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Virial expansion bounds. (English) Zbl 1296.82039
Summary: In the 1960s, the technique of using cluster expansion bounds in order to achieve bounds on the virial expansion was developed by J. L. Lebowitz and O. Penrose [“Convergence of virial expansions”, J. Math. Phys. 5, 841 (1964)] and D. Ruelle [Statistical mechanics: rigorous results. New York-Amsterdam: W. A. Benjamin (1969; Zbl 0177.57301)]. This technique is generalised to more recent cluster expansion bounds by S. Poghosyan and D. Ueltschi [J. Math. Phys. 50, No. 5, 053509, 17 p. (2009; Zbl 1187.82009)], which are related to the work of A. Procacci [J. Stat. Phys. 129, No. 1, 171–188 (2007; Zbl 1206.82149)] and the tree-graph identity, detailed by D. C. Brydges [Critical phenomena, random systems, gauge theories, Proc. Summer Sch. Theor. Phys., Sess. 43, Les Houches/France 1984, Pt. 1, 129–183 (1986; Zbl 0659.60136)]. The bounds achieved by Lebowitz and Penrose [loc. cit. ]can also be sharpened by doing the actual optimisation and achieving expressions in terms of the Lambert W-function. The different bound from the cluster expansion shows some improvements for bounds on the convergence of the virial expansion in the case of positive potentials, which are allowed to have a hard core.

MSC:
82C22 Interacting particle systems in time-dependent statistical mechanics
82D05 Statistical mechanical studies of gases
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References:
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