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Annales Henri Poincaré

, Volume 19, Issue 4, pp 1115–1150 | Cite as

Convergence of Density Expansions of Correlation Functions and the Ornstein–Zernike Equation

  • Tobias Kuna
  • Dimitrios Tsagkarogiannis
Open Access
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Abstract

We prove absolute convergence of the multi-body correlation functions as a power series in the density uniformly in their arguments. This is done by working in the context of the cluster expansion in the canonical ensemble and by expressing the correlation functions as the derivative of the logarithm of an appropriately extended partition function. In the thermodynamic limit, due to combinatorial cancellations, we show that the coefficients of the above series are expressed by sums over some class of two-connected graphs. Furthermore, we prove the convergence of the density expansion of the “direct correlation function” which is based on a completely different approach and it is valid only for some integral norm. Precisely, this integral norm is suitable to derive the Ornstein–Zernike equation. As a further outcome, we obtain a rigorous quantification of the error in the Percus–Yevick approximation.

Notes

Acknowledgements

We acknowledge support from the London Mathematical Society via a research in pairs Scheme 4 Grant. T. K. acknowledges support from J. Lebowitz via NSF Grant DMR 1104501 and AFOSR Grant F49620-01-0154. We would like to thank the anonymous referees for their valuable and detailed suggestions.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of ReadingReadingUK
  2. 2.Department of MathematicsUniversity of SussexBrightonUK

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