Skip to main content
SpringerLink
Log in

Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Equivalence (allowing for convex combinations) of microcanonical, canonical and grandcanonical ensembles for states of classical systems is established under very mild assumptions on the limiting state. We introduce the notion of conditional equilibrium (C.E.), a property of states of infinite systems which characterizes convex combinations of limits of microcanonical ensembles. It is shown that C.E. states are, under quite general conditions, mixtures of Gibbs states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ruelle, D.: Statistical mechanics. New York: Benjamin 1969

    Google Scholar 

  2. Lanford III, O.E., Ruelle, D.: Commun. math. Phys.13, 194 (1969)

    Google Scholar 

  3. Dobrushin, R.L.: Funct. Anal. Appl.2, 31 (1968)

    Google Scholar 

  4. Bogoliubov, N.N.: J. Phys. USSR10, 257, 265 (1946)

    Google Scholar 

  5. Thompson, R.L.: Mem. A. M. S.150 (1974)

  6. Georgii, H.: Z. Wahrscheinlichkeitstheorie verw. Gebiete32, 272 (1975);33, 331 (1976)

    Google Scholar 

  7. Martin-Löf, A.: The equivalence of ensembles and Gibbs phase rule for classical lattice systems: J. Stat. Phys. (to appear)

  8. Georgii, H.: Commun. math. Phys.48, 37 (1976)

    Google Scholar 

  9. Aizenman, M., Gallavotti, G., Goldstein, S., Lebowitz, J.L.: Commun. math. Phys.48, 1 (1976)

    Google Scholar 

  10. Haag, R., Kastler, D., Trych-Pohlmeyer, E.: Commun. math. Phys.38, 173 (1974)

    Google Scholar 

  11. Gallavotti, G., Verboven, E.: Nuovo Cimento28B, 274 (1975)

    Google Scholar 

  12. Aizenman, M., Goldstein, S., Gruber, C., Lebowitz, J.L., Martin, P.: Commun. math. Phys.53, 209 (1977)

    Google Scholar 

  13. Preston, C.J.: Canonical and microcanonical Gibbs states. Preprint (1977)

  14. Schwartz, L.: Radon measures. London, New York: Oxford University Press (1973)

    Google Scholar 

  15. Lanford III O.E.: Time evolution of large classical systems. In: Dynamical systems, theory, and applications. In: Lecture notes in physics, Vol. 38. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  16. Föllmer, H.: In: Seminaire de probabilites IX. Lecture Notes in Mathematics, Vol. 465. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  17. Dynkin, E.B.: Actes, Congrès intern. Math., 1970, Tome 2, 507 (1971)

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Princeton University, 08540, Princeton, New Jersey, USA

    Michael Aizenman

  2. Department of Mathematics, Rutgers University, 08903, New Brunswick, New Jersey, USA

    Sheldon Goldstein & Joel L. Lebowitz

Authors
  1. Michael Aizenman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sheldon Goldstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joel L. Lebowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J. Glimm

Supported in part by NSF Grant No. MCS 75-21684 A02

Supported in part by NSF Grant No. MPS 72-04534

Supported in part by NSF Grant No. Phy 77-22302

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aizenman, M., Goldstein, S. & Lebowitz, J.L. Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit. Commun.Math. Phys. 62, 279–302 (1978). https://doi.org/10.1007/BF01202528

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01202528

Keywords

Search

Navigation