Advertisement

Letters in Mathematical Physics

, Volume 108, Issue 2, pp 285–306 | Cite as

Fréchet differentiability of molecular distribution functions I. \(L^\infty \) analysis

  • Martin HankeEmail author
Article

Abstract

For a grand canonical ensemble of classical point-like particles at equilibrium in continuous space, we investigate the functional relationship between a stable and regular pair potential describing the interaction of the particles and the corresponding molecular distribution functions. For certain admissible perturbations of the pair potential and sufficiently small activity, we rigorously establish Frechet differentiability with respect to the supremum norm in the image space—both for bounded domains and in the thermodynamical limit.

Keywords

Statistical mechanics Molecular distribution function Fréchet derivative Kirkwood–Salsburg equations 

Mathematics Subject Classification

82B21 82B80 

References

  1. 1.
    Fisher, M.E., Ruelle, D.: The stability of many-particle systems. J. Math. Phys. 7, 260–270 (1966)MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Hanke, M.: Fréchet differentiability of molecular distribution functions II. The Ursell function, preprint. Lett. Math. Phys. doi: 10.1007/s11005-017-1010-7 (2017)
  3. 3.
    Hansen, J.-P., McDonald, I.R.: Theory of Simple Liquids, 4th edn. Academic Press, Oxford (2013)zbMATHGoogle Scholar
  4. 4.
    Henderson, R.L.: A uniqueness theorem for fluid pair correlation functions. Phys. Lett. A 49, 197–198 (1974)CrossRefADSGoogle Scholar
  5. 5.
    Koralov, L.: An inverse problem for Gibbs fields with hard core potential. J. Math. Phys. 48, 053301 (2007)MathSciNetCrossRefzbMATHADSGoogle Scholar
  6. 6.
    Lyubartsev, A.P., Laaksonen, A.: Calculation of effective interaction potentials from radial distribution functions: a reverse Monte Carlo approach. Phys. Rev. E 52, 3730–3737 (1995)CrossRefADSGoogle Scholar
  7. 7.
    Rühle, V., Junghans, C., Lukyanov, A., Kremer, K., Andrienko, D.: Versatile object-oriented toolkit for coarse-graining applications. J. Chem. Theory Comput. 5, 3211–3223 (2009)CrossRefGoogle Scholar
  8. 8.
    Ruelle, D.: Cluster property of the correlation functions of classical gases. Rev. Mod. Phys. 36, 580–584 (1964)MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. W.A. Benjamin Publ, New York (1969)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany

Personalised recommendations