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

Gay–Balmaz, François; Yoshimura, Hidetoshi

doi:10.1016/j.geomphys.2016.08.018

Cited by 74 publications

(123 citation statements)

References 25 publications

(34 reference statements)

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“…The constraint (1.2) on the variations (δq, δS) follows from (1.3) by formally replacing the velocity by the corresponding virtual displacement, and by removing the contribution from the exterior of the system. Such a simple correspondence between the phenomenological and variational constraints still holds for more general thermodynamic systems, see Gay-Balmaz and Yoshimura [2017a]. Taking variations in the variational condition (1.1) and using the constraints (1.2) and (1.3), we obtain the following system of differential equations:…”

Section: Variational Formulation Of Thermodynamicsmentioning

confidence: 97%

Variational discretization of thermodynamical simple systems on Lie groups

Couéraud¹,

Gay–Balmaz²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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This paper presents the continuous and discrete variational formulations of simple thermodynamical systems whose configuration space is a (finite dimensional) Lie group. We follow the variational approach to nonequilibrium thermodynamics developed in Gay-Balmaz and Yoshimura [2017a,b], as well as its discrete counterpart whose foundations have been laid in Gay-Balmaz and Yoshimura [2018]. In a first part, starting from this variational formalism on the Lie group, we perform an Euler-Poincaré reduction in order to obtain the reduced evolution equations of the system on the Lie algebra of the configuration space. We obtain as corollaries the energy balance and a Kelvin-Noether theorem. In a second part, a compatible discretization is developed resulting in discrete evolution equations that take place on the Lie group. Then, these discrete equations are transported onto the Lie algebra of the configuration space with the help of a group difference map. Finally we illustrate our framework with a heavy top immersed in a viscous fluid modeled by a Stokes flow and proceed with a numerical simulation.

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Section: Variational Formulation Of Thermodynamicsmentioning

confidence: 97%

Variational discretization of thermodynamical simple systems on Lie groups

Couéraud¹,

Gay–Balmaz²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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“…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

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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...

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“…As above, µ a and T a denote the chemical potential and temperature at the a-th port and T b denotes the temperature of the b-th heat source. An essential ingredient for our variational formulation of thermodynamics is the concept of thermodynamic displacements (Gay-Balmaz and Yoshimura [2017aYoshimura [ ,b, 2018c). By definition, the thermodynamic displacement associated with an irreversible process is given by the primitive in time of a thermodynamic force (or affinity) of the process.…”

Section: Variational Formulation For Open Simple Systemsmentioning

confidence: 99%

Dirac structures in nonequilibrium thermodynamics

Gay–Balmaz

Yoshimura

2018

Journal of Mathematical Physics

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Dirac structures are geometric objects that generalize Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems and play an essential role in structuring a dynamical system through the energy flow between its subsystems and elements. In this paper, we show that the evolution equations for open thermodynamic systems, i.e., systems exchanging heat and matter with the exterior, admit an intrinsic formulation in terms of Dirac structures. We focus on simple systems, in which the thermodynamic state is described by a single entropy variable. A main difficulty compared to the case of closed systems lies in the explicit time dependence of the constraint associated to the entropy production. We overcome this issue by working with the geometric setting of time-dependent nonholonomic mechanics. We define three type of Dirac dynamical systems for the nonequilibrium thermodynamics of open systems, based either on the generalized energy, the Lagrangian, or the Hamiltonian. The variational formulations associated to the Dirac systems formulations are also presented.

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

Cited by 74 publications

References 25 publications

Variational discretization of thermodynamical simple systems on Lie groups

Variational discretization of thermodynamical simple systems on Lie groups

Quantum measurements and thermodynamics

Dirac structures in nonequilibrium thermodynamics

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