Phenomenological Macroscopic Symmetry in Dissipative Nonlinear Systems

  • J. S. Shiner
  • Stanislaw Sieniutycz
Part of the Understanding Chemical Reactivity book series (UCRE, volume 18)


We show that systems whose equations of motion are derivable from a Lagrangian formulation extended to allow for dissipative effects by the inclusion of dissipative forces of the form (resistance X flow) and subject to constraints expressing conservation of quantities such as mass and charge possess a symmetry property at stationary states arbitrarily far from thermodynamic equilibrium. Examples of such systems are electrical networks] discrete mechanical systems and systems of chemical reactions. The symmetry is in general only an algebraic one, not the differential Onsager reciprocity valid close to equilibrium; the more general property does reduce to the Onsager result in the equilibrium limit, however. No restrictions are placed on the form of the resistances for dissipative processes; they may have arbitrary dependencies on the state of the system and/or its flows. Thus, the symmetry property is not limited to linear systems. The approach presented here leads naturally to the proper set of flows and forces for which the Onsager-like symmetry holds.


Dissipative Process Electrical Network Dissipation Function Dissipative Force Internal State Variable 
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  1. 1.
    Callen, H. (1985) Thermodynamics and Introduction to Thermostatistics, Wiley, New York.zbMATHGoogle Scholar
  2. 2.
    Onsager, L. (1931) Phys. Rev. 37,405–426.CrossRefADSzbMATHGoogle Scholar
  3. 3.
    Onsager, L. (1931) Phys. Rev. 38,2265–2279.CrossRefADSzbMATHGoogle Scholar
  4. 4.
    deGroot, S. R. and Mazur, P. (1984) Non-Equilibrium Thermodynamics, Dover, New York.Google Scholar
  5. 5.
    Keizer, J. (1987) Statistical Thermodynamics of Nonequilibrium Processes, Springer, New York.Google Scholar
  6. 6.
    Essig, A. and Caplan, S. R. (1981) Proc. Natl. Acad. Sci. (USA) 78,1647–1651.ADSCrossRefGoogle Scholar
  7. 7.
    Rothschild, K. J., Elias, S. A., Essig, A. and Stanley, H. E. (1980) Biophys. J. 30, 209–230.CrossRefGoogle Scholar
  8. 8.
    Stucki, J. W., Compiani, M. and Caplan, S. R. (1983) Biophys. Chem. 18, 101–109.CrossRefGoogle Scholar
  9. 9.
    Keizer, J. (1979) Acc. Chem. Res. 12,243–249.CrossRefADSGoogle Scholar
  10. 10.
    Keizer, J. (1979) in H. Haken (ed.), Pattern formation by dynamic systems and pattern recognition, Springer, New York, pp. 266–277.Google Scholar
  11. 11.
    Prigogine, I. (1967) Introduction to thermodynamics of irreversible processes, Wiley, New York.Google Scholar
  12. 12.
    Glansdorff, P. and Prigogine, I. (1971) Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley, New York.zbMATHGoogle Scholar
  13. 13.
    Nicolis, G. and Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems, Wiley, New York.zbMATHGoogle Scholar
  14. 14.
    Shiner, J. S. (1987) J. Chem. Phys. 87,1089–1094.CrossRefADSGoogle Scholar
  15. 15.
    Shiner, J. S. (1992) in S. Sieniutycz and P. Salamon (eds.) Flow, Diffusion and Rate Processes (Adv. Thermodyn. Vol. 6), Taylor and Francis, New York, pp. 248–282.Google Scholar
  16. 16.
    Rayleigh, J. W. S. (1945) The Theory of Sound, Dover, New York.zbMATHGoogle Scholar
  17. 17.
    Goldstein, H. (1980) Classical Mechanics, Addison-Wesley, Reading.zbMATHGoogle Scholar
  18. 18.
    Whittaker, E. T. (1988) A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, University Press, Cambridge.zbMATHGoogle Scholar
  19. 19.
    Casimir, H.B.G. (1945) Rev. Mod. Phys. 17,342–350.CrossRefADSGoogle Scholar
  20. 20.
    MacFarlane, A. G. J. (1970) Dynamical System Models, Harrap, London.zbMATHGoogle Scholar
  21. 21.
    Ross, J., Hunt, K.L.C. and Hunt, P.M. (1988) J. Chem. Phys. 88,2719–2729.CrossRefADSGoogle Scholar
  22. 22.
    Grabert, H., Hänggi, P. and Oppenheim, I. (1983) Physica 117A, 300–316.ADSGoogle Scholar
  23. 23.
    Sieniutycz, S. (1987) Chem. Eng. Sci. 42,2697–2711.CrossRefGoogle Scholar
  24. 24.
    Hill, T. L. (1977) Free Energy Transduction in Biology, Academic Press, New York.Google Scholar
  25. 25.
    Sieniutycz, S. and Shiner, J.S. (1992) Open Sys. Information Dyn. 1,149–182.zbMATHCrossRefGoogle Scholar
  26. 26.
    Sieniutycz, S. and Shiner, J.S. (1992) Open Sys. Information Dyn. 1,327–348.zbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • J. S. Shiner
    • 1
  • Stanislaw Sieniutycz
    • 2
  1. 1.Physiologisches InstitutUniversität BernBernSwitzerland
  2. 2.Department of Chemical EngineeringWarsaw Technical UniversityWarsawPoland

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