Czechoslovak Journal of Physics B

, Volume 30, Issue 8, pp 841–861 | Cite as

On measures, convex cones, and foundations of thermodynamics I. Systems with vector-valued actions

  • M. šilhavý
Article
Received:

Abstract

In this two-part paper, a theory of non-equilibrium thermodynamic systems (“with memory”) is developed. The emphasis is laid upon the possibility of presenting the non-equilibrium thermodynamics deductively starting from the basic laws in a form which is capable of a direct experimental verification. This first part introduces the concept of a system with a vector-valued action which underlies the concept of a thermodynamic system (introduced in the second part). A special case of a vector-valued action is a real-valued action; the theory of real-valued actions with the Clausius property due to Coleman and Owen is briefly sketched. A subintegrating functional for a vector-valued action is a non-zero linear functional whose composition with the action gives a real-valued action with the Clausius property. For special, actions which have relevance to thermodynamics the existence of a subintegrating functional with some extra properties is equivalent to the existence of the absolute temperature scale or to the existence of the mechanical equivalent of a unit of heat. The main purpose of the first part of the paper is to give conditions for the existence of a subintegrating functional which is positive on a given set of vectors. These conditions will be shown in the second part to be the abstract analogues of the verbal statements of the laws of thermodynamics.

Keywords

Temperature Scale Absolute Temperature Extra Property Experimental Verification Verbal Statement 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Fermi E., Thermodynamics. Prentice-Hall, New York 1937.Google Scholar
  2. [2]
    Sommerfeld A., Thermodynamik und Statistik. Wiesbaden 1952.Google Scholar
  3. [3]
    Kvasnica J., Termodynamika. SNTL, Praha 1965.Google Scholar
  4. [4]
    Born M., Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik. Physik. Zschr.22 (1920), 218, 249, 282.Google Scholar
  5. [5]
    Buchdahl H. A., A formal treatment of the consequences of the second law of thermodynamics in Carathéodory's formulation. Z. Phys.152 (1958), 425.Google Scholar
  6. [6]
    Giles R., Mathematical foundations of thermodynamics. Pergamon, London 1964.Google Scholar
  7. [7]
    Serrin J., Foundations of classical thermodynamics. Lecture notes. University of Chicago, Chicago 1975.Google Scholar
  8. [8]
    Truesdell C., Tragicomedy of classical thermodynamics. Springer, Berlin-Heidelberg-New York 1971.Google Scholar
  9. [9]
    Truesdell C., Rational thermodynamics. Mc Graw-Hill, New York 1969.Google Scholar
  10. [10]
    Truesdell C., Absolute temperatures as a consequence of Carnot's general axiom. Arch. Hist. Exact Sci.20 (1979), 357.Google Scholar
  11. [11]
    Truesdell C., Bharatha S., The concepts and logic of classical thermodynamics as a theory of heat engines. Springer, New York-Heidelberg-Berlin 1977.Google Scholar
  12. [12]
    Perzyna P., Thermodynamic theory of viscoplasticity. Advan. Appl. Mech.11 (1971), 313.Google Scholar
  13. [13]
    Hutter K., The foundations of thermodynamics, its basic postulates and implications. A review of modern thermodynamics. Acta Mechanica27 (1977), 1.Google Scholar
  14. [14]
    Meixner J., Processes in simple thermodynamic materials. Arch. Rational Mech. Anal.33 (1969), 33.Google Scholar
  15. [15]
    Planck M., Wissenschaftliche Selbstbiographie. Johan Ambrosius Barth, Leipzig 1970.Google Scholar
  16. [16]
    Carathéodory C., Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann.67 (1909), 355.Google Scholar
  17. [17]
    Boyling J. B., Carathéodory's principle and the existence of global integrating factors. Commun. Math. Phys.10 (1968), 52.Google Scholar
  18. [18]
    Boyling J. B., An axiomatic approach to classical thermodynamics. Proc. R. Soc. Lond. A329 (1972), 35.Google Scholar
  19. [19]
    Boyling J. B., Thermodynamics of non-differentiable systems. Internat. J. Theoret. Phys.9 (1974), 379.Google Scholar
  20. [20]
    Rastall P., Classical thermodynamics simplified. J. Math. Phys.11 (197), 2955.Google Scholar
  21. [21]
    Bree J., Beevers C. E., Non-equilibrium thermodynamics of continuous media. J. Non-Equilib. Thermodyn.4 (1979), 159.Google Scholar
  22. [22]
    Coleman B. D., Noll W., The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal.13 (1963), 167.Google Scholar
  23. [23]
    Coleman B. D., Thermodynamics of materials with memory. Arch. Rational Mech. Anal.17 (1964), 1.Google Scholar
  24. [24]
    Day W. A., The thermodynamics of simple materials with fading memory. Springer, Berlin-Heidelberg-New York 1972.Google Scholar
  25. [25]
    Coleman B. D., Owen D. R., A mathematical foundation for thermodynamics. Arch. Rational Mech. Anal.54 (1974), 1.Google Scholar
  26. [26]
    Fosdick R., Serrin J., Global properties of continuum thermodynamic processes. Arch. Rational Mech. Anal.59 (1975), 97.Google Scholar
  27. [27]
    Day W. A., A theory of thermodynamics for materials with memory. Arch. Rational Mech. Anal.34 (1969), 85.Google Scholar
  28. [28]
    Day W. A., Entropy and hidden variables in continuum thermodynamics. Arch. Rational Mech. Anal.62 (1976), 367.Google Scholar
  29. [29]
    Coleman B. D., Owen D. R., Thermodynamics and elastic-plastic materials. Arch. Rational Mech. Anal.59 (1975), 25.Google Scholar
  30. [30]
    Coleman B. D.,Owen D. R., On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses. To appear in Arch. Rational Mech. Anal.Google Scholar
  31. [31]
    Müller I., Entropy in non-equilibrium — A challenge to mathematicians. In Trends in application of pure mathematics to mechanic. Vol. 2. (Ed. H. Zorski.) Pitman Publ. London, San Francisco, Melbourne 1979.Google Scholar
  32. [32]
    Serrin J., The concepts of thermodynamics. In Contemporary developments in Continuum mechanics and partial differential equations. (Ed. G. M. de La Penha, and L. A. Medeiros). North-Holland, Amsterdam 1978.Google Scholar
  33. [33]
    šilhavý M., A condition equivalent to the existence of non-equilibrium entropy and temperature for materials with internal variables. Arch. Rational Mech. Anal.68 (1978), 299.Google Scholar
  34. [34]
    šilhavý M., On the Clausius inequality. Submitted to Arch. Rational Mech. Anal.Google Scholar
  35. [35]
    Gurtin M. E., Williams W. O., An axiomatic foundation for continuum thermodynamics. Arch. Rational Mech. Anal.26 (1967), 83.Google Scholar
  36. [36]
    Müller I., On the entropy inequality. Arch. Rational Mech. Anal.26 (1967), 118.Google Scholar
  37. [37]
    Müller I., The coldness, a universal function in thermoeleastic bodies. Arch. Rational Mech. Anal.41 (1971), 319.Google Scholar
  38. [38]
    Meixner J., Thermodynamik der Vorgänge in einfachen fluiden Medien und die Charakterisierung der Thermodynamik irreversibler Prozesse. Z. Physik219 (1969), 79.Google Scholar
  39. [39]
    Wilmański K., On thermodynamics and functions of states of nonisolated systems. Arch. Rational Mech. Anal.45 (1972), 251.Google Scholar
  40. [40]
    Wilmański K., Podstawy termodynamiki fenomenologicznej. Panstwowe wydawnictwo naukowe, Warszawa 1974.Google Scholar
  41. [41]
    Day W. A., Continuum thermodynamics based on a notion of rate of loss of information. Arch. Rational Mech. Anal.59 (1975), 53.Google Scholar
  42. [42]
    Day W. A., An information theory approach to the symmetry, monotonicity and concavity of the creep function in linear viscoelasticity. Arch. Rational Mech. Anal.65 (1977), 47.Google Scholar
  43. [43]
    Beevers C. E., Bree J., A thermodynamic theory of isotropic elastic-plastic materials. Archives of Mechanics31 (1979), 385.Google Scholar
  44. [44]
    Green A. E., Naghdi P. M., On thermodynamics and the nature of the second law. Proc. R. Soc. Lond. A357 (1977), 253.Google Scholar
  45. [45]
    šilhavý M., On measures, convex cones, and foundations of thermodynamics. II. Thermodynamic systems. Czech. J. Phys. B30 (1980), 961.Google Scholar
  46. [46]
    Noll W., A new mathematical theory of simple materials. Arch. Rational Mech. Anal.48 (1972), 1.Google Scholar
  47. [47]
    Gurtin M. E., Thermodynamics and stability. Arch. Rational Mech. Anal.59 (1975), 63.Google Scholar
  48. [48]
    Halmos P. R., Measure theory. Van Nostrand, New York 1950.Google Scholar
  49. [49]
    Day M. M., Normed linear spaces. Springer, Berlin 1962.Google Scholar

Copyright information

© Academia, Publishing House of the Czechoslovak Academy of Sciences 1980

Authors and Affiliations

  • M. šilhavý
    • 1
  1. 1.Mathematical InstituteCzechosl. Acad. Sci., PraguePraha 1Czechoslovakia

Personalised recommendations