Journal of Statistical Physics

, Volume 18, Issue 6, pp 585–632 | Cite as

Dynamics of the open BCS model

  • Emmanuel Buffet
  • Philippe A. Martin
Articles
Received:

Abstract

The dynamics of the strong coupling BCS model, considered as an open system interacting with a thermal bath, is solved rigorously and explicitly in the weak coupling limit and in the infinite-volume limit. The BCS system goes from the normal phase to the ordered phase by bifurcation. Fluctuations around trajectories of intensive observables are Gaussian and Markovian. Thermodynamic phases are global attractors in the physical domain. Structural stability is discussed. The model provides an example of a nonequilibrium statistical mechanical system with phase transition whose irreversible macroscopic dynamics can be calculated exactly from the underlying Hamiltonian quantum mechanics.

Key words

Strong coupling BCS model macroscopic states heat bath dissipative semigroup bifurcation Liapunov function stability nonequilibrium thermodynamics fluctuations 
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References

  1. 1.
    E. B. Davies,Comm. Math. Phys. 39:91 (1974).Google Scholar
  2. 2.
    E. B. Davies,Math. Ann. 219:147 (1976).Google Scholar
  3. 3.
    Ph. A. Martin and G. G. Emch,Helv. Phys. Acta 48:59 (1975).Google Scholar
  4. 4.
    H. Spohn, Derivation of the transport equation for electrons moving through random impurities, Princeton Univ., preprint (1977).Google Scholar
  5. 5.
    K. Hepp and E. Lieb, Dynamical systems, inLecture Notes in Physics,38 (Springer, 1975), p. 178.Google Scholar
  6. 6.
    K. Hepp and E. Lieb, inProceedings of the 1973 Erice Summer School, Lecture Notes in Physics,25 (Springer, 1973), p. 298.Google Scholar
  7. 7.
    O. E. Lanford, Dynamical systems, inLecture Notes in Physics,38 (Springer, 1975), p. 70.Google Scholar
  8. 8.
    W. Braun and K. Hepp,Comm. Math. Phys. 56:101 (1977).Google Scholar
  9. 9.
    Ph. A. Martin,J. Stat. Phys. 16:149 (1977).Google Scholar
  10. 10.
    M. Giuffre, Univ. of Rochester, preprint (1977); G. G. Emch, Phase transitions, approach to equilibrium and structural stability, to appear inSpringer Lecture Notes in Physics,79 (1978).Google Scholar
  11. 11.
    H. Spohn and J. Lebowitz, Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs,Adv. Phys. Chem., to appear.Google Scholar
  12. 12.
    W. Thirring and A. Wehrl,Comm. Math. Phys. 4:303 (1967).Google Scholar
  13. 13.
    W. Thirring,Comm. Math. Phys. 7:181 (1968).Google Scholar
  14. 14.
    L. van Hemmen, Linear fermion systems, molecular field models and the KMS condition,Fort, der Phys., to appear.Google Scholar
  15. 15.
    P. C. Hohenberg and B. I. Halperin,Rev. Mod. Phys. 49:435 (1977).Google Scholar
  16. 16.
    E. B. Davies, Master equations, a survey of rigorous results, Oxford Univ., preprint (1977).Google Scholar
  17. 17.
    V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and G. C. G. Sudarshan, Properties of quantum markovian master equations, Univ. of Texas, preprint (1976).Google Scholar
  18. 18.
    G. Nicolis and I. Prigogine,Self-Organization in Non-Equilibrium Systems (Wiley, New York, 1977).Google Scholar
  19. 19.
    H. Haken,Synergetics (Springer, Berlin, 1977).Google Scholar
  20. 20.
    D. J. Thouless,The Quantum Mechanics of Many Body Systems (Academic Press, 1972).Google Scholar
  21. 21.
    K. Hepp and E. Lieb,Helv. Phys. Acta 46:573 (1973).Google Scholar
  22. 22.
    R. W. Zwanzig, inLectures in Theoretical Physics III, Boulder (Interscience, 1960), p. 106.Google Scholar
  23. 23.
    F. Haake,Statistical Treatment of Open Systems by Generalized Master Equations (Springer, 1973).Google Scholar
  24. 24.
    E. Balslev and A. Verbeure,Comm. Math. Phys. 7:55 (1968).Google Scholar
  25. 25.
    W. Hirsch and S. Smale,Differential Equations, Dynamical Systems and Linear Algebra (Academic Press, 1974).Google Scholar
  26. 26.
    R. Holley,Comm. Math. Phys. 23:87 (1971).Google Scholar
  27. 27.
    Hartmann,Ordinary Differential Equations (Wiley, 1964).Google Scholar
  28. 28.
    L. Pontryagin,Equations differentielles ordinaires (Mir, 1969).Google Scholar
  29. 29.
    H. Spohn,Rep. Math. Phys. 10:189 (1976).Google Scholar
  30. 30.
    A. Frigerio,Lett. Math. Phys. 2:79 (1977).Google Scholar
  31. 31.
    Marsden and MacCracken,The Hopf Bifurcation and Its Applications (Springer, 1976).Google Scholar
  32. 32.
    S. R. De Groot and P. Mazur,Non-Equilibrium Thermodynamics (North-Holland, 1961).Google Scholar
  33. 33.
    M. Luban, Generalized Landau theories, in Domb and Green,Phase Transitions and Critical Phenomena Va (Academic Press, 1975).Google Scholar
  34. 34.
    N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, Univ. of Heidelberg, preprint.Google Scholar

Copyright information

© Plenum Publishing Corporation 1978

Authors and Affiliations

  • Emmanuel Buffet
    • 1
  • Philippe A. Martin
    • 1
  1. 1.Laboratoire de Physique ThéoriqueEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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