Research

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Section: Lorentz Invariant Thermal Energy Propagationmentioning

confidence: 99%

Section: Lorentz Invariant Thermal Energy Propagationmentioning

confidence: 99%

Section: Lorentz Invariant Thermal Energy Propagationmentioning

confidence: 99%

“…Since, for the cases when the Lagrangian contains second order time derivatives the HamiltonianH must be expressed as follows (Gambár & Márkus, 1994;Márkus & Gambár, 1991),…”

Section: The Coupling Of the Fieldsmentioning

confidence: 99%

See 2 more Smart Citations

Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?

Márkus¹

2011

Heat Conduction - Basic Research

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“…The propoused method permits to obtain all the dynamical infomation of the system allowing the quantization in terms of conserved quantities prescibed for the differential equation. On the other hand, a variational formalism has been devised by Gambár and Márkus (see for example [2,3,4]) as groundwork for proposing a field theory for nonequilibrium thermodynamical systems, giving valuable information about the entropy in terms of current density and thermodynamic forces. Effective actions can be found by means of the Martin-Siggia-Rose formalism [5], a perturbative procedure that makes use of both physical and "conjugate" (auxiliary) fields.…”

mentioning

confidence: 99%

Variational and potential formulation for stochastic partial differential equations

Munoz¹,

Ojeda²,

Sierra³

et al. 2006

J. Phys. A: Math. Gen.

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There is recent interest in finding a potential formulation for Stochastic Partial Differential Equations (SPDEs). The rationale behind this idea lies in obtaining all the dynamical information of the system under study from one single expression. In this Letter we formally provide a general Lagrangian formalism for SPDEs using the Hojman et al. method. We show that it is possible to write the corresponding effective potential starting from an s-equivalent Lagrangean, and that this potential is able to reproduce all the dynamics of the system, once a special differential operator has been applied. This procedure can be used to study the complete time evolution and spatial inhomogeneities of the system under consideration, and is also suitable for the statistical mechanics description of the problem. The inverse problem of variational calculus have been used by Hojman et al. [1] in the study of systems associated with first and second order deterministic differential equations. The propoused method permits to obtain all the dynamical infomation of the system allowing the quantization in terms of conserved quantities prescibed for the differential equation. On the other hand, a variational formalism has been devised by Gambár and Márkus (see for example [2, 3, 4]) as groundwork for proposing a field theory for nonequilibrium thermodynamical systems, giving valuable information about the entropy in terms of current density and thermodynamic forces. Effective actions can be found by means of the Martin-Siggia-Rose formalism [5], a perturbative procedure that makes use of both physical and "conjugate" (auxiliary) fields. Also, Hochberg et al. have proposed in a series of papers [6,7,8] a "direct" approach, finding effective actions and potentials for SPDEs using a functional integral formalism similar in structure to those of quantum field theory. Last year Ao [9] has reported a worth consulting novel approach for constructing potentials associated with first order SPDEs. All this efforts conduct to the idea that the variational formalism, in its direct or inverse form, could be the mathematical mechanism with the necessary tools for exploring the inherent dynamics of physical, biological and chemical systems. In particular, structural development in cosmology and biology, pattern-formation, symmetry breaking, *

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Variational time integrators for finite‐dimensional thermo‐elasto‐dynamics without heat conduction

Mata

Lew

2011

Numerical Meth Engineering

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SUMMARYThis paper focuses on the formulation of variational integrators (VIs) for finite-dimensional thermo-elastic systems without heat conduction. The dynamics of these systems happen to have a Hamiltonian structure, after thermal displacements are introduced. It is then possible to formulate integrators by taking advantage of standard methods in VI. The class of integrators we construct have some remarkable features: (a) they are symplectic, (b) they exactly conserve the entropy of the system, or in other words, they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes, (c) they nearly exactly conserve the value of the energy for very long times and (d) they exactly conserve linear and angular momentum. We first describe how to adapt any VI for the classical mechanical systems to integrate adiabatic thermo-elastic ones, and then formulate three new types of integrators. The first class, based on a generalized trapezoidal rule, gives rise to two first order, explicit integrators, and a second order, implicit one. By composing then the two first-order integrators we construct a second order, explicit one. Finally, we formulate a fourth order, implicit integrator, which is a symplectic partitioned Runge-Kutta method. The performance of these new algorithms is showcased through numerical examples.

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Hamilton-Lagrange formalism of nonequilibrium thermodynamics

Cited by 58 publications

References 25 publications

Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?

Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?

Variational and potential formulation for stochastic partial differential equations

Variational time integrators for finite‐dimensional thermo‐elasto‐dynamics without heat conduction

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