Advertisement

Communications in Mathematical Physics

, Volume 330, Issue 2, pp 801–817 | Cite as

Multispecies Virial Expansions

  • Sabine Jansen
  • Stephen J. Tate
  • Dimitrios Tsagkarogiannis
  • Daniel Ueltschi
Article

Abstract

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange–Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs.

Keywords

Connected Graph Formal Power Series Colour Graph Cluster Expansion Articulation Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abd03.
    Abdesselam A.: A physicist’s proof of the Lagrange–Good multivariable inversion formula. J. Phys. A 36, 9471–9477 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. Bru83.
    de Bruijn N.G.: The Lagrange–Good inversion formula and its application to integral equations. J. Math. Anal. Appl. 92, 397–409 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  3. BL64.
    Baert S., Lebowitz J.L.: Convergence of fugacity expansion and bounds on molecular distributions for mixtures. J. Chem. Phys. 40, 3474–3478 (1964)ADSCrossRefMathSciNetGoogle Scholar
  4. BLL98.
    Bergeron F., Labelle G., Leroux P.: Combinatorial Species and Tree-like Structures, Encyclopaedia of Mathematics and its Applications, Vol. 67. Cambridge University Press, Cambridge (1998)Google Scholar
  5. BF38.
    Born M., Fuchs K.: The statistical mechanics of condensing systems. Proc. R. Soc. A 166, 391 (1938)ADSCrossRefGoogle Scholar
  6. EM94.
    Ehrenborg R., Méndez M.: A bijective proof of infinite variated Good’s inversion. Adv. Math. 103, 221–259 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  7. Far10.
    Faris W.G.: Combinatorics and cluster expansions. Probab. Surv. 17, 157–206 (2010)MathSciNetGoogle Scholar
  8. Far12.
    Faris W.G.: Biconnected graphs and the multivariate virial expansion. Markov Proc. Rel. Fields 18, 357–386 (2012)zbMATHMathSciNetGoogle Scholar
  9. Fuc42.
    Fuchs K.: The statistical mechanics of many component gases. Proc. R. Soc. Lond. A. 179, 408–432 (1942)ADSCrossRefGoogle Scholar
  10. Ges87.
    Gessel I.M.: A combinatorial proof of the multivariable Lagrange inversion formula. J. Combin. Theory 45, 178–195 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  11. Good60.
    Good I.J.: Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes. Proc. Cambridge Philos. Soc. 56, 367–380 (1960)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. Good65.
    Good I.J.: The generalization of Lagrange’s expansion and the enumeration of trees. Proc. Cambridge Philos. Soc. 61, 499–517 (1965)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. HM38.
    Harrison S.F., Mayer J.E.: The statistical mechanics of condensing systems. IV. J. Chem. Phys. 6, 101 (1938)ADSCrossRefGoogle Scholar
  14. HL70.
    Henderson D., Leonard P.J.: One- and two-fluid van der Waals theories of liquid mixtures, I. Hard sphere mixtures. Proc. Natl. Acad. Sci. USA 67, 1818–1823 (1970)ADSCrossRefzbMATHGoogle Scholar
  15. Hil56.
    Hill T.L.: Statistical Mechanics: Principles and Selected Applications. McGraw-Hill Series in Advanced Chemistry, New York (1956)zbMATHGoogle Scholar
  16. Jan12.
    Jansen S.: Mayer and virial series at low temperature. J. Stat. Phys. 147, 678–706 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. LP64.
    Lebowitz J.L., Penrose O.: Convergence of virial expansions. J. Math. Phys. 7, 841–847 (1964)ADSCrossRefMathSciNetGoogle Scholar
  18. Ler04.
    Leroux, P.: Enumerative problems inspired by Mayer’s theory of cluster integrals. Electr. J. Combin. 11 (2004) (Research Paper 32)Google Scholar
  19. LR64.
    Lebowitz J.L., Rowlinson J.S.: Thermodynamic properties of mixtures of hard spheres. J. Chem. Phys. 41, 133 (1964)ADSCrossRefGoogle Scholar
  20. May37.
    Mayer J.E.: The statistical mechanics of condensing systems. I. J. Chem. Phys. 5, 67 (1937)ADSCrossRefzbMATHGoogle Scholar
  21. May39.
    Mayer J.E.: Statistical mechanics of condensing systems V. Two-component systems. J. Phys. Chem. 43, 71–95 (1939)CrossRefGoogle Scholar
  22. MA37.
    Mayer J.E., Ackermann P.G.: The statistical mechanics of condensing systems. II. J. Chem. Phys. 5, 74 (1937)ADSCrossRefzbMATHGoogle Scholar
  23. MH38.
    Mayer J.E., Harrison S.F.: The statistical mechanics of condensing systems. III. J. Chem. Phys. 6, 87 (1938)ADSCrossRefGoogle Scholar
  24. MN93.
    Méndez M., Nava O.: Colored species, c-monoids, and plethysm. I. J. Combin. Theory Ser. A 64, 102–129 (1993)CrossRefzbMATHGoogle Scholar
  25. MP13.
    Morais T., Procacci A.: Continuous particles in the canonical ensemble as an abstract polymer gas. J. Stat. Phys. 151, 830–849 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. PU09.
    Poghosyan S., Ueltschi D.: Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50, 053509 (2009)ADSCrossRefMathSciNetGoogle Scholar
  27. PT12.
    Pulvirenti E., Tsagkarogiannis D.: Cluster expansion in the canonical ensemble. Commun. Math. Phys. 316, 289–306 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. Tate13.
    Tate S.: Virial expansion bounds. J. Stat. Phys. 153, 325–338 (2013)ADSCrossRefMathSciNetGoogle Scholar
  29. Uel04.
    Ueltschi D.: Cluster expansions and correlation functions. Moscow Math. J. 4, 511–522 (2004)zbMATHMathSciNetGoogle Scholar
  30. UK38.
    Uhlenbeck G.E., Kahn B.: On the theory of condensation. Physica 5, 399 (1938)ADSCrossRefGoogle Scholar
  31. Zei95.
    Zeidler E.: Applied Functional Analysis, Applied Mathematical Sciences, vol. 109. Springer, New York (1995)Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Sabine Jansen
    • 1
  • Stephen J. Tate
    • 2
  • Dimitrios Tsagkarogiannis
    • 3
  • Daniel Ueltschi
    • 2
  1. 1.Leiden UniversityLeidenThe Netherlands
  2. 2.Department of MathematicsUniversity of WarwickCoventryUK
  3. 3.Department of Applied MathematicsUniversity of CreteHeraklionGreece

Personalised recommendations