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

Authors

Abstract

In this paper, we present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation is an extension of Hamilton’s principle of classical mechanics that allows the inclusion of irreversible phenomena. The irreversibility is encoded into a nonlinear phenomenological constraint given by the expression of the entropy production associated to all the irreversible processes involved. From a mathematical point of view, our variational formulation may be regarded as a generalization to nonequilibrium thermodynamics of the Lagrange–d’Alembert principle used in nonlinear nonholonomic mechanics, where the conventional Lagrange–d’Alembert principle cannot be applied since the nonlinear phenomenological constraint and its associated variational constraint must be treated separately. 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 to systematically 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 formulation of discrete systems to the case of continuum systems.

Keywords

Lagrangian formulation; Nonequilibrium thermodynamics; Variational formulation; Nonholonomic constraints; Irreversible processes; Discrete systems

Citation

Journal: Journal of Geometry and Physics
Year: 2017
Volume: 111
Issue:
Pages: 169–193
Publisher: Elsevier BV
DOI: 10.1016/j.geomphys.2016.08.018

BibTeX

@article{Gay_Balmaz_2017,
  title={{A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems}},
  volume={111},
  ISSN={0393-0440},
  DOI={10.1016/j.geomphys.2016.08.018},
  journal={Journal of Geometry and Physics},
  publisher={Elsevier BV},
  author={Gay-Balmaz, François and Yoshimura, Hiroaki},
  year={2017},
  pages={169--193}
}

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References

Gibbs, (1902)
Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. Rev. 37, 405–426 (1931) – 10.1103/physrev.37.405
de Groot, (1969)
Truesdell, (1969)
Glansdorff, (1971)
Stueckelberg, (1974)
Biot, A virtual dissipation principle and Lagrangian equations in non-linear irreversible thermodynamics. Acad. Roy. Belg. Bull. Cl. Sci. (1975)
Woods, (1975)
Lavenda, (1978)
Kondepudi, (1998)
Onsager, L. & Machlup, S. Fluctuations and Irreversible Processes. Phys. Rev. 91, 1505–1512 (1953) – 10.1103/physrev.91.1505
Onsager, L. & Machlup, S. Fluctuations and Irreversible Processes. Phys. Rev. 91, 1505–1512 (1953) – 10.1103/physrev.91.1505
Prigogine, (1947)
Ziegler, A possible generalization of Onsager’s theory. (1968)
Gyarmati, (1970)
Ichiyanagi, M. Variational principles of irreversible processes. Physics Reports 243, 125–182 (1994) – 10.1016/0370-1573(94)90052-3
Biot, M. A. New Variational-Lagrangian Irreversible Thermodynamics with Application to Viscous Flow, Reaction–Diffusion, and Solid Mechanics. Advances in Applied Mechanics 1–91 (1984) doi:10.1016/s0065-2156(08)70042-5 – 10.1016/s0065-2156(08)70042-5
Fukagawa, H. & Fujitani, Y. A Variational Principle for Dissipative Fluid Dynamics. Progress of Theoretical Physics 127, 921–935 (2012) – 10.1143/ptp.127.921
Green, A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. (1885)
Gay-Balmaz, F. & Yoshimura, H. A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems. Journal of Geometry and Physics 111, 194–212 (2017) – 10.1016/j.geomphys.2016.08.019
Gruber, C. Thermodynamics of systems with internal adiabatic constraints: time evolution of the adiabatic piston. Eur. J. Phys. 20, 259–266 (1999) – 10.1088/0143-0807/20/4/303
Ferrari, C. & Gruber, C. Friction force: from mechanics to thermodynamics. Eur. J. Phys. 31, 1159–1175 (2010) – 10.1088/0143-0807/31/5/017
Gruber, C. & Brechet, S. D. Lagrange Equations Coupled to a Thermal Equation: Mechanics as Consequence of Thermodynamics. Entropy 13, 367–378 (2011) – 10.3390/e13020367
Bloch, (2003)
Arnold, (1988)
Jiménez, F. & Yoshimura, H. Dirac structures in vakonomic mechanics. Journal of Geometry and Physics 94, 158–178 (2015) – 10.1016/j.geomphys.2014.11.002
Chetaev, On Gauss principle. Izv. Fiz-Mat. Obsc. Kazan Univ. (1934)
Appell, Sur les liaisons exprimées par des relations non linéaires entre les vitesses. C. R. Math. Acad. Sci. Paris (1911)
Marle, C.-M. Various approaches to conservative and nonconservative nonholonomic systems. Reports on Mathematical Physics 42, 211–229 (1998) – 10.1016/s0034-4877(98)80011-6
Cendra, H., Ibort, A., de León, M. & Martı́n de Diego, D. A generalization of Chetaev’s principle for a class of higher order nonholonomic constraints. Journal of Mathematical Physics 45, 2785–2801 (2004) – 10.1063/1.1763245
Gruber, (1997)
Chua, L. & McPherson, J. Explicit topological formulation of Lagrangian and Hamiltonian equations for nonlinear networks. IEEE Trans. Circuits Syst. 21, 277–286 (1974) – 10.1109/tcs.1974.1083849
Yoshimura, H. & Marsden, J. E. Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. Journal of Geometry and Physics 57, 133–156 (2006) – 10.1016/j.geomphys.2006.02.009
Yoshimura, H. & Marsden, J. E. Dirac structures in Lagrangian mechanics Part II: Variational structures. Journal of Geometry and Physics 57, 209–250 (2006) – 10.1016/j.geomphys.2006.02.012
Oster, G. F., Perelson, A. S. & Katchalsky, A. Network thermodynamics: dynamic modelling of biophysical systems. Quart. Rev. Biophys. 6, 1–134 (1973) – 10.1017/s0033583500000081
von Helmholtz, Studien zur Statik monocyklischer Systeme. (1884)
Podio-Guidugli, P. A virtual power format for thermomechanics. Continuum Mech. Thermodyn. 20, 479–487 (2009) – 10.1007/s00161-009-0093-5
O. Jacobs, H. & Yoshimura, H. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics 6, 67–98 (2014) – 10.3934/jgm.2014.6.67
Gay-Balmaz, F. & Yoshimura, H. Dirac reduction for nonholonomic mechanical systems and semidirect products. Advances in Applied Mathematics 63, 131–213 (2015) – 10.1016/j.aam.2014.10.004