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Open Systems & Information Dynamics

, Volume 1, Issue 2, pp 149–182 | Cite as

Variational and extremum properties of homogeneous chemical kinetics. I. Lagrangian- and Hamiltonian-like formulations

  • S. Sieniutycz
  • J. S. Shiner
Article

Conclusion

In this work we have constructed and compared various action-based variational approaches leading to nonstationary chemical kinetics and its asymptotic steady limit, Guldberg-Waage kinetics. We have shown that Guldberg-Waage kinetics is consistent with the chemical Lagrangian, eq. (14), at the steady state, but that it should be modified by an additional term (with the time derivative of the reaction rate) for very fast transients of the frequency greater than the ratio of the mean molecular velocity to the mean free path. These approaches developed here are consistent with the dissipative Lagrangian equations (with a Rayleigh dissipation function) and the Onsager-Machlup and Casimir formulations. The new action approach with the action explicit in the Lagrangian proposed here provides for the conservation of an energy-like integral which was missing in earlier approaches. At the same time it preserves a dissipation inequality by proving that an extended free energy, allowing for a dependence on the reaction rates, decreases in time.

Keywords

Steady State Free Energy System Theory Free Path Time Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Casimir, H. G. B., Rev. Mod. Phys.17, 342 (1945).Google Scholar
  2. [2]
    Chapman, S., and T.G. Cowling,The Mathematical Theory of Non-Uniform Gases, University Press, Cambridge, 1952.Google Scholar
  3. [3]
    Corso, P.L., J. Phys. Chem.87, 2416 (1983).Google Scholar
  4. [4]
    Courant, R., and D. Hilbert,Methods of Mathematical Physics, vol. II:Partial Differential Equations, Interscience Publishers, New York, 1962.Google Scholar
  5. [5]
    Essex, Ch., The Astrophysical Journal285, 279 (1984).Google Scholar
  6. [6]
    Fan, L.T.,The Continuous Maximum Principle, Wiley, New York, 1966.Google Scholar
  7. [7]
    Goldstein, H.,Klassische Mechanik, Akademische Verlagsgesellschaft, Wiesbaden, 1983.Google Scholar
  8. [8]
    Grabert, H., P. Hänggi and I. Oppenheim, Physica117A, 300 (1983)Google Scholar
  9. [9]
    de Groot, S.R., and P. Mazur,Non-Equilibrium Thermodynamics, North-Holland, London, 1969.Google Scholar
  10. [10]
    Huang, K.,Statistical Mechanics, Wiley, New York, 1967.Google Scholar
  11. [11]
    Jezierski, J., and J. Kijowski, C. R. Acad. Sc. Paris301, serie II, no. 4, 221 (1985).Google Scholar
  12. [12]
    Jou, D., J. Casas-Vazquez, and G. Lebon, Rep. Progr. Phys.51, 1105 (1989).Google Scholar
  13. [13]
    Landau, L.D., and E.M. Lifshitz,Mechanics, Pergamon, Oxford, 1960.Google Scholar
  14. [14]
    Landau, L.D., and E.M. Lifshitz,Electrodynamics of Continuous Media, Pergamon, Oxford, 1960.Google Scholar
  15. [15]
    Landau, L.D., and E.M. Lifshitz,The Classical Theory of Fields, Pergamon, Oxford, 1962.Google Scholar
  16. [16]
    MacFarlane, A.G.J.,Dynamical System Models, George G. Harrap, London, 1970.Google Scholar
  17. [17]
    Marion, J.B.,Classical Dynamics of Particles and Systems, Academic Press, London, 1969.Google Scholar
  18. [18]
    Meixner, J., J. Math. Phys.4, 154 (1963).Google Scholar
  19. [19]
    Miljah, A.N., and A.K. Shidlovskij,Reciprocity Principle and Reversibility of Electotechnical Phenomena (in Russian), Naukova Dumka, Kiev, 1967.Google Scholar
  20. [20]
    Mitura, E., S. Michalowski and W. Kaminski, Drying Technol.6, 113 (1988).Google Scholar
  21. [21]
    Nonnenmacher, T.F., Functional Poisson Brackets for Nonlinear Fluid Mechanics Equations, inRecent Developments in Nonequilibrium Thermodynamics: Fluids and Related Topics, ed. by J. Casas-Vazquez, D. Jou and J.M. Rubi, Springer, Berlin, 1986.Google Scholar
  22. [22]
    Ono, S., Adv. Chem. Phys. III, 267 (1961).Google Scholar
  23. [23]
    Onsager, L., Phys. Rev.37, 405, ibid.38, 2265 (1931).Google Scholar
  24. [24]
    Onsager, L., and S. Machlup, Phys. Rev.91, 1505, 1512 (1953).Google Scholar
  25. [25]
    Pontryagin, L.S., V.A. Boltyanski, R.V. Gamkrelidze, and E.F. Mischenko,The Mathematical Theory of the Optimal Processes, Wiley, New York, 1962.Google Scholar
  26. [26]
    Prigogine, I.,Etude Thermodynamique des Phenomenes Irreversibles, Desoer, Liege, 1947.Google Scholar
  27. [27]
    Ray, J.R., Am. J. Phys.47, 626 (1979).Google Scholar
  28. [28]
    Selinger, R.L., and G.B. Whitham, Proc. Roy. Soc.302A, 1 (1968).Google Scholar
  29. [29]
    Shiner, J.S., J. Chem. Phys.81, 1455 (1984).Google Scholar
  30. [30]
    Shiner, J.S., J. Chem. Phys.87, 1089 (1987).Google Scholar
  31. [31]
    Shiner, J.S., A Lagrangian Formulation of Chemical Reaction Dynamics and Mechano-Chemical Coupling, unpublished, 1989.Google Scholar
  32. [32]
    Shiner, J.S.,A Lagrangian Formulation of Chemical Reaction Dynamics Far From Equilibrium, inAdvances in Thermodynamics, vol. 6:Flow, Diffusion and Rate Processes, Taylor & Francis, New York, 1991, pp. 248–282.Google Scholar
  33. [33]
    Sieniutycz, S., Chem. Engng. Sci.44, 727 (1989).Google Scholar
  34. [34]
    Sieniutycz, S., Chem. Engng. Sci.42, 2697 (1987).Google Scholar
  35. [35]
    Sieniutycz, S., Thermal Momentum, Heat Inertia and a Macroscopic Extension of de Broglie Micro-Thermodynamics I. The Multicomponent Fluids with Sourceless Continuity Constraints, inAdvances in Thermodynamics, vol. 3:Nonequilibrium Theory and Extremum Principles, Taylor & Francis, New York, 1990, pp. 328–368.Google Scholar
  36. [36]
    Sieniutycz, S., unpublished, 1990.Google Scholar
  37. [37]
    Sieniutycz, S., Thermal Momentum, Heat Inertia and a Macroscopic Extension of de Broglie Micro-Thermodynamics II. The Conservation Laws for Continuity Equations with Sources, inAdvances in Thermodynamics, vol. 7:Extended Thermodynamic Systems, Taylor & Francis, New York, 1991, pp. 408–447.Google Scholar
  38. [38]
    Sieniutycz, S., and R.S. Berry, Phys. Rev. A40, 348 (1989).Google Scholar
  39. [39]
    Sieniutycz, S., and R.S. Berry, Phys. Rev. A43, 2807 (1991).Google Scholar
  40. [40]
    Sieniutycz, S., and P. Salamon, Diversity of Nonequilibrium Theories and Extremum Principles, inAdvances in Thermodynamics, vol. 3:Nonequilibrium Theory and Extremum Principles, Taylor & Francis, New York, 1990, pp. 1–38.Google Scholar
  41. [41]
    Sieniutycz, S., and P. Salamon, Thermodynamics and Optimization, inAdvances in Thermodynamics, vol. 4:Finite-Time Thermodynamics and Thermoeconomics, Taylor & Francis, New York, 1990, pp. 1–21.Google Scholar
  42. [42]
    Sieniutycz, S., and J.S. Shiner, Action Variables, Inertial Terms and Lagrangian Thermodynamics of Chemical Networks, in preparation for J. Non-Equilib. Thermodyn., 1991.Google Scholar
  43. [43]
    Sieniutycz, S., and J.S. Shiner, Variational and Extremum Properties of Homogeneous Chemical Kinetics. II. Minimum Dissipation Approaches, part II of this series, 1992.Google Scholar
  44. [44]
    Vujanovic, B., and S.E. Jones,Variational Methods in Nonconservative Phenomena, Academic Press, New York, 1988.Google Scholar
  45. [45]
    Wyatt, J.L., Computer Programs in Biomedicine8, 180 (1978).Google Scholar

Copyright information

© Nicholas Copernicus University Press 1992

Authors and Affiliations

  • S. Sieniutycz
    • 1
  • J. S. Shiner
    • 1
  1. 1.Physiologisches InstitutUniversität BernBernSwitzerland

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