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Journal of Statistical Physics

, Volume 147, Issue 4, pp 678–706 | Cite as

Mayer and Virial Series at Low Temperature

  • Sabine JansenEmail author
Article

Abstract

We analyze the Mayer and virial series (pressure as a function of the activity resp. the density) for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series’ radii of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results are consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson model.

Keywords

Classical statistical mechanics Mayer and virial series Phase transitions 

Notes

Acknowledgements

The author gratefully acknowledges very useful discussion with D. Ueltschi, D. Tsagkarogiannis, E. Presutti, E. Pulvirenti, B. Metzger and W. König, and also thanks E. Presutti for hospitality during a visit at the University of Rome “Tor Vergata”. This work was supported by the DFG Forschergruppe 718 “Analysis and stochastics in complex physical systems”.

References

  1. 1.
    Au Yeung, Y., Friesecke, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial Differ. Equ. 44, 81–100 (2012) zbMATHCrossRefGoogle Scholar
  2. 2.
    Brydges, D., Federbush, P.: A new form of the Mayer expansion in classical statistical mechanics. J. Math. Phys. 19, 2064–2067 (1978) MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Brydges, D., Martin, Ph.A.: Coulomb systems at low density: a review. J. Stat. Phys. 96, 1163–1330 (1999) MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Chayes, J.T., Chayes, L., Kotecký, R.: The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569 (1995) ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Collevecchio, A., König, W., Mörters, P., Sidorova, N.: Phase transitions for dilute particle systems with Lennard-Jones potential. Commun. Math. Phys. 299, 603–630 (2010) ADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Conlon, J.G., Lieb, E.H., Yau, H.-T.: The Coulomb gas at low temperature and low density. Commun. Math. Phys. 125, 153–180 (1989) MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Fefferman, C.L.: The atomic and molecular nature of matter. Rev. Mat. Iberoam. 1, 1–44 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Hill, T.L.: Statistical Mechanics: Principles and Selected Applications. The McGraw-Hill Series in Advanced Chemistry. McGraw-Hill, New York (1956) zbMATHGoogle Scholar
  9. 9.
    Hill, T.L.: An Introduction to Statistical Thermodynamics. Addison-Wesley Series in Chemistry, Addison-Wesley, Reading–London (1960) Google Scholar
  10. 10.
    Jansen, S., König, W., Metzger, B.: Large deviations for cluster size distributions in a continuous classical many-body system. arXiv:1107.3670v1 [math.PR]
  11. 11.
    Lebowitz, J.L., Penrose, O.: Convergence of virial expansions. J. Math. Phys. 5, 841–847 (1964) MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Lenard, A.: Exact statistical mechanics of a one-dimensional system with Coulomb forces. J. Math. Phys. 2, 682–693 (1961) MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Mayer, J.E., Mayer, M.G.: Statistical Mechanics. Wiley, New York (1940) zbMATHGoogle Scholar
  14. 14.
    Penrose, O.: Convergence of fugacity expansions for fluids and lattice gases. J. Math. Phys. 4, 1312–1320 (1963) MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Poghosyan, S., Ueltschi, D.: Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50, 053509 (2009, 17 pp.) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Pulvirenti, E., Tsagkarogiannis, D.: Cluster expansion in the canonical ensemble. arXiv:1105.1022v4 [math-ph]
  17. 17.
    Radin, C.: The ground state for soft disks. J. Stat. Phys. 26, 365–373 (1981) MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. Benjamin, New York–Amsterdam (1969) zbMATHGoogle Scholar
  19. 19.
    Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27, 1040–1041 (1971) MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Šamaj, L.: Widom-Rowlinson model (continuum and lattice). arXiv:0709.0617v1 [cond-mat.stat-mech]
  21. 21.
    Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262, 209–236 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1684 (1970) ADSCrossRefGoogle Scholar
  23. 23.
    Zeidler, E.: Applied Functional Analysis. Main Principles and Their Applications. Springer, New York (1995) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and Stochastics, Leibniz Institute in Forschungsverbund Berlin e.V.BerlinGermany

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