Authors

Jochen Merker, Matthias Krüger

Abstract

The dynamic principle is proposed that a thermodynamic system evolves in time so that the total energy balance including the energy drawn from the environment becomes stationary for all admissible variations of thermodynamic states. It is shown that this principle allows to obtain variational characterizations of contact Hamiltonian equations (even in presence of ports), reaction equations and doubly nonlinear reaction–diffusion equations. Further, examples are discussed which support this principle.

Keywords

contact hamiltonian equations, reaction–diffusion equations, variational principle

Citation

  • Journal: Continuum Mechanics and Thermodynamics
  • Year: 2013
  • Volume: 25
  • Issue: 6
  • Pages: 779–793
  • Publisher: Springer Science and Business Media LLC
  • DOI: 10.1007/s00161-012-0277-2

BibTeX

@article{Merker_2012,
  title={{On a variational principle in thermodynamics}},
  volume={25},
  ISSN={1432-0959},
  DOI={10.1007/s00161-012-0277-2},
  number={6},
  journal={Continuum Mechanics and Thermodynamics},
  publisher={Springer Science and Business Media LLC},
  author={Merker, Jochen and Krüger, Matthias},
  year={2012},
  pages={779--793}
}

Download the bib file

References

  • R. Hermann, Geometry, Physics and Systems (1973)
  • Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry. Springer New Yor – 10.1007/978-0-387-21792-5
  • Lacomba EA, Losco L (1982) Variational characterization of contact vector fields in the group of contact diffeomorphisms. Physica A: Statistical Mechanics and its Applications 114(1–3):124–128. https://doi.org/10.1016/0378-4371(82)90270- – 10.1016/0378-4371(82)90270-9
  • Hildebrandt S (1994) Contact transformations, Huygens’s principle, and calculus of variations. Calc Var 2(3):249–281. https://doi.org/10.1007/bf0123553 – 10.1007/bf01235531
  • Bloch AM (2003) Nonholonomic Mechanics and Control. Springer New Yor – 10.1007/b97376
  • Mrugala R, Nulton JD, Christian Schön J, Salamon P (1991) Contact structure in thermodynamic theory. Reports on Mathematical Physics 29(1):109–121. https://doi.org/10.1016/0034-4877(91)90017- – 10.1016/0034-4877(91)90017-h
  • B.C. Eu, Generalized Thermodynamics (2002)
  • Eberard D, Maschke BM, van der Schaft AJ (2007) An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics 60(2):175–198. https://doi.org/10.1016/s0034-4877(07)00024- – 10.1016/s0034-4877(07)00024-9
  • Eberard D, Maschke BM, van der Schaft AJ Port contact systems for irreversible thermodynamical systems. Proceedings of the 44th IEEE Conference on Decision and Control 5977–598 – 10.1109/cdc.2005.1583118
  • Mielke A (2011) A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24(4):1329–1346. https://doi.org/10.1088/0951-7715/24/4/01 – 10.1088/0951-7715/24/4/016
  • Otto F (2001) THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION. Communications in Partial Differential Equations 26(1–2):101–174. https://doi.org/10.1081/pde-10000224 – 10.1081/pde-100002243
  • Onsager L (1931) Reciprocal Relations in Irreversible Processes. I. Phys Rev 37(4):405–426. https://doi.org/10.1103/physrev.37.40 – 10.1103/physrev.37.405
  • I. Prigogine, Bulletin de la Classe des Sciences, Academie Royale de Belgique (1945)
  • Gyarmati I (1969) On the “Governing Principle of Dissipative Processes” and its Extension to Non‐linear Problems. Annalen der Physik 478(7–8):353–378. https://doi.org/10.1002/andp.1969478070 – 10.1002/andp.19694780707
  • P. Glansdorff, Thermodynamic Theory of Structure, Stability and Fluctuations (1971)
  • Lebon G, Jou D, Casas-Vázquez J (2008) Understanding Non-equilibrium Thermodynamics. Springer Berlin Heidelber – 10.1007/978-3-540-74252-4
  • W. Muschik, Technische Mechanik (2000)
  • W. Muschik, J. Non-Equilib. Thermodyn. (2004)
  • Muschik W (2009) Contact Quantities and Non-Equilibrium Entropy of Discrete Systems. Journal of Non-Equilibrium Thermodynamics 34(1). https://doi.org/10.1515/jnetdy.2009.00 – 10.1515/jnetdy.2009.005
  • (2006) Chapter 5 Contact geometry. Handbook of Differential Geometry 315–38 – 10.1016/s1874-5741(06)80008-7
  • Arnold VI (1989) Mathematical Methods of Classical Mechanics. Springer New Yor – 10.1007/978-1-4757-2063-1