Time in Statistical Physics and Special Relativity

  • P. T. Landsberg


The effort to meet the Society’s request for a talk on this topic has been a humbling experience. The literature is vast; it contains the names of the ancient philosophers, of Newton, Leibniz, Bergson, Planck, Einstein, and of many contemporary men of learning. Yet some deep doubts and differences remain which, in the case of statistical physics, may be at least hinted at by quotations which are at variance with each other (Appendices A, B). A survey of this field is therefore out of the question (but see [5]). Here it will be easier for all concerned if, instead, I present a point of view.


Special Relativity Inertial Frame World Line Entropy Increase Thermodynamic Entropy 
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  1. 1.
    Ageno, M.: Does quantum mechanics exclude life. Nature 205 (1965) 1306–1307.CrossRefGoogle Scholar
  2. 2.
    Aron, J. C.: Quantum laws in connection with Stochastic Processes. Prog. Theoretical Physics 33 (1965) 726–754.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Balescu, R.: Velocity inversion in Statistical mechanics. Physica 36 (1967) 433–456.CrossRefGoogle Scholar
  4. 4.
    Balescu, R.: Some comments on irreversibility. Phys. Letters 27 A (1968), 249.CrossRefGoogle Scholar
  5. 5.
    Beauregard, O. Costa de: Irreversibility problems. In: Logic. Methodology and Philosophy of Science. Ed. Y. Bar-Hillel, p. 313–342. Amsterdam: North Holland 1965.Google Scholar
  6. 6.
    Bergmann, P. G.: Foundations Research in Physics. In: Delaware Seminar in the Foundations of Physics (Ed. M. Bunge) p. 1–14. Berlin: Springer 1967.Google Scholar
  7. 7.
    Bilaniuk, O. M., Sudarshan, E. C. G.: Particles beyond the light barrier. Physics Today 22 (May, 1969) 43–51.CrossRefGoogle Scholar
  8. 8.
    Bilaniuk, O. M., Deshpande, V. K., Sudarshan, E. C. G.: “Meta” relativity. Am. J. Phys. 30 (1962) 718–723.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Blatt, J. M.: An alternative approach to the ergodic problem. Prog. Theoret. Phys. 22 (1959) 745–756.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Boltzmann, L.: Vorlesungen über Gastheorie (Leipzig: Barth 1896). English translation by S. G. Brush, Univ. of California Press 1964.Google Scholar
  11. 11.
    Bondi, H.: Physics and Cosmology. Observatory, 82 (1962) 133–143 (Halley Lecture).Google Scholar
  12. 12.
    Bopp, F.: Der Wechselwirkungsoperator im Gitterraum. Z. Phys. 200 (1967) 142–157.CrossRefGoogle Scholar
  13. 13.
    Born, M.: Die Quantenmechanik und der zweite Hauptsatz der Thermodynamik. Ann. Phys. (Lpz) 3 (1948) 107–114.MATHCrossRefGoogle Scholar
  14. 14.
    Born, M.: Continuity, Determinism, and Reality. K. Danske Vidensk. Selsk. mat.-fys. Medd. 30 (1955) No. 2.Google Scholar
  15. 15.
    Bridgman, P. W.: The Nature of Thermodynamics. Chapter III (Harper Torchbook, p. 213). Harvard 1941.Google Scholar
  16. 16.
    Büchel, W.: Entropy and information in the universe, Nature 213 (1967) 319–320.CrossRefGoogle Scholar
  17. 17.
    Caldirola, P. (Ed): Ergodic Theories. New York: Academic Press 1961.Google Scholar
  18. 18.
    Campbell, B.: Biological Entropy. Nature 215 (1967) 1308.CrossRefGoogle Scholar
  19. 19.
    Chandrasekhar, S.: Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys. 15 (1943) 1–89.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Cohn, J.: Special Relativity in the Quantum Domain. Am. J. Phys. 36 (1968) 749–751.CrossRefGoogle Scholar
  21. 21.
    Conklin, E. K.: Velocity of the earth with respect to the cosmic background radiation. Nature 222 (1969) 971–972.CrossRefGoogle Scholar
  22. 22.
    Darling, B. T.: The irreducible volume character of events. IA Theory of the elementary particles and of fundamental length. Phys. Rev. 80 (1950) 460–466.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Earman, J.: Irreversibility and temporal asymmetry. Journal of Philosophy 64 (1967) 543–549.CrossRefGoogle Scholar
  24. 24.
    Einstein, A.: Prinzipielles zur allgemeinen Relativitätstheorie. Ann. Phys. 55 (1918) 241–244.CrossRefGoogle Scholar
  25. 25.
    Feinberg, G.: Possibility of faster-than-light particles. Phys. Rev. 159 (1967) 1089–1105.CrossRefGoogle Scholar
  26. 26.
    Fock, V.: The theory of Space. Time and Gravitation. p. 4. 2nd revised ed. Pergamon 1964.Google Scholar
  27. 27.
    Foerster, H. von: Memory without record. In: The Anatomy of Memory. Ed. D. P. Kimble, Palo Alto 1965.Google Scholar
  28. 28.
    Fraser, J. T. (Ed.): The Voices of Time. New York: Braziller 1966.Google Scholar
  29. 29.
    Gale, R. M.: Why a cause cannot be later than its effect. Rev. of Metaphysics 19 (1965) 209–234.Google Scholar
  30. 30.
    Gold, T. (Ed.): The Nature of Time. Ithaca: Cornell Univ. Press 1967.Google Scholar
  31. 31.
    Gold, T. (Ed.): The arrow of time. 11th International Physics Congress, Solvay 1958. Am. J. Phys. 30 (1962) 403–410 (Richtmyer Lecture).Google Scholar
  32. 32.
    Grünbaum, A.: Modern Science and Zeno’s Paradoxes. London: Allen and Unwin 1968.MATHGoogle Scholar
  33. 33.
    Harrison, E. R.: The early universe. Physics Today 21 June, (1968) 31–39.CrossRefGoogle Scholar
  34. 34.
    Harrison, E. R.: Cosmology without general relativity. Annals of Physics 35 (1965) 437–446.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Harrison, E. R.: Quantum cosmology. Nature 215 (1967) 151–152.CrossRefGoogle Scholar
  36. 36.
    Hellund, E. J., Tanaka, K.: Quantized Space-Time. Phys. Rev. 94 (1954) 192–195.MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Hobson, A.: Irreversibility in Simple Systems. Am. J. Phys. 34 (1966) 411–416.CrossRefGoogle Scholar
  38. 38.
    Hobson, A.: Irreversibility in Finite Quantum-Statistical-Mechanical Systems. J. Chem. Phys. 46 (1967) 1365–1372.CrossRefGoogle Scholar
  39. 39.
    Hobson, A.: Further comments on irreversibility. Physics Lett. 28 A (1968) 183–184.CrossRefGoogle Scholar
  40. 40.
    Hoyle, F., Narlikar, J. V.: Electrodynamics of direct inter par tick action I The quantum mechanical response of the universe. Annals of Physics. 54 (1969) 207–239.MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Ingraham, R. L.: Stochastic space-time. Nuovo Cimento 34 (1964) 182–197.MathSciNetMATHCrossRefGoogle Scholar
  42. 43.
    Jaynes, E. T.: Gibbs vs Boltzmann Entropies. Am. J. Phys. 33 (1965) 391–398.MATHCrossRefGoogle Scholar
  43. 43.
    Kac, M.: Probability and Related Topics in Physical Sciences. New York: Interscience 1959.MATHGoogle Scholar
  44. 44.
    Kampen, N. G. van.: Fundamental problems in the statistical mechanics of irreversible processes. In: Fundamental Problems in Statistical Mechanics I, p. 173–202. (Ed. E. G. D. Cohen) Amsterdam: North Holland 1962.Google Scholar
  45. 45.
    Katz, A.: Principles of Statistical Mechanics. San Francisco: Freeman 1967.Google Scholar
  46. 46.
    Kretschmann, E.: Physical Significance of the relativity postulate. Einstein’s new and original theories of relativity. Ann. Phys. 53 (1918) 575–614.CrossRefGoogle Scholar
  47. 47.
    Kurth, R.: Axiomatic s of Classical Statistical Mechanics. New York: Pergamon 1960.Google Scholar
  48. 48.
    Landsberg, P. T.: The uncertainty principle as a problem in philosophy. Mind 56 (1947) 250–256.CrossRefGoogle Scholar
  49. 49.
    Landsberg, P. T.: On Bose-Einstein condensation. Proc. Camb. Phil. Soc. 50 (1954) 65–76.MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Landsberg, P. T.: Entropy and the Unity of Knowledge. Univ. of Wales Press, 1961 p. 18. Inaugural Lecture.MATHGoogle Scholar
  51. 51.
    Landsberg, P. T.: Thermodynamics with Quantum Statistical Illustrations. New York: Wiley 1961.MATHGoogle Scholar
  52. 52.
    Landsberg, P. T.: Two relativistic paradoxes. Mathematical Gazette 47 (1964) 197–202.CrossRefGoogle Scholar
  53. 53.
    Landsberg, P. T.: Does quantum mechanics exclude life? Nature 203 (1964) 928–930.CrossRefGoogle Scholar
  54. 54.
    Landsberg, P. T., Johns, K. A.: Energy flux from the 3°K radiation. Nature 220 (1968) 1120.CrossRefGoogle Scholar
  55. 55.
    Landsberg, P. T.: Time reversibility. In: Solid State Theory: Methodsand Applications. (Ed; P. T. Landsberg) New York: Wiley, 1969.Google Scholar
  56. 56.
    Landsberg, P. T.: Special Theory of Relativity. Nature 220 (1968) 1182–1183.CrossRefGoogle Scholar
  57. 57.
    Layzer, D.: A unified approach to cosmology. In: Lectures in Applied Mathematics. Vol. 8, p. 237 American Mathematical Society 1967; Ed. J. Ehlers.Google Scholar
  58. 58.
    Lewis, G. N.: The Symmetry of time in physics. Science 71 (1930) 569–577.CrossRefGoogle Scholar
  59. 59.
    Linden, J. van der, Mazur, P.: The asymptotic problem of statistical thermodynamics for a real system. Physica 36 (1967) 491–508.CrossRefGoogle Scholar
  60. 60.
    Marshall, T. W.: Random electrodynamics. Proc. Roy. Soc. A. 276 (1963) 475–491.MATHCrossRefGoogle Scholar
  61. 61.
    Marshall, T. W.: Statistical electrodynamics. Proc. Camb. Phil. Soc. 61 (1965) 537–546.MATHCrossRefGoogle Scholar
  62. 62.
    Mayer, J. E.: Approach to thermodynamic equilibrium. J. Chem. Phys. 34 (1961) 1207–1223.MathSciNetCrossRefGoogle Scholar
  63. 63.
    McCrea, W. H.: Relativity theory and the creation of matter. Proc. Roy. Soc. A206 (1951) 562–575.MathSciNetGoogle Scholar
  64. 64.
    Ono, S.: The long time behaviour of quantum mechanical systems. Mem. Fac. Eng. Kyushu Univ. 11 (1949) 125–140.Google Scholar
  65. 65.
    Orban, J., Bellemans, A.: Velocity-Inversion and irreversibility in a dilute gas of hard discs. Physics. Lett. 24A (1967) 620–621.CrossRefGoogle Scholar
  66. 66.
    Penrose, O., Percival, I. C: The direction of time. Proc. Phys. Soc. 79 (1962) 605–616.MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Planck, M.: The Theory of Heat Radiation. Dover, 2nd Ed. of 1912, published in 1959, 175.MATHGoogle Scholar
  68. 68.
    Pompe, W., Schöpf, H.-G.: Kosmologische und statistische Überlegungen zur Irreversibilität. Annalend. Phys. 21 (1968) 26–30.CrossRefGoogle Scholar
  69. 69.
    Popper, K.: Time’s arrow and entropy. Nature 207 (1965) 233–234.CrossRefGoogle Scholar
  70. 70.
    Popper, K.: Time’s arrow and feeding on negentropy. Nature 213 (1967) 320.CrossRefGoogle Scholar
  71. 71.
    Popper, K.: Structural information and the arrow of time. Nature 214 (1967) 322.CrossRefGoogle Scholar
  72. 72.
    Popper, K., Prigogine, I.: Non-equilibrium Statistical Mechanics. New York: Interscience 1962.Google Scholar
  73. 73.
    Popper, K.,: Dynamic Foundations of Thermodynamics and Statistical Mechanics. In: A Critical Review of Thermodynamics Ed. E. B. Stuart, B. Gal-Or, A. J. Brainard). Baltimore: Mono Book Corp. 1970Google Scholar
  74. 74.
    Reichenbach, H.: The Direction of Time. Berkely: Univ. of California Press 1956.Google Scholar
  75. 75.
    Rietdijk, C. W.: A rigorous proof of determinism derived from the special theory of relativity. Phil, of Science 33 (1966) 341–344.CrossRefGoogle Scholar
  76. 76.
    Schmidt, H.: Eine makroskopisch reale Quantentheorie. Zeit. für Natur, f. 18a (1963) 265–275.Google Scholar
  77. 77.
    Snyder, H. S.: Quantised space-time. Phys. Rev. 71 (1947) 38–41.MATHCrossRefGoogle Scholar
  78. 78.
    Snyder, H. S.: The electromagnetic field in quantised space-time. Phys. Rev. 72 (1947) 68–71.MATHCrossRefGoogle Scholar
  79. 79.
    Stannard, F. R.: Symmetry of the time axis. Nature 211 (1966) 693–694.CrossRefGoogle Scholar
  80. 80.
    Terletskii, Y. P.: Paradoxes in the theory of relativity. New York: Plenum 1968.Google Scholar
  81. 81.
    Tolman, R. C.: Relativity, Thermodynamics and Cosmology. Oxford 1934.Google Scholar
  82. 82.
    Treder, H. J.: Kosmische und subjektive Zeit. Forschungen und Fortschritte. 39 (1965) 325–327.Google Scholar
  83. 83.
    Treder, H. J.: (Ed.): Einstein Symposium. Berlin: Akademie Verlag 1966.MATHGoogle Scholar
  84. 84.
    Tribus, M.: Thermostatics and Thermodynamics. Princeton: van Nostrand 1961.Google Scholar
  85. 85.
    Uhlenbeck, G. E.: An outline of statistical mechanics. In: Fundamental Problems in Statistical Mechanics II p. 27. (Ed. E. G. D. Cohen). Amsterdam: North Holland 1968.Google Scholar
  86. 86.
    Verboven, E. J.: Quantum Thermodynamics of an infinite system of harmonic Oscillators. In: Statistical Mechanics (Ed. T. A. Bak), p. 49–54. New York: Benjamin 1967.Google Scholar
  87. 87.
    Viswanadham, C. R.: Entropy, evolution and living systems. Nature 219 (1968) 653.CrossRefGoogle Scholar
  88. 88.
    Weizsäcker, C. F. von: Der zweite Hauptsatz und der Unterschied von Vergangenheit und Zukunft. Annalen d. Phys. 36 (1939) 275–283.MATHCrossRefGoogle Scholar
  89. 89.
    Weyer, E. M. (Ed.): Interdisciplinary Perspectives of Time. Annals of the New York Academy of Sciences 138 (1967) 367–915.Google Scholar
  90. 90.
    Whitrow, G. J.: The Natural Philosophy of Time. London: Nelson 1961.MATHGoogle Scholar
  91. 91.
    Whittaker, E. T.: From Euclid to Eddington. Cambridge: Univ. Press 1949.MATHGoogle Scholar
  92. 92.
    Wigner, E. P.: The probability of the existence of a self-reproducing unit. In: The Logic of Personal Knowledge. London: Rutledge and Kegan Paul 1961.Google Scholar
  93. 93.
    Wigner, E. P., Landsberg, P. T.: Reply to Ageno. Nature 205 (1965) 1307.CrossRefGoogle Scholar
  94. 94.
    Wilson, J. A.: Increasing entropy of biological systems. Nature 219 (1968) 534–535.CrossRefGoogle Scholar
  95. 95.
    Wiener, N.: Cybernetics. New York: Wiley 1948.Google Scholar
  96. 96.
    Woolhouse, H. W.: Negentropy, information and the feeding of organisms. Nature 213 (1967) 952.CrossRefGoogle Scholar
  97. 97.
    Woolhouse, H. W.: Entropy and Evolution. Nature 216 (1967) 200.CrossRefGoogle Scholar
  98. 98.
    Zanstra, H.: On the pulsating or expanding universe and its thermodynamical aspect. Kon. Nederl. Akad. van Wetenschappen 60 (1957) 285–307.Google Scholar
  99. 99.
    Bilaniuk, O. M., and others: More about tachyons. Physics Today (December 1969) 47–52.Google Scholar
  100. 100.
    Bilaniuk, O. M., Causality and space-like signals. Nature 223 (1969) 386–387.CrossRefGoogle Scholar
  101. 101.
    Thouless, D. J.: Causality and tachyons. Nature 224 (1969) 506.CrossRefGoogle Scholar
  102. 102.
    Root, R. G., and Trefil, J.S.: An amusing paradox involving tachyons. Lett. Nuovo Cimento 3 (1970) 412–415.CrossRefGoogle Scholar
  103. 103.
    Baltay, C., Feinberg G., Yeh, N., and Linsker, R.: Search for uncharged faster-than-light particles. Phys. Rev. D 1 (1970) 759–770.CrossRefGoogle Scholar
  104. 104.
    Watanabe, S.: Conditional Probability in Physics. Progress of Theoretical Physics Suppl. Extra Number (1965) 135–160.Google Scholar
  105. 105.
    Cocke, W. J.: Statistical Time Symmetry and two-time boundary conditions in Physics and Cosmology. Physical Review 160 (1967), 1165–1170.CrossRefGoogle Scholar

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© Springer-Verlag, Berlin · Heidelberg 1972

Authors and Affiliations

  • P. T. Landsberg
    • 1
  1. 1.Department of Applied Mathematics and Mathematical PhysicsUniversity CollegeCardiffGreat Britain

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