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.
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Abdesselam A.: A physicist’s proof of the Lagrange–Good multivariable inversion formula. J. Phys. A 36, 9471–9477 (2003)
de Bruijn N.G.: The Lagrange–Good inversion formula and its application to integral equations. J. Math. Anal. Appl. 92, 397–409 (1983)
Baert S., Lebowitz J.L.: Convergence of fugacity expansion and bounds on molecular distributions for mixtures. J. Chem. Phys. 40, 3474–3478 (1964)
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)
Born M., Fuchs K.: The statistical mechanics of condensing systems. Proc. R. Soc. A 166, 391 (1938)
Ehrenborg R., Méndez M.: A bijective proof of infinite variated Good’s inversion. Adv. Math. 103, 221–259 (1994)
Faris W.G.: Combinatorics and cluster expansions. Probab. Surv. 17, 157–206 (2010)
Faris W.G.: Biconnected graphs and the multivariate virial expansion. Markov Proc. Rel. Fields 18, 357–386 (2012)
Fuchs K.: The statistical mechanics of many component gases. Proc. R. Soc. Lond. A. 179, 408–432 (1942)
Gessel I.M.: A combinatorial proof of the multivariable Lagrange inversion formula. J. Combin. Theory 45, 178–195 (1987)
Good I.J.: Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes. Proc. Cambridge Philos. Soc. 56, 367–380 (1960)
Good I.J.: The generalization of Lagrange’s expansion and the enumeration of trees. Proc. Cambridge Philos. Soc. 61, 499–517 (1965)
Harrison S.F., Mayer J.E.: The statistical mechanics of condensing systems. IV. J. Chem. Phys. 6, 101 (1938)
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)
Hill T.L.: Statistical Mechanics: Principles and Selected Applications. McGraw-Hill Series in Advanced Chemistry, New York (1956)
Jansen S.: Mayer and virial series at low temperature. J. Stat. Phys. 147, 678–706 (2012)
Lebowitz J.L., Penrose O.: Convergence of virial expansions. J. Math. Phys. 7, 841–847 (1964)
Leroux, P.: Enumerative problems inspired by Mayer’s theory of cluster integrals. Electr. J. Combin. 11 (2004) (Research Paper 32)
Lebowitz J.L., Rowlinson J.S.: Thermodynamic properties of mixtures of hard spheres. J. Chem. Phys. 41, 133 (1964)
Mayer J.E.: The statistical mechanics of condensing systems. I. J. Chem. Phys. 5, 67 (1937)
Mayer J.E.: Statistical mechanics of condensing systems V. Two-component systems. J. Phys. Chem. 43, 71–95 (1939)
Mayer J.E., Ackermann P.G.: The statistical mechanics of condensing systems. II. J. Chem. Phys. 5, 74 (1937)
Mayer J.E., Harrison S.F.: The statistical mechanics of condensing systems. III. J. Chem. Phys. 6, 87 (1938)
Méndez M., Nava O.: Colored species, c-monoids, and plethysm. I. J. Combin. Theory Ser. A 64, 102–129 (1993)
Morais T., Procacci A.: Continuous particles in the canonical ensemble as an abstract polymer gas. J. Stat. Phys. 151, 830–849 (2013)
Poghosyan S., Ueltschi D.: Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50, 053509 (2009)
Pulvirenti E., Tsagkarogiannis D.: Cluster expansion in the canonical ensemble. Commun. Math. Phys. 316, 289–306 (2012)
Tate S.: Virial expansion bounds. J. Stat. Phys. 153, 325–338 (2013)
Ueltschi D.: Cluster expansions and correlation functions. Moscow Math. J. 4, 511–522 (2004)
Uhlenbeck G.E., Kahn B.: On the theory of condensation. Physica 5, 399 (1938)
Zeidler E.: Applied Functional Analysis, Applied Mathematical Sciences, vol. 109. Springer, New York (1995)
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Jansen, S., Tate, S.J., Tsagkarogiannis, D. et al. Multispecies Virial Expansions. Commun. Math. Phys. 330, 801–817 (2014). https://doi.org/10.1007/s00220-014-2026-9
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DOI: https://doi.org/10.1007/s00220-014-2026-9