Skip to main content

Caratheodory and the Foundations of Thermodynamics and Statistical Physics

Abstract

Constantin Caratheodory offered the first systematic and contradiction free formulation of thermodynamics on the basis of his mathematical work on Pfaff forms. Moreover, his work on measure theory provided the basis for later improved formulations of thermodynamics and physics of continua where extensive variables are measures and intensive variables are densities. Caratheodory was the first to see that measure theory and not topology is the natural tool to understand the difficulties (ergodicity, approach to equilibrium, irreversibility) in the Foundations of Statistical Physics. He gave a measure-theoretic proof of Poincaré's recurrence theorem in 1919. This work paved the way for Birkhoff to identify later ergodicity as metric transitivity and for Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert spaces. Mixing provided an explanation of the approach to equilibrium but not of irreversibility. The recent extension of spectral theory of dynamical systems to locally convex spaces, achieved by the Brussels–Austin groups, gives new nontrivial time asymmetric spectral decompositions for unstable and/or non-integrable systems. In this way irreversibility is resolved in a natural way.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. E. Spandagos, The Life and Works of Constantine Caratheodory (Aithra, Athens, 2000).

    Google Scholar 

  2. D. Caratheodory-Rodopoulou and D. Vlahostergiou-Vasbateki, Constantine Caratheodory: The Wise Greek of Munich (Athens, 2000). The picture on p. 242 refers to the 2nd Solvay conference held in Brussels in 1913.

  3. Constantine Caratheodory: 125 Years from His Birth, special issue (Aristoteles University of Thessaloniki, 1999).

  4. Th. Rassias, ed., C. Caratheodory: An International Tribute (World Scientific, Singapore, 1991).

    Google Scholar 

  5. A. Sommerfeld, Thermodynamics and Statistical Mechanics (Academic, New York, 1955).

    Google Scholar 

  6. M. Zemansky, Heat and Thermodynamics, 5th edn. (McGraw–Hill, London, 1968).

    Google Scholar 

  7. A. Pippard, The Elements of Classical Thermodynamics (Cambridge University Press, London, 1997).

    Google Scholar 

  8. H. Callen, Thermodynamics (John Wiley, New York, 1960).

    Google Scholar 

  9. C. Caratheodory, “Untersuchungen über die Grundlagen der Thermodynamik,“ Math. Ann. 67, 355 (1909).

    Google Scholar 

  10. C. Caratheodory, Sitzber Preuss. Acad. Wiss. Physik-Math. Kl. 39 (1925).

  11. P. Rastal, “Classical thermodynamics simplified,” J. Math. Phys. 11, 2955–2965 (1970).

    Google Scholar 

  12. F. Weinhold, “Thermodynamics and geometry,” Physics Today 3, 23–30 (1976).

    Google Scholar 

  13. M. Peterson, “Analogy between thermodynamics and mechanics,” Am. J. Phys. 47, 488–490 (1979).

    Google Scholar 

  14. P. Landsberg, Thermodynamics (Interscience, New York, 1961).

    Google Scholar 

  15. G. Giannakopoulos, Chemical Thermodynamics (University of Athens, 1974).

  16. O. Redlich, “Fundamental thermodynamics since Caratheodory,” Rev. Mod. Phys. 40, 556–563 (1968).

    Google Scholar 

  17. C. Caratheodory, Vorlesungen uber reelle Functionen (Teubner, Leipzig, 1918, 1927).

    Google Scholar 

  18. C. Caratheodory, Algebraic Theory of Measure and Integration, 2nd edn. (Chelsea, New York, 1986).

    Google Scholar 

  19. H. Royden, Real Analysis, 3rd edn. (McMillan, New York, 1988). Caratheodory's way to define measurable sets is considered (p. 58) to be the best. A clear discussion of Caratheodory's extension theorem is presented in p. 295.

    Google Scholar 

  20. M. Gurtin and W. Williams, “An axiomatic formulation for continuous thermodynamics,” Arch. Rat. Mech. Anal. 26, 83–117 (1967).

    Google Scholar 

  21. M. Gurtin, W. Williams, and W. Ziemer, “Geometric measure theory and the axioms of continuum thermodynamics,” Arch. Rat. Mech. Anal. 92, 1–22 (1986).

    Google Scholar 

  22. I. Antoniou and Z. Suchanecki, “Densities of singular measures and generalized spectral decomposition,” in Generalized Functions, Operator Theory, and Dynamical Systems, Res. Not. Math. 399, I. Antoniou and G. Lumer, eds. (Chapman &; Hall/CCR, 1999), pp. 56–67.

  23. C. Truesdell, The Elements of Continuum Mechanics (Springer, Berlin, 1966).

    Google Scholar 

  24. A. Bressan, Relativistic Theories of Materials (Springer, Berlin, 1978).

    Google Scholar 

  25. J. Serrin, ed., New Perspectives on Thermodynamics (Springer, Berlin, 1986).

    Google Scholar 

  26. E. Schroedinger, What is Life? (Cambridge University Press, Cambridge, 1944).

    Google Scholar 

  27. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Wiley, New York, 1967).

    Google Scholar 

  28. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations(Wiley, New York, 1971).

    Google Scholar 

  29. I. Prigogine, From Being to Becoming (Freeman, New York, 1981).

    Google Scholar 

  30. D. Kondepudi and I. Prigogine, “sensitivity of non-equilibrium systems,” Physica A 107, 1–24 (1981).

    Google Scholar 

  31. D. Kondepudi, “sensitivity of chemical dissipative structures to external fields: Formation of propagating bands,” Physica A 115, 552–566 (1982).

    Google Scholar 

  32. C. Papaseit, N. Pochon, and J. Tabony, “Microtubule self-organization is gravity-dependent,” Proc. Natl. Acad. Sci. USA 97, 8364–8368 (2000).

    Google Scholar 

  33. L. Boltzmann, J. Reine Angew. Math. 100, 201 (1887).

    Google Scholar 

  34. L. Boltzmann, Lectures on Gas Theory (English translation of the original, 1896, 1898; University of California Press, Berkeley, 1964; Dover reprint, New York, 1995).

    Google Scholar 

  35. I. Farquhar, Ergodic Theory in Statistical Mechanics (Wiley, New York, 1964).

    Google Scholar 

  36. S. Brush, The Kind of Motion we Call Heat: A History of the Kinetic Theory of Gases (North-Holland, Amsterdam, 1976).

    Google Scholar 

  37. M. Plancherel, Arch. Sci. Phys. 33, 254 (1912); iAnn. Phys. 42, 1061–1063 (1913).

    Google Scholar 

  38. A. Rosenthal, Ann. Phys. 42, 796–806; 43, 894–904 (1913).

    Google Scholar 

  39. L. Boltzmann, Akad. Wiss. Wien 66, 275–370 (1872).

    Google Scholar 

  40. J. Loschmidt, Akad. Wiss. Wien 73, 128–142 (1876).

    Google Scholar 

  41. E. Zermelo, Ann. Phys. 57, 485–494 (1896).

    Google Scholar 

  42. H. Poincaré, C. R. Acad. Sci. 108, 550–553 (1889); Acta Math. 13, 1 (1890).

    Google Scholar 

  43. C. Caratheodory, Sitz. Preuss. Akad. Wiss. Phys. Math. 580–584 (1919).

  44. J. Gibbs, Elementary Principles of Statistical Mechanics (Yale University Press, 1902; Dover reprint, New York, 1960).

  45. G. Birkhoff, Proc. Natl. Acad. Sci. USA 17, 650–660 (1931).

    Google Scholar 

  46. I. Cornfeld, S. Fomin, and Ya. Sinai, Ergodic Theory (Springer, Berlin, 1982).

    Google Scholar 

  47. J. von Neumann, Ann. Math. 33, 587 (1932).

    Google Scholar 

  48. E. Hopf, J. Math. and Phys. 13, 51–102 (1934).

    Google Scholar 

  49. G. Birkhoff and B. Koopman, Proc. Natl. Acad. Sci. USA 18, 279–282 (1932).

    Google Scholar 

  50. M. Stone, Linear Transformations in Hilbert Space and their Applications in Analysis (American Mathematical Society, New York, 1932).

    Google Scholar 

  51. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Springer, 1932; English translation, Princeton University Press, New Jersey, 1955).

    Google Scholar 

  52. B. Koopman, “Hamiltonian systems and transformations in Hilbert spaces,” Proc. Nat. Acad. Sci. USA 17, 315–318 (1931).

    Google Scholar 

  53. R. Goodrich, K. Gustafson, and B. Misra, “On a converse to Koopman Lemma and irreversibility,” J. Statist. Phys. 43, 317–320 (1986).

    Google Scholar 

  54. M. Nadkarni, Spectral Theory of Dynamical Systems (Birkhäuser, Basel, Switzerland, 1998).

    Google Scholar 

  55. B. Koopman and J. von Neumann, “Dynamical systems of continuous spectra,” Proc. Nat. Acad. Sci. USA 18, 255–266 (1932).

    Google Scholar 

  56. I. Antoniou and Z. Suchanecki, in Evolution Equations and their Applications in Physical and Life Sciences, G. Lumer and L. Weiss, eds. (Marcel Dekker, New York, 2001), pp. 301–310.

    Google Scholar 

  57. I. Antoniou and S. A. Shkarin, “Decay spectrum and decay subspace of normal operators,” Proc. Roy. Edinb. Soc. 131A, 1245–1255 (2001).

    Google Scholar 

  58. I. Antoniou and S. Shkarin, “Decaying measures,” Russ. Math. Doclady 61, 24–27 (2000); Proc. Roy. Edinb. Soc. 131A, 1257–1273 (2001).

    Google Scholar 

  59. A. Lasota and M. Mackey, Chaos, Fractals, and Noise (Springer, New York, 1994).

    Google Scholar 

  60. N. S. Krylov, Works on the Foundations of Statistical Physics (Princeton University Press, 1979).

  61. B. Misra and I. Prigogine, “Time probability and dynamics,” in Long Time Predictions in Dynamical Systems, C. Horton, L. Reichl, and V. Szebehely, eds. (Wiley, New York, 1983), pp. 21–43.

    Google Scholar 

  62. B. Misra, “Nonequilibrium entropy, Lyapunov variables, and ergodic properties of classical systems,” Proc. Natl. Acad. USA 75, 1627–1631 (1978).

    Google Scholar 

  63. I. Antoniou, “Internal Time and Irreversibility of Relativistic Dynamical Systems,” Thesis (Free University of Brussels, 1988).

  64. I. Antoniou and B. Misra, “Non-unitary transformation of conservative to dissipative evolutions,” J. Phys. A Math. Gen. 24, 2723–2729 (1991).

    Google Scholar 

  65. I. Antoniou and B. Misra “Relativistic internal time operator,” Int. J. Theor. Phys. 31, 119–136 (1992).

    Google Scholar 

  66. O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. I, II (Springer, New York, 1979, 1981).

    Google Scholar 

  67. B. Misra, I. Prigogine, and M. Courbage, “Liapunov variable: entropy and measurements in quantum mechanics,” Proc. Natl. Acad. Sci. USA 76, 4768–4772 (1979).

    Google Scholar 

  68. M. Courbage, “On necessary and sufficient conditions for the existence of time and entropy operators in quantum mechanics,” Lett. Math. Phys. 4, 425–432 (1980).

    Google Scholar 

  69. C.M. Lockhart and B. Misra, “Irreversibility and measurement in quantum mechanics,” Physica A 136, 47–76 (1986).

    Google Scholar 

  70. G. Ordó ñez, T. Petrosky, E. Karpov, and I. Prigogine, “Explicit construction of a time superoperator for quantum unstable systems,” Chaos Solitons and Fractals 12, 2591–2601 (2001).

    Google Scholar 

  71. I. Prigogine, The End of Certainty (Free Press, New York, 1997).

    Google Scholar 

  72. I. Antoniou and S. Tasaki, “Generalized spectral decomposition of mixing dynamical systems,” Int. J. Quantum Chemistry 46, 425–474 (1993).

    Google Scholar 

  73. I. Antoniou and S. Shkarin, “Extended spectral decomposition of evolution operators,” in Generalized Functions, Operator Theory and Dynamical Systems, Research Notes in Mathematics 399 (Chapman and Hall/CRC, London, 1999).

    Google Scholar 

  74. I. Antoniou and I. Prigogine, “Intrinsic irreversibility and integrability of dynamics,” Physica A 192, 443–464 (1993).

    Google Scholar 

  75. A. Bohm, Quantum Mechanics, Foundations and Applications, 3rd edn. (Springer, Berlin, 1993).

    Google Scholar 

  76. A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors and Gelfand Triplets, Lecture Notes on Physics 348 (Springer, Berlin, 1989).

    Google Scholar 

  77. I. Antoniou and Z. Suchanecki, “The fuzzy logic of chaos and probabilistic inference,” Found. Phys. 27, 333–362 (1997).

    Google Scholar 

  78. I. Antoniou and Z. Suchanecki, “Logics associated with complex systems,” in Proceedings of the First Panhellenic Symposium on Logic, A. Kakas and A. Synachopoulou, eds. (University of Cyprus editions, ISBN 9963-607-11-X, Nicosia, 1997), pp. 261–274.

  79. J. Lighthill, “The recently recognized failure of predictability in Newtonian dynamics,” Proc. Roy. Soc. London A407, 35–50 (1986).

    Google Scholar 

  80. T. Bedford, N. Keane, and C. Series, Ergodic Theory Symbolic Dynamics and Hyperbolic Spaces (Oxford University Press, New York, 1991).

    Google Scholar 

  81. A. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957).

    Google Scholar 

  82. Y. Kakihara, Abstract Methods in Information Theory (World Scientific, Singapore, 1999).

    Google Scholar 

  83. I. Antoniou, F. Bosco, and Z. Suchanecki, “spectral decomposition of expanding probabilistic dynamical systems,” Phys. Lett. A 239, 153–158 (1998).

    Google Scholar 

  84. I. Antoniou and F. Bosco, “spectral decomposition of contracting probabilistic dynamical systems,” Chaos, Solitons and Fractals 9, 401–418 (1998).

    Google Scholar 

  85. I. Antoniou and F. Bosco, “On the spectral properties of a Markov model for learning processes,” J. Mod. Phys. C 11, 213–220 (2000).

    Google Scholar 

  86. I. Antoniou, V. Basios, and F. Bosco, “Probabilistic control of chaos: chaotic maps under control,” Computers Math. Applic. 34, 373–389 (1997).

    Google Scholar 

  87. I. Antoniou, V. Basios, and F. Bosco, “Absolute controllability condition for probabilistic control of chaos,” J. Bifurcation Chaos 8, 409–413 (1998).

    Google Scholar 

  88. I. Antoniou and Z. Suchanecki, “Non-uniform Time operator, chaos and wavelets on the interval,” Chaos, Solitons and Fractals 11, 423–435 (2000).

    Google Scholar 

  89. I. Antoniou, V. Sadovnichii, and S. Shkarin, “Time operators and shift representation of dynamical systems,” Physica A 299, 299–313 (1999).

    Google Scholar 

  90. I. Antoniou, I. Prigogine, V. Sadovnichii, and S. Shkarin “Time operator for diffusion,” Chaos, Solitons and Fractals 11, 465–477 (2000).

    Google Scholar 

  91. E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Comm. Pure Appl. Math. 13, 1–14 (1960).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Antoniou, I.E. Caratheodory and the Foundations of Thermodynamics and Statistical Physics. Foundations of Physics 32, 627–641 (2002). https://doi.org/10.1023/A:1015040501205

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1015040501205

  • Caratheodory
  • thermodynamics
  • statistical physics
  • irreversibility
  • dynamical dystems
  • spectral analysis