Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Thermodynamic potentials and evolution towards the stationary state in open systems of far-from-equilibrium chemical reactions: The affinity squared minimum function

  • 59 Accesses

  • 2 Citations


The “evolutionary” characteristics of several quadratic functions (based both on the affinities and on the reaction velocities), and of the entropy production per unit time, have been studied for a number of 2-variable open systems of far-from-equilibrium chemical reactions. The first and second order systems were chosen to include: straight line, loop (network), autocatalytic and disproportionate kinetic features. All of the functions examined are closely related in form to the entropy production though they differ qualitatively and quantitatively in the nonlinear domain. By a combination of analytical and computational methods one function, called ℳ, is seen to have the variational properties of a “thermodynamic potential” for all of the systems, relative to their non-equilibrium stationary states. The “homeostatic-like” stability criterion, d ℳ/dt ≤0 is also seen to hold for these systems. The function, a composite property of the system, may be interpreted as a “kinetically-weighted system-free-energy” quantity, which would tend to be minimized during the evolution, in configuration and in time space, of a constrained biochemical system.

This is a preview of subscription content, log in to check access.


  1. [1]

    De Donder, Th.: L'Affinité, Paris: Gauthier-Villars 1927; De Donder, Th., Van Rysselberghe, P.: Affinity. Stanford, Calif.: Stanford Univ. Press 1936.

  2. [2]

    Finlayson, B. A., Scriven, L. E.: Int. J. Heat Mass Transfer 10, 799 (1967).

  3. [3]

    Gardner, M. R., Ashby, W. R.: Nature 228, 784 (1970).

  4. [4]

    Glansdorff, P., Prigogine, I.: Physica 20, 773 (1954).

  5. [5]

    Goodwin, B. C.: Temporal Organization in Cells. London: Academic Press 1963.

  6. [6]

    Katchalsky, A., Curran, P. F.: Nonequilibrium Thermodynamics in Biophysics. Cambridge, Mass.: Harvard University Press 1965.

  7. [7]

    Klein, M. J.: in: Proc. of Int. Symp. on Transport Processes in Statistical Mechanics. Prigogine, I. (ed.). p. 311. New York: Interscience 1958.

  8. [8]

    Mel, H. C.: Bull. Acad. roy. Belgique Cl. Sc. 40, 834 (1954).

  9. [9]

    Oster, G., Desoer, C. A: Univ. of Calif., Lawrence Radiation Lab. UCRL-19421. Nov. 1969.

  10. [10]

    Oster, G., Perelson, A., Katchalsky, A.: Nature 234, 393 (1971).

  11. [11]

    Pardee, A. B.: in: Proc. of Symp. in Applied Mathematics 14 (R. E. Bellman, ed.), p. 69. Providence, R. I.: Amer. Math. Soc. 1962.

  12. [12]

    Payne, W. J.: Ann. Rev. Microbiol. 24, 17 (1970).

  13. [13]

    Prigogine, I.: Etude Thermodynamique des Phénomènes Irreversibles. Liege: Desoer 1947.

  14. [14]

    Prigogine, I.: Introduction to Thermodynamics of Irreversible Processes, Ch. 7. New York: Interscience 1967.

  15. [15]

    Tykodi, R. J.: Thermodynamics of Steady States, Ch. 15. New York: Macmillan 1967.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mel, H.C., Ewald, D.A. Thermodynamic potentials and evolution towards the stationary state in open systems of far-from-equilibrium chemical reactions: The affinity squared minimum function. J. Math. Biology 1, 133–151 (1974). https://doi.org/10.1007/BF00275799

Download citation


  • Stationary State
  • Entropy Production
  • Velocity Potential
  • Thermodynamic Potential
  • Minimum Function