Journal of Statistical Physics

, Volume 99, Issue 1–2, pp 273–312 | Cite as

Wall-Induced Density Profiles and Density Correlations in Confined Takahashi Lattice Gases

  • J. Buschle
  • P. Maass
  • W. Dieterich


We propose a general formalism to study the static properties of a system composed of particles with nearest neighbor interactions that are located on the sites of a one-dimensional lattice confined by walls (“confined Takahashi lattice gas”). Linear recursion relations for generalized partition functions are derived, from which thermodynamic quantities, as well as density distributions and correlation functions of arbitrary order can be determined in the presence of an external potential. Explicit results for density profiles and pair correlations near a wall are presented for various situations. As a special case of the Takahashi model we consider in particular the hard rod lattice gas, for which a system of nonlinear coupled difference equations for the occupation probabilities has been presented by Robledo and Varea. A solution of these equations is given in terms of the solution of a system of independent linear equations. Moreover, for zero external potential in the hard-rod system we specify various central regions between the confining walls, where the occupation probabilities are constant and the correlation functions are translationally invariant in the canonical ensemble. In the grand canonical ensemble such regions do not exist.

density functional theory nonuniform lattice gases nonlinear difference equations density correlations confined systems Takahashi interaction hard rods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fluid Interfacial Phenomena, C. A. Croxton, ed. (Wiley, New York, 1986).Google Scholar
  2. 2.
    Liquids at Interfaces, Les Houches Summer School Lectures, Vol. XLVIII, J. Chavrolin, J. F. Joanny, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1990).Google Scholar
  3. 3.
    E. Delamarche, B. Michel, H. A. Biebuyck, and Christoph Gerber, Adv. Mater. 8:719 (1996).Google Scholar
  4. 4.
    K. Binder, in Phase Transitions and Critical Phenomena, C. Domb and J. Lebowitz, eds., Vol. 8 (Academic Press, London, 1986).Google Scholar
  5. 5.
    S. Dietrich, in Phase Transitions and Critical Phenomena, C. Domb and J. Lebowitz, eds., Vol. 12 (Academic Press, London, 1988), p. 1.Google Scholar
  6. 6.
    M. Schick, in Liquids at Interfaces, Les Houches Summer School Lectures, Vol. XLVIII, J. Chavrolin, J. F. Joanny, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1990), p. 415.Google Scholar
  7. 7.
    S. Puri and H. L. Frisch, J. Phys. Condens. Matter 9, 2109 (1997).Google Scholar
  8. 8.
    R. Evans, in Fundamentals of Inhomogeneous Fluids, D. Henderson, ed. (Marcel Dekker, New York, 1992), p. 85.Google Scholar
  9. 9.
    R. Kikuchi, Phys. Rev. 81:988 (1951).Google Scholar
  10. 10.
    R. Kikuchi, Prog. Theor. Phys. (Kyoto) Suppl. 35:1 (1966).Google Scholar
  11. 11.
    S. J. Salter and H. T. Davies, J. Chem. Phys. 63:157 (1975).Google Scholar
  12. 12.
    Y. Rosenfeld, M. Schmidt, H. Löwen, P. Tarazona, J. Phys. Condensed Matter 8:L577 (1996).Google Scholar
  13. 13.
    J. K. Percus, J. Stat. Phys. 15:505 (1976).Google Scholar
  14. 14.
    J. K. Percus, J. Stat. Phys. 28:67 (1982).Google Scholar
  15. 15.
    J. K. Percus, J. Phys. Condensed Matter 1:2911 (1989).Google Scholar
  16. 16.
    A. Robledo and C. Varea, J. Stat. Phys. 26:513 (1981).Google Scholar
  17. 17.
    J. K. Percus, Acc. Chem. Rev. 27:8 (1994).Google Scholar
  18. 18.
    J. Buschle, Diplomarbeit (Universität Konstanz, 1999), unpublished.Google Scholar
  19. 19.
    H. Takahashi, Proceedings of the Physico-Mathematical Society of Japan (Nippon Suugaku-Buturigakkwai Kizi Tokyo) 24:60 (1942). For a translation from the German see Mathematical Physics in One Dimension, E. H. Lieb and D. C. Mattis, eds. (Academic Press, New York, 1966), p. 25.Google Scholar
  20. 20.
    R. P. Stanley, Enumerative Combinatorics, Vol. I (Wadsworth & Brooks, Belmont/California, 1986), p. 202.Google Scholar
  21. 21.
    H. S. Leff and M. H. Coopersmith, J. Math. Phys. 8:306 (1967).Google Scholar
  22. 22.
    M. Flicker, J. Math. Phys. 9:171 (1968).Google Scholar
  23. 23.
    N. D. Mermin, Phys. Rev. B 137:1441 (1965).Google Scholar
  24. 24.
    A. Lenard, J. Math. Phys. 2:682 (1961), see Lemma 3.Google Scholar
  25. 25.
    W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I (Wiley & Sons, New York, 1968), p. 330.Google Scholar
  26. 26.
    I. J. Good, Ann. Math. Stat. 28:861 (1957).Google Scholar
  27. 27.
    J. Riordan, Combinatorial Identities (Wiley & Sons, New York, 1968).Google Scholar
  28. 28.
    I. P. Goulden and D. M. Jackson, Combinatorial Enumerations (Wiley & Sons, New York, 1983), p. 17.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • J. Buschle
    • 1
  • P. Maass
    • 1
  • W. Dieterich
    • 1
  1. 1.Fakultät für PhysikUniversität KonstanzKonstanzGermany

Personalised recommendations