Non-Equilibrium Statistical Mechanics: Dynamics of Macroscopic Observables

  • Geoffrey L. Sewell
Part of the NATO ASI Series book series (NSSB, volume 258)

Abstract

We present an approach to non-equilibrium statistical thermodynamics within the framework of the quantum theory of infinite systems. This framework is natural for a mathematically precise formulation of irreversible processes, since it is only by idealising many-particle systems as infinite that Poincare’ cycles can be eliminated. We start by illustrating how this idealisation can indeed lead to both irreversibility and macroscopic causality by a treatment of a simple, exactly solvable model, corresponding to a ‘heavy’ particle, that interacts with a ‘heat bath’. Specifically, we show that the motion of the heavy particle conforms to a classical, deterministic, irreversible law, in the limit where the ratio of its mass to that of an atom of the bath tends to infinity: this limit serves to characterise the macroscopicality of the large particle. We next explain why, for a general treatment of infinite systems, it is neccessary to extend the standard quantum-mechanical framework, designed for finite assemblies of particles, to a form based on the algebraic structure of its observables (cf. [1–4]). We provide a brief, self-contained account of this generalised quantum mechanical framework and demonstrate how, by contrast with the traditional one for finite systems, it accommodates a precise distinction between microscopic and macroscopic quantities and even between different levels of macroscopicality. We employ it to obtain a general quantum statistical thermodynamical formulation both of equilibrium states and phases and of irreversible deterministic processes. In this way, we obtain a non-linear generalisation of the Onsager reciprocity relations for a class of irreversible processes in continuum mechanics.

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References

  1. 1.
    D. Ruelle: ‘Statistical Mechanics: Rigorous Results’, W. A. Benjamin, New York, 1969Google Scholar
  2. 2.
    G. G. Emch: ‘Algebraic Methods in Statistical Mechanics and Quantum Field Theory’, Wiley, New York, London, 1972Google Scholar
  3. 3.
    W. Thirring: ‘Quantum Mechanics of Large Systems’, Springer, New York, Vienna, 1980Google Scholar
  4. 4.
    G. L. Sewell: ‘Quantum Theory of Collective Phenomena’, Clarendon Press, Oxford, 1989Google Scholar
  5. 5.
    E. H. Lieb and W. Thirring: Phys. Rev. Lett. 35, 687, 1975CrossRefGoogle Scholar
  6. 6.
    N. G. Van Kampen: Can. J. Phys. 39, 551, 1961CrossRefGoogle Scholar
  7. 7.
    K. Hepp and E. H. Lieb: Helv. Phys. Acta 45, 237, 1973Google Scholar
  8. 8.
    G. L. Sewell: J. Math. Phys. 26, 2324, 1985CrossRefGoogle Scholar
  9. 9.
    J. Dixmier: Ann. Math. 51, 387, 1950CrossRefGoogle Scholar
  10. 10.
    L. Onsager, Phys. Rev. 37, 405, 1931; and 38, 2265, 1931CrossRefGoogle Scholar
  11. 11.
    L. Onsager and S. Maschlup: Phys. Rev. 18, 1505, 1953; and 91, 1512, 1953CrossRefGoogle Scholar
  12. 12.
    G. W. Ford, M. Kac and P. Mazur: J. Math. Phys. 6, 504, 1965CrossRefGoogle Scholar
  13. 13.
    J. T. Lewis and L. C. Thomas: ‘How to Make a Heat Bath’, pp. 97–123 of ‘Functional Analysis and its Applications’, Ed. A. M. Arthur, Oxford, Clarendon, 1975Google Scholar
  14. 14.
    G. L. Sewell: Ann. Phys. 85, 336, 1974CrossRefGoogle Scholar
  15. 15.
    E. B. Davies: Commun. Math. Phys. 39, 91, 1974CrossRefGoogle Scholar
  16. 16.
    P. L. Torres: J. Math. Phys. 18, 301, 1977CrossRefGoogle Scholar
  17. 17.
    G. L. Sewell: ‘Statistical Mechanical Considerations of Local Equilibrium and Hydrodynamics’, pp. 1–14 of ‘Local Equilibrium in Strong Interaction Physics’, Ed. D. K. Scott and R. M. Weiner, World Scientific Publ. Co., 1985Google Scholar
  18. 18.
    G. E. Uhlenbeck and L. S. Ornstein: Phys. Rev. 36, 823, 1930CrossRefGoogle Scholar
  19. 19.
    P. A. M. Dirac: ‘Principles of Quantum Mechanics’, Clarendon Press, Oxford, 1958Google Scholar
  20. 20.
    J. Von Neumann: ‘Mathematical Foundations of Quantum Mechanics’, Princeton University Press, 1955Google Scholar
  21. 21.
    J. Von Neumann: Math. Annalen 104, 570, 1931CrossRefGoogle Scholar
  22. 22.
    C. Radin: Commun. Math. Phys. 54, 69, 1977CrossRefGoogle Scholar
  23. 23.
    G. L. Sewell: Lett. Math. Phys. 6, 209, 1982CrossRefGoogle Scholar
  24. 24.
    G. L. Sewell: J. Math. Phys. 11, 1868, 1970CrossRefGoogle Scholar
  25. 25.
    G. F. Dell’ Antonio, S. Doplicher and D. Ruelle: Commun. Math. Phys. 2, 223Google Scholar
  26. 26.
  27. 27.
    R. Kubo: J. Phys. Soc. Japan 12, 570, 1957CrossRefGoogle Scholar
  28. 28.
    P.C. Martin and J. Schwinger: Phys. Rev. 115, 1342, 1977CrossRefGoogle Scholar
  29. 29.
    R. Haag, N. M. Hugenholtz and M. Winnink: Commun. Math. Phys 5, 215, 1967CrossRefGoogle Scholar
  30. 30.
    A. Kossakowski, A. Frigerio, V. Gorini and M. Verri: Commun. Math. Phys. 57, 97, 1977CrossRefGoogle Scholar
  31. 31.
    H. Araki and G. L. Sewell: Commun Math. Phys. 52, 103, 1977CrossRefGoogle Scholar
  32. 32.
    G. L. Sewell: Commun. Math. Phys. 55, 53, 1977CrossRefGoogle Scholar
  33. 33.
    L. D. Landau and E. M. Lifschitz: ‘Statistical Physics’, Pergamon, London, New York, Paris, 1959Google Scholar
  34. 34.
    D. Goderis and P. Vets: Commun. Math. Phys. 122, 249, 1989CrossRefGoogle Scholar
  35. 35.
    G. L. Sewell: ‘Quantum Macrostatistics and Irreversible Thermodynamics’, to be Published in the Proceedings of ‘Quantum Probability and Applications, V’, held at Heidelberg in 1988Google Scholar
  36. 36.
    G. L. Sewell: ‘Macrostatistics and Nonequilibrium Thermodynamics’, to be Published in the Proceedings of the Symposium on ‘Stochastic Processes, Physics and Geometry’, held at Ascona, 1988Google Scholar
  37. 37.
    L. Schwartz: ‘Theorie des Distrbutions’, Tome I, Hermann, Paris, 1950; Tome II, Hermann, Paris, 1951Google Scholar
  38. 38.
    R. F. Streater and A. S. Wightman: ‘PCT, Spin and Statistics and All That’, W. A. Benjamin, New York, Amsterdam, 1964Google Scholar
  39. 39.
    H. B. G. Casimir: Rev. Mod. Phys. 17, 343, 1945CrossRefGoogle Scholar
  40. 40.
    H. Narnhofer and G. L. Sewell: Commun. Math. Phys. 71, 1, 1980CrossRefGoogle Scholar
  41. 41.
    D. Goderis, A. Verbeure and P. Vets: J. Stat. Phys. 56, 721, 1989CrossRefGoogle Scholar
  42. 42.
    D. Goderis, A. Verbeure and P. Vets: ‘Glauber Dynamics of Fluctuations and the Onsager Theory’, Preprint, 1989Google Scholar
  43. 43.
    A. De Masi, N. Janiro, A. Pellegrinotti and E. Presutti: ‘A Survey of the Hydrody-namical Properties of Many-Particle Systems’, pp. 123–294 of ‘Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics’, Ed. J. L. Lebowitz and E. W. Montroll, North Holland, Amsterdam, 1984Google Scholar
  44. 44.
    W. D. Wick: J. Stat. Phys. 38, 1015, 1985CrossRefGoogle Scholar
  45. 45.
    R. B. Griffiths and D. Ruelle: Commun. Math. Phys. 23, 169, 1971CrossRefGoogle Scholar
  46. R. T. Rockafeller: ‘Convex Analysis’, Princeton University Press, Princeton, 1970Google Scholar
  47. 46.
    H. Araki: Publ. R.I.M.S. 9, 165, 1973CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Geoffrey L. Sewell
    • 1
  1. 1.Department of PhysicsQueen Mary and Westfield CollegeLondon E1UK

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