Skip to main content
SpringerLink
Log in

Penrose-Stable Interactions in Classical Statistical Mechanics

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

For a pair potential \(\Phi \) in a general underlying space X satisfying some natural and sufficiently general conditions in the sense of Penrose (J Math Phys 4:1312, 1963) and Poghosyan and Ueltschi (J Math Phys 50:053509, 2009) together with a locally finite measure \(\varrho \) on X we define by means of the so-called Ursell kernel a function r which is shown to be the correlation function of a unique process \(\mathrm{G}\), the limiting Gibbs process for \((\Phi ,\varrho )\) with empty boundary conditions. This process is exhibited as a Gibbs process in the sense of Dobrushin, Lanford and Ruelle for a class of pair potentials, which contains classical stable and hard-core potentials that are called Penrose potentials here. Particularly, a class of positive potentials is included. Finally, for some class of Penrose potentials, we show that \(\mathrm{G}\) is the unique Gibbs process for \(\Phi \). We use the classical method of Kirkwood–Salsburg equations. A decisive role is played by a generalization of Ruelle’s estimate for correlation functions.

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

Notes

  1. This explains the choice of our title.

  2. This notion is weaker than Penrose-stability and stronger than the classical notion of stability.

  3. i.e., Dobrushin, Lanford and Ruelle.

  4. Usually Radon measures are considered on locally compact and second countable Hausdorff topological spaces.

  5. This can be realized by the \(1-1\) correspondence \(\mu \leftrightarrow \kappa =\sum _{x\in \mu ^*}\sum _{i=1}^{\mu \{x\}}\varepsilon _{(x,i)}\) between \(\mu \in {\mathscr {M}}^{\cdot \cdot }(X)\) and \(\kappa \in \mathscr {M}^{\cdot }(X\times \mathbb {N})\).

  6. This notion will play a role in the definition of \(\mathscr {P}\)-stability and in the proof of Lemma 14.

  7. Also called factorial moment measure.

  8. This notion goes back to Oliver Penrose [29]. (Cf. also Charles Morrey [25] and Groeneveld [9]).

  9. The definition of this regularity condition can be found at the beginning of Sect. 10.

  10. Thus, we realize Minlos’ program on limiting Gibbs processes from [22] by means of methods from point process theory which had been developed later.

  11. Although the function \(g+\Phi _x\) is not an element of U we speak, in an abuse of language, of the Laplace transform evaluated in \(g+\Phi _x\).

References

  1. Dobrushin, R.L.: Gibbsian probability field. Funkts. Anal. Ego Pril. 2, 31–43 (1968)

    Google Scholar 

  2. Dobrushin, R.L.: Gibbsian probability field. Funkts. Anal. Ego Pril. 2, 44–57 (1968)

    Google Scholar 

  3. Dobrushin, R.L.: Gibbsian probability field. Funkts. Anal. Ego Pril. 3, 27–35 (1969)

    Google Scholar 

  4. Gallavotti, G., Miracle-Solé, S., Ruelle, D.: Absence of phase transitions in one-dimensional systems with hard cores. Phys. Lett. 26A, 350–352 (1968)

    Article  ADS  Google Scholar 

  5. Gallavotti, G., Miracle-Solé, S.: Absence of phase transitions in one-dimensional systems with long-range interactions. J. Math. Phys. 11, 147–154 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  6. Ginibre, J.: Reduced density matrices of quantum gases III. Hard-core potentials. J. Math. Phys. 6, 1432–1446 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  7. Ginibre, J.: Continuity of the pressure as a function of the density for some quantum systems. Phys. Rev. Lett. 24(26), 1473–1475 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  8. Ginibre, J.: Some applications of functional integration in statistical mechanics, in: Statistical Mechanics and Field Theory, C. de Witt and R. Stora (eds.), Gordon and Breach (1971)

  9. Groeneveld, J.: Two theorems on classical many-particle systems. Phys. Letters 3, 50–51 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  10. Jansen, S.: Cluster expansions for Gibbs point processes. Adv. Appl. Prob. 51, 1129–1178 (2019)

    Article  MathSciNet  Google Scholar 

  11. Kac, M.: On the partition function of a one-dimensional gas. Phys. Fluids 2, 8–12 (1959)

    Article  ADS  Google Scholar 

  12. Kallenberg, O.: Random measures. Theory and Applications, Springer (2017)

  13. Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491–498 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  14. Krickeberg, K.: Point processes. Classical lectures, Walter Warmuth Verlag (2014)

  15. Lanford, O., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 174–215 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  16. Lenard, A.: Correlation functions and the uniqueness of the state in classical statistical mechanics. Commun. Math. Phys. 30, 35–44 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  17. Lenard, A.: States of classical statistical mechanical systems of infinitely many particles. II, Characterization of correlation measures. Arch. Rational Mech. Anal. 59, 242–256 (1975)

    ADS  MathSciNet  Google Scholar 

  18. Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)

    MATH  Google Scholar 

  19. Matthes, K., Kerstan, J., Mecke, J.: Infinitely Divisible Point Processes. Wiley, New York (1978)

    MATH  Google Scholar 

  20. Mecke, J.: Stationäre Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitstheorie verw. Gebiete 9, 36–58 (1967)

    Article  MathSciNet  Google Scholar 

  21. Mecke, J.: Random Measures. Classical Lectures, Walter Warmuth Verlag (2011)

  22. Minlos, R.A.: Limiting Gibbs distribution. translated from Funktsional’nyi Analiz i Ego Prilozheniya 1, 60–73 (1967)

  23. Minlos, R.A.: Lectures on statistical physics. Russian Math. Surv. 23, 137–194 (1968)

    Article  ADS  MathSciNet  Google Scholar 

  24. Minlos, R.A., Poghosyan, S.: Estimates of Ursell functions, group functions, and their derivatives. Theor. Math. Phys. 31, 1–62 (1980)

    Google Scholar 

  25. Morrey, ChB: On the derivation of the equations of hydrodynamics from statistical mechanics. Commun. Pure Appl. Math. 8, 279–326 (1955)

    Article  MathSciNet  Google Scholar 

  26. Nehring, B.: Construction of classical and quantum gases. The method of cluster expansions, Mathematical lessons, WalterWarmuth Verlag (2013)

  27. Nehring, B., Zessin, H.: A representation of the moment measures of the general ideal Bose gas. Math. Nachr. 285, 878–888 (2012)

    Article  MathSciNet  Google Scholar 

  28. Nguyen Xuan Xanh, Zessin, H.: Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105–115 (1979)

  29. Penrose, O.: Convergence of fugacity expansions for fluids and lattice gases. J. Math. Phys. 4, 1312 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  30. Poghosyan, S., Ueltschi, D.: Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50, 053509 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  31. Poghosyan, S., Zessin, H.: Cluster representation of classical and quantum processes. Moscow Math. J. 19, 1–19 (2019)

    Article  MathSciNet  Google Scholar 

  32. Procacci, A.: Abstract polymer models with general pair interactions. J. Stat. Phys. 129, 171–188 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  33. Ruelle, D.: Statistical mechanics. W.A. Benjamin, Massachusetts (1969)

  34. Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  35. Stanley, R.P.: Enumerative Combinatorics I. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  36. Ueltschi, D.: Cluster expansions and correlation functions. Moscow Math. J. 4, 511–522 (2004)

    Article  MathSciNet  Google Scholar 

  37. Ueltschi, D.: An improved tree-graph bound, arXiv: 1705.05353v1 [math-phys], (15 May 2017)

  38. Van Hove, L.: Sur l’intégrale de configuration pour les systèmes de particules à une dimension. Physica 16, 137–143 (1950)

    Article  ADS  MathSciNet  Google Scholar 

  39. Zessin, H.: The method of moments for random measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete 62, 395–409 (1983)

    Article  MathSciNet  Google Scholar 

  40. Zessin, H.: Point proceses in general position. J. Contemp. Math. Anal. 43, 81–88 (2008)

    MathSciNet  Google Scholar 

  41. Zessin, H.: Der Papangelou Prozess [in German]. Izvestija NAN Armenii: Matematika 44(1), 61–72 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank one of the referees for his valuable remarks which lead to substantial improvements of this paper. S.P. is grateful to Prof. Sylvie Roelly for her hospitality at Potsdam University. H.Z. thanks Sylvie Roelly and Alexander Zass for illuminating discussions on the uniqueness problem.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Armenian National Academy of Sciences, Bagramian Av. 24 - 5, 0019, Yerevan, Armenia

    Suren Poghosyan

  2. Fakultät für Mathematik der Universität Bielefeld, Postfach 10 01 31, 33501, Bielefeld, Germany

    Hans Zessin

Authors
  1. Suren Poghosyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans Zessin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suren Poghosyan.

Additional information

Communicated by Christian Maes.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

In memoriam Jean Ginibre.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Poghosyan, S., Zessin, H. Penrose-Stable Interactions in Classical Statistical Mechanics. Ann. Henri Poincaré 23, 739–771 (2022). https://doi.org/10.1007/s00023-021-01098-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-021-01098-1

Search

Navigation