WJET  Vol.7 No.2 , May 2019
New Formulas for the Mayer and Ree-Hoover Weights of Infinite Families of Graphs
Author(s) Amel Kaouche
ABSTRACT
The virial expansion, in statistical mechanics, makes use of the sums of the Mayer weight of all 2-connected graphs on n vertices. We study the Second Mayer weight ωM(c) and the Ree-Hoover weight ωRH(c) of a 2-connected graph c which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph c. In the present work, we use the method of graph homomorphisms, to give new formulas of Mayer and Ree-Hoover weights for special infinite families of 2-connected graphs.
Cite this paper
Kaouche, A. (2019) New Formulas for the Mayer and Ree-Hoover Weights of Infinite Families of Graphs. World Journal of Engineering and Technology, 7, 283-292. doi: 10.4236/wjet.2019.72019.
References
[1]   Kaouche, A. (2010) Invariants de graphes liés aux gaz imparfaits. Publications du Laboratoire de Combinatoire et d’Informatique Mathematique (LaCIM), 42.

[2]   Kaouche, A. and Leroux, P. (2008) Graph Weights Arising from Mayer and Ree-Hoover Theories. DMTCS Proceedings, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 259-270.

[3]   Kaouche, A. and Leroux, P. (2009) Mayer and Ree-Hoover Weights of Infinite Families of 2-Connected Graphs. Séminaire Lotharingien de Combinatoire 61A, Article B61Af.

[4]   Labelle, G., Leroux, P. and Ducharme, M.G. (2007) Graph Weights Arising from Mayer’s Theory of Cluster Integrals. Séminaire Lotharingien de Combinatoire, 54, Article B54m.

[5]   Uhlenbeck, G.E. and Ford, G.W. (1963) Lectures in Statistical Mechanics. American Mathematical Society, Providence, 181 p.

[6]   Ree, F.H. and Hoover, W.G. (1964) Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Discs. The Journal of Chemical Physics, 40, 939-950. https://doi.org/10.1063/1.1725286

[7]   Ree, F.H. and Hoover, W.G. (1964) Reformulation of the Virial Series for Classical Fluids. The Journal of Chemical Physics, 41, 1635-1645. https://doi.org/10.1063/1.1726136

[8]   Ree, F.H. and Hoover, W.G. (1967) Seventh Virial Coefficients for Hard Spheres and Hard Discs. The Journal of Chemical Physics, 46, 4181-4196. https://doi.org/10.1063/1.1840521

[9]   Clisby, N. and McCoy, B.M. (2004) Negative Virial Coefficients and the Dominance of Loose Packed Diagrams for D-Dimensional Hard Spheres. Journal of Statistical Physics, 114. https://doi.org/10.1023/B:JOSS.0000013960.83555.7d

[10]   Clisby, N. and McCoy, B.M. (2006) Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions. Journal of Statistical Physics, 122, 15-57. https://doi.org/10.1007/s10955-005-8080-0

[11]   Kaouche, A. (2016) Valeurs des poids de Mayer et des poids de Ree-Hoover pour tous les graphes 2-connexes de taille au plus 8 et leurs parametres descriptifs. http://professeure.umoncton.ca/umce-kaouche_amel/files/umce-kaouche_amel/wf/wf/TableauRH7.pdf http://professeure.umoncton.ca/umce-kaouche_amel/files/umce-kaouche_amel/wf/wf/TableauRH8.pdf

[12]   Kaouche, A. and Labelle, G. (2008) Mayer Polytopes and divided differences, Congrès Combinatorial Identities and Their Applications in Statistical Mechanics, Isaac Newton Institute de l’Université de Cambridge, en Angleterre.

[13]   Kaouche, A. and Labelle, G. (2013) Mayer and Ree-Hoover Weights, Graphs Invariants and Bipartite Complete Graphs. Pure Mathematics and Applications, 24, 19-29.

[14]   Kaouche, A., Labelle, G. (2014) Poids de Mayer et transformées de Fourier. Annales Mathématiques du Québec Springer-Verlag, 38, 37-59. https://doi.org/10.1007/s40316-014-0018-y

 
 
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