A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems

Gay–Balmaz, François; Yoshimura, Hidetoshi

doi:10.1016/j.geomphys.2016.08.019

Cited by 62 publications

(139 citation statements)

References 15 publications

Supporting

Mentioning

138

Contrasting

Order By: Relevance

“…In this section we present the fundamental laws of thermodynamics following the approach of Stueckelberg, we recall the definition of simple and discrete systems in thermodynamics, and we review the Lagrange-d'Alembert principle in nonholonomic mechanics. These notions are fundamental for our developments in Part II of the present paper, Gay-Balmaz and Yoshimura [2016].…”

Section: Some Preliminariesmentioning

confidence: 95%

“…The first formulation follows from Theorem 3.1 and uses the degree of advancement as a generalized coordinate. The second formulation, that we will present in Definition 3.7 below, needs the introduction of new variables, but it has the advantage to allow us to develop a corresponding version in the continuum case, which will be of crucial use in the case of a multicomponent fluid with chemical reactions in Part II, Gay-Balmaz and Yoshimura [2016].…”

mentioning

confidence: 99%

See 1 more Smart Citation

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

Gay–Balmaz

Yoshimura

2017

Journal of Geometry and Physics

Self Cite

120

View full text Add to dashboard Cite

In this paper, we present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation is an extension of the Hamilton principle in classical mechanics that allows the inclusion of irreversible phenomena. The irreversibility is encoded into a nonlinear phenomenological constraint given by the expression of entropy production associated to all the irreversible processes involved. Hence from a mathematical point of view, our variational formulation may be regarded as a generalization of the Lagrange-d'Alembert principle used in nonlinear nonholonomic mechanics to the nonequilibrium thermodynamics, where the conventional Lagrange-d'Alembert principle cannot be applied since the nonlinear phenomenological constraint and its associated variational constraint must be separately treated. In our approach, to deal with the nonlinear nonholonomic constraint, we introduce a variable called the thermodynamic displacement associated to each irreversible process. This allows us systematically to define the corresponding variational constraint. In Part I, our variational theory is illustrated with various examples of discrete systems such as mechanical systems with friction, matter transfer, electric circuits, chemical reactions, and diffusion across membranes. In Part II of the present paper, we will extend our variational theory of discrete systems to the case of of continuum systems. Contents 1 Introduction 2 2 Some preliminaries 5 3 Variational formulation for nonequilibrium thermodynamics of simple systems 9 3.

show abstract

Section: Some Preliminariesmentioning

confidence: 95%

mentioning

confidence: 99%

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

Gay–Balmaz

Yoshimura

2017

Journal of Geometry and Physics

Self Cite

120

View full text Add to dashboard Cite

show abstract

“…In our case, the passage from (2.5) to (2.6) can be formally seen as a generalization of this approach, to the case of a nonlinear constraint. We refer to Gay-Balmaz and Yoshimura [2017b] for an extensive discussion of the principle (2.4)-(2.6).…”

Section: Variational Derivation Of the Thermodynamic Of A Dry Atmospherementioning

confidence: 99%

A variational derivation of the thermodynamics of a moist atmosphere with rain process and its pseudoincompressible approximation

Gay–Balmaz

2019

Geophysical & Astrophysical Fluid Dynamics

Self Cite

View full text Add to dashboard Cite

Irreversible processes play a major role in the description and prediction of atmospheric dynamics. In this paper, we present a variational derivation of the evolution equations for a moist atmosphere with rain process and subject to the irreversible processes of viscosity, heat conduction, diffusion, and phase transition. This derivation is based on a general variational formalism for nonequilibrium thermodynamics which extends Hamilton's principle to incorporates irreversible processes. It is valid for any state equation and thus also covers the case of the atmosphere of other planets. In this approach, the second law of thermodynamics is understood as a nonlinear constraint formulated with the help of new variables, called thermodynamic displacements, whose time derivative coincides with the thermodynamic force of the irreversible process. The formulation is written both in the Lagrangian and Eulerian descriptions and can be directly adapted to oceanic dynamics. We illustrate the efficiency of our variational formulation as a modeling tool in atmospheric thermodynamics, by deriving a pseudoincompressible model for moist atmospheric thermodynamics with general equations of state and subject to the irreversible processes of viscosity, heat conduction, diffusion, and phase transition.

show abstract

“…A mathematical model is introduced which permits the description of the situation, in particular, their mutual influence on each other. The classical apparatus can be thought of as a trajectory in a manifold that has coordinates x = ( x 1 , …, x n ) , so the path is described by a classical Lagrangian depending on x ( t ) and its velocity tangent vector defined by v i ( t ) = dx i / dt if there was no interaction with the quantum object,

ℒ_{A} ((), boldx, boldv) = \frac{1}{2} m_{jk} ((), x) v^{j} v^{k} - V ((), x) .

…”

Section: Environment and Measurementmentioning

confidence: 99%

Quantum measurements and thermodynamics

Bracken

2019

Int J of Quantum Chemistry

View full text Add to dashboard Cite

Thermodynamics was developed when it was still unclear whether the microworld had an atomic substructure. The objective here is to investigate the relationship between the process of a quantum measurement and the second law of thermodynamics. It is necessary to make the ideas of heat and work explicit in this approach. It is possible to study the correspondence between information entropy and thermodynamic entropy. Finally, a model is proposed in such a way as to resemble an experimental example for the discussion. K E Y W O R D S density matrix, entropy, heat bath, measurement, thermodynamics P A C S 3.65.Ta; 4.20.Cv; 5.30.-d; 5.70.-a 1 | INTRODUCTIONThermodynamics is a fundamental part of classical physics that was developed when the atomic nature of matter was still undecided. Yet the basis of thermodynamics at the microscopic level remains incomplete. It is the belief now that quantum mechanics determines the behavior of matter at very small scales of the microscopic realm. The correspondence between these two subjects is of great interest in general. One of the more important reasons is that what appears to be the occurrence of time reversal symmetry at microscopic scales appears to be the occurrence of time reversal asymmetry as one moves to macroscopic scales. The entire basis for what intuition for this separation dictates is connected to the observations of the second law of thermodynamics. This means the forward direction of time is related to the experience of observing an increase in entropy. [1,2] Suppose we examine a quantum measurement according to the rules of Bohr. Then, a measurement may be thought of as an interaction between a classical apparatus and a quantum object such that the data may be thought of as classical and reside in the dynamics of a classical device. Thus, there is no direct observation of the quantum object. To ask the question what is the quantum object really doing is to restrict the quantum particle to behave in a classical manner and yields the appearance of a classical apparatus. [2][3][4][5][6] Landau and Lifshitz conjectured that second law entropy increases are required for quantum measurements, and quantum measurements are also responsible for the increase in entropy dictated by the second law itself. One implication of this is the notion that the second law has no grounding or basis in classical statistical thermodynamics, and in fact, quantum mechanics is required. These investigators never proved their conjecture, and this will not be carried out here either. However, some of the formal detail and existing problems will be described in detail.Once a typical classical apparatus and quantum object are defined, the manner in which the classical apparatus is to interact with the quantum object is established. The quantum object may exert nonadiabatic and adiabatic forces on the classical device so that quantum measurements can be discussed. These ideas will be developed more completely by formulating the equation of interaction with a heat bath. The idea of work and hea...

show abstract

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems

Cited by 62 publications

References 15 publications

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

A variational derivation of the thermodynamics of a moist atmosphere with rain process and its pseudoincompressible approximation

Quantum measurements and thermodynamics

Contact Info

Product

Resources

About