Foundations of Physics

, Volume 34, Issue 1, pp 21–57 | Cite as

Time Evolution in Macroscopic Systems. II. The Entropy

  • W. T. GrandyJr.


The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined.

nonequilibrium thermodynamics time-dependent entropy 


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  1. 1.
    R. Clausius, “Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanische Wärmetheorie,” Ann. Phys. [2] 125, 390(1865).Google Scholar
  2. 2.
    L. Boltzmann, “On certain questions of the theory of gases,” Nature 51, 413(1895).Google Scholar
  3. 3.
    J. W. Gibbs, Elementary Principles in Statistical Mechanics (Yale University Press, New Haven, CT, 1902).Google Scholar
  4. 4.
    W. T. Grandy, Jr., “Time evolution in macroscopic systems. I: Equations of motion,” Found. Phys. 341(2004), foregoing paper.Google Scholar
  5. 5.
    M. Berry, “Singular limits,” Physics Today 55 (5), 10(2002).Google Scholar
  6. 6.
    E. T. Jaynes, “Information theory and statistical mechanics”, in Statistical Physics, K. W. Ford, ed. (Benjamin, New York, 1963).Google Scholar
  7. 7.
    E. T. Jaynes, “Foundations of probability theory and statistical mechanics”, in Delaware Seminar in the Foundations of Physics, M. Bunge, ed. (Springer, New York, 1967).Google Scholar
  8. 8.
    E. T. Jaynes, “Where do we stand on maximum entropy?”, in The Maximum Entropy Formalism, R. D. Levine and M. Tribus, eds. (M.I.T. Press, Cambridge, MA, 1979).Google Scholar
  9. 9.
    R. Evans, “The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids,” Adv. Phys. 28, 143(1979).Google Scholar
  10. 10.
    W. T. Grandy, Jr., Foundations of Statistical Mechanics, Vol. II: Nonequilibrium Phenomena (Reidel, Dordrecht, 1988).Google Scholar
  11. 11.
    U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74(1957).Google Scholar
  12. 12.
    S. Nakajima, “On quantum theory of transport phenomena,” Prog. Theor. Phys. 20, 948(1958).Google Scholar
  13. 13.
    R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II (Springer, Berlin, 1985).Google Scholar
  14. 14.
    S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics (North-Holland, Amsterdam, 1962).Google Scholar
  15. 15.
    L. Onsager, “Reciprocal relations in irreversible processes. I,” Phys. Rev. 37, 405(1931).Google Scholar
  16. 16.
    W. C. Mitchell, “Statistical mechanics of thermally driven Systems,” Ph.D. Thesis (Washington University, Saint Louis, 1967).Google Scholar
  17. 17.
    W. T. Grandy, Jr., Foundations of Statistical Mechanics, Vol. 1: Equilibrium Theory (Reidel, Dordrecht, 1987).Google Scholar
  18. 18.
    R. J. Tykodi, Thermodynamics of Steady States (Macmillan, New York, 1967).Google Scholar
  19. 19.
    A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971).Google Scholar
  20. 20.
    R. D. Puff and N. S. Gillis, “Fluctuations and transport properties of many-particle systems,” Ann. Phys. (N.Y.) 6, 364(1968).Google Scholar
  21. 21.
    P. Palffy-Muhory, “Comment on ‘A check of Prigogine's theorem of minimum entropy production in a rod in a nonequilibrium stationary state’; by Irena Danielewicz-Ferchmin and A. Ryszard Ferchmin [Am. J. Phys. 68(10), 962-965 (2000)]”, Am. J. Phys. 69, 825(2001).Google Scholar
  22. 22.
    S. P. Heims and E. T. Jaynes, “Theory of gyromagnetic effects and some related magnetic phenomena,” Rev.Mod.Phys. 34, 143 (1962).Google Scholar
  23. 23.
    W. T. Grandy, Jr., “Time evolution in macroscopic systems.III.Selected applications,” Found.Phys. (to appear)Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • W. T. GrandyJr.
    • 1
  1. 1.Department of Physics & AstronomyUniversity of WyomingLaramie

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