Journal of Statistical Physics

, Volume 61, Issue 5–6, pp 1187–1201 | Cite as

Moments of the Percus-Yevick hard-sphere correlation function

  • N. E. Berger
  • V. Twersky


A simple recursive relation is derived for the momentsM n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheMn are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theMn(w) include testing different approximations forh by numerical integration ofh(r) r n . We compare exact moments with shell approximationsM n [h s ] corresponding to integration fromr=0 tos+1 fors=3−8, and with hybrid approximationsM n [h s +h a ] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to ∞. For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM n [h3+h a ] is even better thanM n [h8]

Key words

Percus-Yevick correlation function moments shell expansions asymptotic forms residue series hybrid approximations 


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  1. 1.
    J. K. Percus and G. J. Yevick, Analysis of classical statistical mechanics by means of collective coordinates,Phys. Rev. 110:1–13 (1958).Google Scholar
  2. 2.
    M. S. Wertheim, Exact solution of the Percus-Yevick integral equation for hard spheres,Phys. Rev. Lett. 10:321–323 (1963).Google Scholar
  3. 3.
    E. Thiele, Equation of state for hard spheres,J. Chem. Phys. 39:474–479 (1963).Google Scholar
  4. 4.
    W. R. Smith and D. Henderson, Analytical representation of the Percus-Yevick hard-sphere radial distribution function,Mol. Phys. 19:411–415 (1970).Google Scholar
  5. 5.
    D. Henderson and W. R. Smith, Exact analytical formulas for the distribution functions of charged hard spheres in the mean spherical approximation,J. Stat. Phys. 19:191–200 (1978).Google Scholar
  6. 6.
    G. J. Throop and R. J. Bearman, Numerical solution of the Percus-Yevick equation for the hard sphere potential,J. Chem. Phys. 42:2408–2411 (1965); F. Mandel, R. J. Bearman, and M. Y. Bearman, Numerical solution of the Percus-Yevick equation for the Lennard- Jones (6–12) and hard sphere potentials,J. Chem. Phys. 52:3315–3323 (1970).Google Scholar
  7. 7.
    V. Twersky, Coherent scalar field in pair-correlated random distributions of aligned scatterers,J. Math. Phys. 18:2468–2486 (1977);J. Acoust. Soc. Am. 64:1710–1719 (1978).Google Scholar
  8. 8.
    H. D. Jones, Method of finding the equation of state of liquid metals,J. Chem. Phys. 55:2640–2642 (1971).Google Scholar
  9. 9.
    I. Nezbeda, Analytical solution of the Percus-Yevick equation for fluid and hard spheres,Czech J. Phys. B 24:55–62 (1974).Google Scholar
  10. 10.
    R. J. Baxter, Orenstein-Zernike relation for a disordered fluid,Aust. J. Phys. 21:563–569 (1968).Google Scholar
  11. 11.
    J. G. Kirkwood, Statistical mechanics of fluid mixtures,J. Chem. Phys. 3:300–313 (1935).Google Scholar
  12. 12.
    B. R. A. Nijboer and L. van Hove, Radial distribution function of a gas of hard spheres and the superposition approximation,Phys. Rev. 85:777–783 (1952).Google Scholar
  13. 13.
    Symbolic manipulation package Reduce, Version 3.2, Rand Corporation, Santa Monica, California (1985).Google Scholar
  14. 14.
    N. W. Ashcroft and J. Lekner, Structure and resistivity of liquid metals,Phys. Rev. 145:83–90 (1966).Google Scholar
  15. 15.
    S. W. Hawley, T. H. Kays, and V. Twersky, Comparison of distribution functions from scattering data on different sets of spheres,IEEE Trans. Antennas Propagation AP-15:118–135 (1967).Google Scholar
  16. 16.
    H. Reiss, H. L. Frisch, and J. L. Lebowitz, Statistical mechanics of rigid spheres,J. Chem. Phys. 31:369–380 (1959).Google Scholar
  17. 17.
    P. Henrici,Applied Computational Complex Analysis, Vol.2 (Wiley, New York, 1977), pp. 442ff.Google Scholar
  18. 18.
    G. David Scott, Packing of spheres,Nature 188:908–909 (1960).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • N. E. Berger
    • 1
  • V. Twersky
    • 1
  1. 1.Mathematics DepartmentUniversity of IllinoisChicago

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