Skip to main content
SpringerLink
Log in

A Representation of Virial Coefficients That Avoids the Asymptotic Catastrophe

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate the notion of an asymptotic catastrophe in representations of Mayer coefficients. The manifestations of the catastrophe and its formal definition are given. The significance of the definition introduced for an asymptotic catastrophe is clarified. Virial-coefficient representations that are free of the asymptotic catastrophe phenomenon are given. Sets of labeled graphs (blocks) nonseparable in the Harary sense are expanded into classes labeled by cycle ensembles satisfying specific conditions, and the representations are based on these expansions. These cycle ensembles are called frame cycle ensembles. The same classes can be labeled by special blocks, which are called frames. The frames are brought into one-to-one correspondence with the frame cycle ensembles. In the block classification, frames play a role similar to the role of trees in the tree classification of connected labeled graphs. A tree classification of frame cycle ensembles is introduced. We prove that the described virial-coefficient representations are free of the asymptotic catastrophe phenomenon.

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. J. Mayer and M. Goeppert Mayer, Statistical Mechanics, Wiley, New York (1940).

    Google Scholar 

  2. D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin, New York (1969).

    Google Scholar 

  3. F. Harary, Graph Theory, Addison–Wesley, Reading, Mass. (1969).

    Google Scholar 

  4. I.I. Ivanchik, “Method of covariant summation of diagrams in classical statistics [in Russian],” Doctoral dissertation, Lebedev Phys. Inst., Moscow (1987); Generalized Master Series in Classical Statistical Mechanics, Nova Science, New York (1993).

    Google Scholar 

  5. I.I. Ivanchik, Theor. Math. Phys., 108, 958 (1996).

    Google Scholar 

  6. G.I. Kalmykov, Theor. Math. Phys., 119, 778 (1999).

    Google Scholar 

  7. O. Penrose, J. Math. Phys., 4, 1488 (1963).

    Google Scholar 

  8. G.E. Uhlenbeck and G.W. Ford, Lectures in Statistical Mechanics (Lect. Appl. Math., Proc. Summer Seminar, Boulder, Colorado,1960, Vol. 1), Am. Math. Soc., Providence, R.I. (1963).

    Google Scholar 

  9. D.N. Zubarev, Sov. Phys. Dokl., 118, 903 (1958).

    Google Scholar 

  10. M. Duneau and B. Souillard, Commun. Math. Phs., 47, 155 (1976).

    Google Scholar 

  11. M. Duneau, B. Souillard, and D. Iagolnitzer, J. Math. Phs., 16, 1662 (1975).

    Google Scholar 

  12. E.A. Arinshtein, Sov. Phys. Dokl., 2, 54 (1957).

    Google Scholar 

  13. E.A. Arinshtein, Sov. Phys. Dokl., 2, 277 (1958).

    Google Scholar 

  14. T. Morita, Progr. Theor. Phys., 20, 920 (1958).

    Google Scholar 

  15. J.M.J. Van Leeuwen, J. Groeneveld, and J. De Boer Jr., Physica, 25, 742 (1959).

    Google Scholar 

  16. T. Morita, Progr. Theor. Phys., 23, 175 (1960).

    Google Scholar 

  17. T. Morita, Progr. Theor. Phys., 23, 829 (1960).

    Google Scholar 

  18. T. Morita and K. Hiroike, Progr. Theor. Phys., 23, 1003 (1960).

    Google Scholar 

  19. E.A. Arinshtein and B.G. Abrosimov, Zh.Strukturn.Khim., 9, 1064 (1968).

    Google Scholar 

  20. G.A. Martynov, Dokl. Akad. Nauk SSSR, 218, 814 (1974).

    Google Scholar 

  21. G.A. Martynov, Theor. Math. Phys., 22, 59 (1975).

    Google Scholar 

  22. A.N. Vasil'ev, Functional Methods in Quantum Field Theory and Statistics [in Russian], Leningrad State Univ. Publ., Leningrad (1976).

  23. I.I. Ivanchik, Trudy Fiz. Inst. Lebedev., 124, 14 (1980).

    Google Scholar 

  24. G.A. Martynov, Mol. Phys., 42, 329 (1981).

    Google Scholar 

  25. I.I. Ivanchik, Trudy Fiz. Inst. Lebedev., 144, 152 (1984).

    Google Scholar 

  26. I.I. Ivanchik, Dokl. Akad. Nauk SSSR, 296, 341 (1987).

    Google Scholar 

  27. I.I. Ivanchik, Dokl. Akad. Nauk SSSR, 300, 596 (1988).

    Google Scholar 

  28. G.I. Kalmykov, Theor. Math. Phys., 84, 869 (1990).

    Google Scholar 

  29. G.I. Kalmykov, “Method of tree sums and its application to solving mathematical problems in classical statistical mechanics [in Russian ],” Doctoral dissertation, Computational Center, RAS, Moscow (1998).

    Google Scholar 

  30. J. Groeneveld, Phys. Lett., 3, 50 (1962).

    Google Scholar 

  31. I.I. Ivanchik, “On nonrepeating enumeration of connected labeled graphs [in Russian],” in: Combinatorial Analysis (Vol. 4, K.A. Rybnikov, ed.), Moscow State Univ. Publ., Moscow (1976), p. 78.

    Google Scholar 

  32. V.A. Malyshev and R.A. Minlos, Gibbs Random Fields: Cluster Expansions [in Russian], Nauka, Moscow (1985); English transl., Dordrecht, Kluwer (1991).

    Google Scholar 

  33. I.I. Ivanchik, Trudy Fiz. Inst. Lebedev., 106, 3 (1979).

    Google Scholar 

  34. A.A. Sapozhenk, “On the number for connected subsets with a given boundary cardinality in bichromatic graphs [in Russian],” in: Discrete Analysis Methods in Solving Combinatorial Problems, Vol. 45, Siberian Branch, RAS, Novosibirsk (1987), p. 42.

    Google Scholar 

  35. G.I. Kalmykov, Theor. Math. Phos., 97, 1405 (1993).

    Google Scholar 

  36. G.I. Kalmykov, Theor. Math. Phys., 101, 1224 (1994).

    Google Scholar 

  37. G.I. Kalmykov, Theor. Math. Phys., 92, 790 (1992).

    Google Scholar 

  38. G.I. Kalmykov, Theor. Math. Phys., 100, 834 (1994).

    Google Scholar 

  39. G.I. Kalmykov, Theor. Math. Phys., 116, 1963 (1998).

    Google Scholar 

  40. G.I. Kalmykov, “Frame classification of graphs [in Russian ],” in:Proc. 4th Intl. Conf. on Discrete Models in the Theory of Control Systems (Krasnovidovo, Moscow Oblast,19–25 June 2000, V.B. Alekseev and V.A. Zakharov, eds.), MAKS Press, Moscow (2000), p. 34.

    Google Scholar 

  41. G.I. Kalmykov, “Frame classification of marked blocks [in Russian ],” in: Materials of 7th Intl. Workshop on Discrete Mathematics and Its Applications (29 January–2 February 2001, V l.2, O.B. Lupanov,ed.), Center for Applied Research, Mechanics–Mathematics Department, Moscow State Univ. Publ., Moscow (2001), p. 221.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lebedev Physical Research Center, Moscow, Russia

    G. I. Kalmykov

Authors
  1. G. I. Kalmykov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kalmykov, G.I. A Representation of Virial Coefficients That Avoids the Asymptotic Catastrophe. Theoretical and Mathematical Physics 130, 432–447 (2002). https://doi.org/10.1023/A:1014775124868

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014775124868

Keywords

Search

Navigation