Journal of Statistical Physics

, Volume 108, Issue 5–6, pp 787–829 | Cite as

Mathematical Theory of Non-Equilibrium Quantum Statistical Mechanics

  • V. Jakšić
  • C.-A. Pillet
Article

Abstract

We review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics recently proposed in refs. 1–7. In the algebraic formalism of quantum statistical mechanics we introduce notions of non-equilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.

Nonequilibrium quantum statistical mechanics Liouvillean open systems entropy production steady state C*-algebra 

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REFERENCES

  1. 1.
    V. Jakši? and C.-A. Pillet, On entropy production in quantum statistical mechanics, Commun. Math. Phys. 217:285 (2001).Google Scholar
  2. 2.
    V. Jakši? and C.-A. Pillet, Non-equilibrium steady states for finite quantum systems coupled to thermal reservoirs, Commun. Math. Phys. 226:131 (2002).Google Scholar
  3. 3.
    V. Jakši? and C.-A. Pillet, in preparation.Google Scholar
  4. 4.
    D. Ruelle, Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys. 98:57 (2000).Google Scholar
  5. 5.
    D. Ruelle, Entropy production in quantum spin systems, Commun. Math. Phys. 224:3 (2001).Google Scholar
  6. 6.
    D. Ruelle, How should one define entropy production for nonequilibrium quantum spin systems? Preprint (2001), mp-arc 01-258.Google Scholar
  7. 7.
    D. Ruelle, Topics in quantum statistical mechanics and operator algebras. Preprint (2001), mp-arc 01-257.Google Scholar
  8. 8.
    O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1 (Springer-Verlag, Berlin, 1987).Google Scholar
  9. 9.
    O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2 (Springer-Verlag, Berlin, 1996).Google Scholar
  10. 10.
    R. Haag, Local Quantum Physics (Springer-Verlag, Berlin, 1993).Google Scholar
  11. 11.
    B. Simon, Statistical Mechanics of Lattice Gases (Princeton University Press, Princeton, 1993).Google Scholar
  12. 12.
    W. Thirring, Quantum Mechanics of Large Systems (Springer-Verlag, Wien, 1980).Google Scholar
  13. 13.
    M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer-Verlag, Berlin, 1993).Google Scholar
  14. 14.
    V. Jakši? and C.-A. Pillet, Spectral theory of thermal relaxation, J. Math. Phys. 38:1757 (1997).Google Scholar
  15. 15.
    V. Jakši? and C.-A. Pillet, On a model for quantum friction III. Ergodic properties of the spin-boson system, Commun. Math. Phys. 178:627 (1996).Google Scholar
  16. 16.
    H. Araki and W. Wyss, Representations of canonical anti-commutation relations, Helv. Phys. Acta 37:136 (1964).Google Scholar
  17. 17.
    N. M. Hugenholz, Derivation of the Boltzmann equation for a Fermi gas, J. Stat. Phys. 32:231 (1983).Google Scholar
  18. 18.
    R. B. Israel, Convexity in the Theory of Lattice Gases (Princeton University Press, Princeton, 1979).Google Scholar
  19. 19.
    D. Ruelle, Statistical Mechanics. Rigorous Results (Benjamin, New York, 1969).Google Scholar
  20. 20.
    H. Araki, On the XY-model on two-sided infinite chain, Publ. Res. Inst. Math. Sci. Kyoto Univ. 20:277 (1984).Google Scholar
  21. 21.
    H. Araki, Dynamic and ergodic properties of the XY-model, in Critical Phenomena (Brasov, 1983), Progr. Phys., Vol. 11 (Birkhäuser, Boston, 1985), p. 287.Google Scholar
  22. 22.
    R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II. Advanced Theory (Academic Press, Orlando, 1986).Google Scholar
  23. 23.
    M. Takesaki, Theory of Operator Algebras I (Springer-Verlag, New York, 1979).Google Scholar
  24. 24.
    H. Araki, Relative Hamiltonian for faithful normal states of a von Neumann algebra, Publ. Res. Inst. Math. Sci. Kyoto Univ. 9:165 (1973).Google Scholar
  25. 25.
    J. Derezinski, V. Jakši?, and C.-A. Pillet, Perturbation theory of Wg-dynamics, Liouvilleans and KMS states. Submitted.Google Scholar
  26. 26.
    V. Bach, J. Fröhlich, and I. Sigal, Return to equilibrium, J. Math. Phys. 41:3985 (2000).Google Scholar
  27. 27.
    J. Derezinski and V. Jakši?, Spectral theory of Pauli-Fierz operators, J. Func. Anal. 180:243 (2001).Google Scholar
  28. 28.
    M. Merkli, Positive commutators in non-equilibrium quantum statistical mechanics, Commun. Math. Phys. 223:327 (2001).Google Scholar
  29. 29.
    O. Bratteli, A. Kishimoto, and D. W. Robinson, Stability properties and the KMS condition, Commun. Math. Phys. 61:209 (1978).Google Scholar
  30. 30.
    R. Haag, D. Kastler, and E. Trych-Pohlmeyer, Stability and equilibrium states, Commun. Math. Phys. 38:213 (1974).Google Scholar
  31. 31.
    R. Haag and E. Trych-Pohlmeyer, Stability properties of equilibrium states, Commun. Math. Phys. 56:273 (1977).Google Scholar
  32. 32.
    H. Narnhofer and W. Thirring, On the adiabatic theorem in quantum statistical mechanics, Phys. Rev. A 26:3646 (1982).Google Scholar
  33. 33.
    D. D. Botvich and V. A. Malyshev, Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi gas, Commun. Math. Phys. 91:301 (1983).Google Scholar
  34. 34.
    D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Stat. Phys. 95:393 (1999).Google Scholar
  35. 35.
    D. Ruelle, One-dimensional Gibbs states and Axiom A diffeomorphisms, J. Diff. Geom. 25:117 (1987).Google Scholar
  36. 36.
    D. Ruelle, Resonances for axiom A flows. Axiom A diffeomorphisms, J. Diff. Geom. 25:99 (1987).Google Scholar
  37. 37.
    D. Ruelle, Resonances of chaotic dynamical systems, Phys. Rev. Lett. 56:405 (1986).Google Scholar
  38. 38.
    J.-P. Eckmann, Resonances in dynamical systems, Proceedings of the IXth International Congress of Mathematical Physics (Swansea 1988) (Hilger, Bristol, 1989).Google Scholar
  39. 39.
    H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci. Kyoto Univ. 11:809 (1975/76).Google Scholar
  40. 40.
    H. Araki, Relative entropy for states of von Neumann algebras, II, Publ. Res. Inst. Math. Sci. Kyoto Univ. 13:173 (1977/78).Google Scholar
  41. 41.
    H. Spohn, Entropy production for quantum dynamical semigroups, J. Math. Phys. 19:227 (1978).Google Scholar
  42. 42.
    J. L. Lebowitz and S. Spohn, Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs, Adv. Chem. Phys. 38:109 (1978).Google Scholar
  43. 43.
    I. Ojima, H. Hasegawa, and M. Ichiyanagi, Entropy production and its positivity in nonlinear response theory of quantum dynamical systems, J. Stat. Phys. 50:633 (1988).Google Scholar
  44. 44.
    I. Ojima, Entropy production and non-equilibrium stationarity in quantum dynamical systems: Physical meaning of van Hove limit, J. Stat. Phys. 56:203 (1989).Google Scholar
  45. 45.
    I. Ojima, Entropy production and non-equilibrium stationarity in quantum dynamical systems, in Proceedings of International Workshop on Quantum Aspects of Optical Communications, Lecture Notes in Physics, Vol. 378 (Springer-Verlag, Berlin, 1991), p. 164.Google Scholar
  46. 46.
    W. Pusz and S. L. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58:273 (1978).Google Scholar
  47. 47.
    P. G. Bargmann and J. L. Lebowitz, New approach to nonequilibrium process, Phys. Rev. 99:578 (1955).Google Scholar
  48. 48.
    J. L. Lebowitz, Stationary nonequilibrium Gibbsian ensembles, Phys. Rev. 114:1192 (1959).Google Scholar
  49. 49.
    J. L. Lebowitz and A. Shimony, Statistical mechanics of open systems, Phys. Rev. 128: 1945 (1962).Google Scholar
  50. 50.
    V. V. Aizenstadt and V. A. Malyshev, Spin interaction with an ideal Fermi gas, J. Stat. Phys. 48:51 (1987).Google Scholar
  51. 51.
    E. B. Davies, Markovian master equations, Commun. Math. Phys. 39:91 (1974).Google Scholar
  52. 52.
    V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. C. G. Sudarshan, Properties of quantum Markovian master equations, Rep. Math. Phys. 13:149 (1978).Google Scholar
  53. 53.
    E. B. Davies and H. Spohn, Open quantum systems with time-dependent Hamiltonians and their linear response, J. Stat. Phys. 19:511 (1978).Google Scholar
  54. 54.
    V. Jakši? and C.-A. Pillet, On a model for quantum friction II. Fermi's golden rule and dynamics at positive temperature, Commun. Math. Phys. 176:619 (1996).Google Scholar
  55. 55.
    A. Frigerio, Quantum dynamical semigroups and approach to equilibrium, Lett. Math. Phys. 2:79 (1977).Google Scholar
  56. 56.
    A. Frigerio, Stationary states of quantum dynamical semigroups, Commun. Math. Phys. 63:269 (1978).Google Scholar
  57. 57.
    H. Spohn, An algebraic condition for the approach to equilibrium of an open N-level system, Lett. Math. Phys. 2:33 (1977/78).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • V. Jakšić
    • 1
  • C.-A. Pillet
    • 2
    • 3
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.PHYMATUniversité de ToulonLa Garde CedexFrance
  3. 3.FRUMAMCPT-CNRS LuminyMarseille Cedex 9France

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