Communications in Mathematical Physics

, Volume 31, Issue 3, pp 171–189 | Cite as

Return to equilibrium

  • Derek W. Robinson


The problem of return to equilibrium is phrased in terms of aC*-algebraU, and two one-parameter groups of automorphisms τ, τ P corresponding to the unperturbed and locally perturbed evolutions. The asymptotic evolution, under τ, of τ P -invariant, and τ P -K.M.S., states is considered. This study is a generalization of scattering theory and results concerning the existence of limit states are obtained by techniques similar to those used to prove the existence, and intertwining properties, of wave-operators. Conditions of asymptotic abelianness provide the necessary dispersive properties for the return to equilibrium. It is demonstrated that the τ P -equilibrium states and their limit states are coupled by automorphisms with a quasi-local property; they are not necessarily normal with respect to one another. An application to theXY model is given which extends previously known results and other applications, and examples, are given for the Fermi gas.


Neural Network Statistical Physic Equilibrium State Complex System Nonlinear Dynamics 
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  1. 1.
    Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer 1966.Google Scholar
  2. 2.
    Ruelle, D.: Statistical Mechanics. New York: Benjamin 1969.Google Scholar
  3. 3.
    Haag, R., Hugenholtz, N. M., Winnink, M.: Commun. math. Phys.5, 215 (1967).Google Scholar
  4. 4.
    Kastler, D., Pool, J., Poulsen, E-Thue: Commun. math. Phys.12, 175 (1969).Google Scholar
  5. 5.
    Winnink, M.: Thesis, Univ. of Groningen (1968).Google Scholar
  6. 6.
    Abraham, D. B., Barouch, E., Gallavotti, G., Martin-Löf, A.: Phys. Rev. Letters25(II), 1449 (1970).Google Scholar
  7. 7.
    Radin, C.: Commun. math. Phys.23, 189 (1971).Google Scholar
  8. 8.
    Robinson, D. W.: Commun. math. Phys.7, 337 (1968).Google Scholar
  9. 9.
    Lanford, O. E.: Cargèse Lectures. New York: Gordon and Breach 1970.Google Scholar
  10. 10.
    Rocca, F., Sirugue, M., Testard, D.: Commun. math. Phys.13, 317 (1969).Google Scholar
  11. 11.
    Hepp, K.: Preprint, Zürich (1972).Google Scholar
  12. 12.
    Emch, G., Radin, C.: J. Math. Phys.12, 2043 (1971).Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • Derek W. Robinson
    • 1
    • 2
  1. 1.Department of PhysicsUniversity of Aix-Marseille, IIMarseille-LuminyFrance
  2. 2.Centre de Physique Théorique C.N.R.S. 31, chemin J. AiguierMarseille Cedex 2France

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