Communications in Mathematical Physics

, Volume 59, Issue 3, pp 235–266 | Cite as

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

  • Jürg Fröhlich
  • Yong Moon Park


We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albeverio, S., Høegh-Krohn, R.: Commun. math. Phys.30, 171 (1973); Edwards, S., Lenard, A.: J. Math. Phys.3, 778 (1962)Google Scholar
  2. 2.
    Brydges, D.: A rigorous approach to Debye screening in dilute classical Coulomb systems. Rockefeller preprint (1977)Google Scholar
  3. 3.
    Brydges, D., Federbush, P.: Commun. math. Phys.49, 233 (1976);53, 19 (1977)Google Scholar
  4. 4.
    Coleman, S.: Phys. Rev. D11, 2088 (1975)Google Scholar
  5. 5.
    Fortuin, C., Kasteleyn, P., Ginibre, J.: Commun. math. Phys.22, 89 (1971)Google Scholar
  6. 6.
    Fröhlich, J.: Commun. math. Phys.47, 233 (1976); Renormalization theory (eds. G. Velo, A. S. Wightman). Nato Adv. St. Inst. Series C. Dordrecht-Boston: Reidel 1976Google Scholar
  7. 7.
    Fröhlich, J.: Proceedings of the International Conference on Mathematical Physics, Rome (1977)Google Scholar
  8. 8.
    Fröhlich, J., Park, Y. M.: Helv. Phys. Acta50, 315 (1977)Google Scholar
  9. 9.
    Fröhlich, J., Park, Y. M.: In preparationGoogle Scholar
  10. 10.
    Fröhlich, J., Seiler, E.: Helv. Phys. Acta49, 889 (1976)Google Scholar
  11. 11.
    Fröhlich, J., Simon, B.: Ann. Math.105, 493 (1977)Google Scholar
  12. 12.
    Ginibre, J.: Some applications of functional integration in statistical mechanics. In: Statistical mechanics and quantum field theory, Les Houches 1970 (eds. C. DeWitt, R. Stora). New York: Gordon and Breach 1971 (See also Refs. to original articles given there)Google Scholar
  13. 13.
    Ginibre, J.: Commun. math. Phys.16, 310 (1970)Google Scholar
  14. 14.
    Guerra, F., Rosen, L., Simon, B.: Ann. Math.101, 111 (1975)Google Scholar
  15. 15.
    Israel, R.: Princeton series in physics, Princeton: University Press (to appear), (based on this author's PhD thesis, 1975)Google Scholar
  16. 16.
    José, J. V., Kadanoff, L. P., Kirkpatrick, S., Nelson, D. R.: IBM Res. Pep. RC 6428 (27401) (1977) and Refs. given thereinGoogle Scholar
  17. 17.
    Lebowitz, J. L.: Commun. math. Phys.28, 313 (1972)Google Scholar
  18. 18.
    Lebowitz, J. L.: Commun. math. Phys.35, 87 (1974)Google Scholar
  19. 19.
    Lebowitz, J. L., Martin-Löf, A.: Commun. math. Phys.25, 276 (1972); Lebowitz, J. L.: Proceedings of the International Conference on Mathematical Physics, Rome (1977) (see Ref. [7])Google Scholar
  20. 20.
    Lebowitz, J. L., Presutti, E.: Commun. math. Phys.50, 195 (1976)Google Scholar
  21. 21.
    Lieb, E. H., Lewbowitz, J. L.: Advanc. Math.9, 316 (1972); Lieb, E. H.: Rev. Mod. Phys.48, 553 (1976)Google Scholar
  22. 22.
    Nelson, E.: J. Math. Phys.5, 332 (1964)Google Scholar
  23. 23.
    Nelson, E.: Constructive quantum field theory (eds. G. Velo, A. S. Wightman). Lecture notes in physics, Vol. 25. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  24. 24.
    Park, Y. M.: J. Math. Phys. (to appear) (1977)Google Scholar
  25. 25.
    Ruelle, D.: Statistical mechanics. Reading-London-Amsterdam-Tokyo: W. A. Benjamin 1969Google Scholar
  26. 26.
    Ruelle, D.: J. Math. Phys.12, 901 (1971); Helv. Phys. Acta45, 215 (1972); Fröhlich, J.: Helv. Phys. Acta48, 355 (1975)Google Scholar
  27. 27.
    Simon, B.: Commun. math. Phys.31, 127 (1973)Google Scholar
  28. 28.
    Sylvester, G.: MIT PhD thesis (1976); J. Stat. Phys. (to appear) (1977)Google Scholar
  29. 29.
    Siegert, A. J. F.: Physica26, 30 (1960)Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Jürg Fröhlich
    • 1
  • Yong Moon Park
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsYonsei UniversitySeoulKorea

Personalised recommendations