Abstract
We consider a binary system of small and large objects in the continuous space interacting via a nonnegative potential. By integrating over the small objects, the effective interaction between the large ones becomes multi-body. We prove convergence of the cluster expansion for the grand canonical ensemble of the effective system of large objects. To perform the combinatorial estimate of hypergraphs (due to the multi-body origin of the interaction), we exploit the underlying structure of the original binary system. Moreover, we obtain a sufficient condition for convergence which involves the surface of the large objects rather than their volume. This amounts to a significant improvement in comparison to a direct application of the known cluster expansion theorems. Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asakura, S., Oosawa, F.: Interaction between particles suspended in solutions of macromolecules. J. Polym. Sci. 33(126), 183–192 (1958)
Brydges, D.C.: A short course on cluster expansions. In: Osterwalder, K., Stora, R. (eds.) Phénomènes critiques. systèmes aléatoires, théories de jauge, part I, II (Les Houches, 1984), pp. 129–183. North-Holland, Amsterdam (1986)
Brydges, D.C.: Lectures on the Renormalisation Group. Statistical Mechanics, IAS/Park City Mathematics Series, vol. 16, pp. 7–93. American Mathematical Society, Providence (2009)
Binder, K., Virnau, P., Statt, A.: Perspective: the Asakura Oosawa model: a colloid prototype for bulk and interfacial phase behavior. J. Chem. Phys. 141(14), 140901 (2014)
Bovier, A., Zahradník, M.: A simple inductive approach to the problem of convergence of cluster expansions of polymer models. J. Stat. Phys. 100(3–4), 765–778 (2000)
Cassandro, M., Olivieri, E.: Renormalization group and analyticity in one dimension: a proof of Dobrushin’s theorem. Comm. Math. Phys. 80(2), 255–269 (1981)
Gallavotti, G.: Statistical Mechanics. A Short Treatise, Theoretical and Mathematical Physics. Springer, Berlin (1999)
Greenberg, W.: Thermodynamic states of classical systems. Comm. Math. Phys. 22, 259–268 (1971)
Hill, T.L.: Statistical Mechanics: Principles and Selected Applications. McGraw-Hill Book Co., Inc., New York (1956)
Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Comm. Math. Phys. 103(3), 491–498 (1986)
Lekkerkerker, H.N.W., Tuinier, R.: Colloids and the Depletion Interaction. Lecture Notes in Physics, vol. 833. Springer, Berlin (2011)
Mao, Y., Cates, M.E., Lekkerkerker, H.N.W.: Depletion force in colloidal systems. Phys. A 222, 10–24 (1995)
Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields. Cluster Expansions, Mathematics and Its Applications (Soviet Series), vol. 44. Kluwer Academic Publishers Group, Dordrecht (1991). Translated from the Russian by R. Kotecký and P. Holický
Moraal, H.: The Kirkwood–Salsburg equation and the virial expansion for many-body potentials. Phys. Lett. A 59(1), 9–10 (1976)
Minlos, R.A., Pogosjan, S.K.: Estimates of Ursell functions, group functions and their derivatives. Teoret. Mat. Fiz. 31, 408–418 (1977)
Penrose, O.: Convergence of fugacity expansions for classical systems. In: Bak, T.A. (ed.) Statistical Mechanics: Foundations and Applications, proceedings of the I. U. P. A. P. Meeting, Copenhagen, 1966, p. 101. W., A. Benjamin, Inc., New York (1967)
Procacci, A., Scoppola, B.: The gas phase of continuous systems of hard spheres interacting via \(n\)-body potential. Comm. Math. Phys. 211(2), 487–496 (2000)
Poghosyan, S., Ueltschi, D.: Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50(5), 053509 (2009)
Rebenko, A.L.: Polymer expansions for continuous classical systems with many-body interaction. Methods Funct. Anal. Topol. 11(1), 73–87 (2005)
Ruelle, D.: Statistical Mechanics: Rigorous Results. World Scientific, Singapore (1969)
Sapozhenko, A.A.: Hypergraph. In: Springer Encyclopedia of Mathematics (2011). https://www.encyclopediaofmath.org/index.php/Hypergraph
Sator, N.: Clusters in simple fluids. Phys. Rep. 376(1), 1–39 (2003)
Stillinger, F.H.: Rigorous basis of the Frenkel–Band theory of association equilibrium. J. Chem. Phys. 38(7), 1486–1494 (1963)
Ueltschi, D.: Cluster expansions and correlation functions. Mosc. Math. J. 4(2), 511–522 (2004)
Widom, B., Rowlinson, J.S.: New model for the study of liquid–vapor phase transitions. J. Chem. Phys. 52(4), 1670–1684 (1970)
Acknowledgements
The main part of this article was completed when both authors were members of the Department of Mathematics at the University of Sussex; the authors acknowledge the department for the nice atmosphere. The authors also wish to acknowledge Stephen J. Tate who contributed at initial stages of this work. S.J. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Abdelmalek Abdesselam.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Jansen, S., Tsagkarogiannis, D. Cluster Expansions with Renormalized Activities and Applications to Colloids. Ann. Henri Poincaré 21, 45–79 (2020). https://doi.org/10.1007/s00023-019-00868-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-019-00868-2