Communications in Mathematical Physics

, Volume 48, Issue 1, pp 1–14 | Cite as

Stability and equilibrium states of infinite classical systems

  • Michael Aizenman
  • Giovanni Gallavotti
  • Sheldon Goldstein
  • Joel L. Lebowitz


We prove that any stationary state describing an infinite classical system which is “stable” under local perturbations (and possesses some strong time clustering properties) must satisfy the “classical” KMS condition. (This in turn implies, quite generally, that it is a Gibbs state.) Similar results have been proven previously for quantum systems by Haag et al. and for finite classical systems by Lebowitz et al.


Neural Network Statistical Physic Equilibrium State Complex System Stationary State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ruelle, D.: Statistical mechanics. New York: Benjamin 1969Google Scholar
  2. 2.
    Lanford, O.: In: Dynamical systems, theory and applications. Lecture Notes in Physics, 38, (ed. J. Moser) Berlin-Heidelberg-New York: Springer 1975Google Scholar
  3. 3.
    Ruelle, D.: Commun. math. Phys.18, 127 (1970)Google Scholar
  4. 4.
    Gruber, C., Lebowitz, J.L.: Commun. math. Phys.41, 11 (1975)Google Scholar
  5. 5.
    Lanford, O.: In: Statistical mechanics and mathematical problems. Lecture Notes in Physics, 20, ed. A. Lenard. Berlin-Heidelberg-New York: Springer 1973;Google Scholar
  6. 5a.
    Thompson, R. L.: Mem. Am. Math. Soc.150 (1974);Google Scholar
  7. 5b.
    Logan, K. G.: Thesis, Cornell Univ. (1974);Google Scholar
  8. 5c.
    Giorgii, H. O.: Z. Wahrscheinlichkeitstheorie verw. Geb.32, 277 (1975): On Canonical Gibbs States and Symmetric and Tail Events (preprint) see also Ref. 16Google Scholar
  9. 6.
    Lebowitz, J. L., Aizenman, M., Goldstein, S.: J. Math. Phys.16, 1284 (1975)Google Scholar
  10. 7.
    Lanford, O. E.: Commun. math. Phys.11, 257 (1969)Google Scholar
  11. 8.
    Sinai, Ya. G.: Vestn. Markov Univ., Ser. I, Mat. Meh.29, 152 (1974)Google Scholar
  12. 9.
    Marchioro, C., Pellegrinotti, A., Presutti, E.: Commun. math. Phys.40, 175–185 (1975)Google Scholar
  13. 10.
    Goldstein, S., Lebowitz, J. L. Aizenman, M.: In Ref. 2, discuss ergodic properties of infinite classical systems. See also Gurevich, B. M., Sukhov, Ju. M.: Stationary Solutions of the Bogoliubov Hierarchy Equations in Classical Statistical Mechanics, 1. (preprint). In this work, received after the present paper was completed, it is shown that for some type of interactions and some assumptions on ω a stationary state has to be GibbsGoogle Scholar
  14. 11.
    Haag, R., Kastler, D., Trych-Pohlmeyer, E.: Commun. math. Phys.38, 173 (1974)Google Scholar
  15. 12.
    Kastler, D.: Equilibrium states of matter and operator algebras, Proc. of the Roma Conf. onC*-Algebras, Inst. Nationale Alta Mat. 1975, to appearGoogle Scholar
  16. 13.
    Bratteli, O, Kastler, D.: Commun. math. Phys.46, 37–42 (1976)Google Scholar
  17. 14.
    Haag, R., Kastler, D.: Stability and equilibrium states II, to appearGoogle Scholar
  18. 15.
    Gallavotti, G., Verboven, E.: Nuevo Cimento28 B, 274 (1975). For further investigations of the classical KMS condition see Gallavotti, G., Pulvirenti, E.: Commun. math. Phys.46, 1–9 (1976)Google Scholar
  19. 16.
    Goldstein, S., Aizenman, M., Lebowitz, J. L.: to appearGoogle Scholar
  20. 17.
    Aizenman, M.: Thesis, Yeshiva University (1975)Google Scholar
  21. 18.
    Gallavotti, G.: Preprint for the series in honor of D. Graffi, in printGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Michael Aizenman
    • 1
  • Giovanni Gallavotti
    • 2
  • Sheldon Goldstein
    • 3
  • Joel L. Lebowitz
    • 4
  1. 1.Department of Physics and MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Istituto MatematicoUniversity di RomaRomaItaly
  3. 3.Department of MathematicsCornell UniversityIthacaUSA
  4. 4.Belfer Graduate School of ScienceYeshiva UniversityNew YorkUSA

Personalised recommendations