Journal of Statistical Physics

, Volume 57, Issue 1–2, pp 267–288 | Cite as

m-Particle correlations in theN-particle McKean model

  • R. Krieg
  • K. -J. Schmitt
  • C. Toepffer


This work extends the comparison between exact and approximate solutions of the McKean model to finite particle numbers. We derive the coupled linear equations of motion for them-body densities (BBGKY hierarchy) and the corresponding nonlinear equations for them-body correlation functions. We calculate the stable fixed points and the subspace admitting a probabilistic interpretation for both descriptions of the model. Neglecting higher correlations withm>n, we obtain approximate solutions, which are compared to the exact one. In this way various truncation effects can be studied, such as the appearance of saddle points and unphysical trajectories. Finally, we linearize the truncated equations for the correlations about the stable fixed point, and calculate the relaxation times up toO(N−1).

Key words

BBGKY hierarchy correlations cluster expansion truncation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Yu. L. Klimontovich,Kinetic Theory of Nonideal Gases and Nonideal Plasmas (Pergamon Press, Oxford, 1982), Chapter 1 and 2.Google Scholar
  2. 2.
    J. P. Hansen and I. R. McDonald,Theory of Simple Liquids (Academic Press, London, 1976), Chapter 5.Google Scholar
  3. 3.
    L. P. Kadanoff and G. Baym,Quantum Statistical Mechanics (Benjamin, New York, 1962), Chapter 6.3 and 10.3; K. Gütter, P.-G. Reinhard, and C. Toepffer,Phys. Rev. A 38:164 (1988).Google Scholar
  4. 4.
    K.-J. Schmitt, P.-G. Reinhard, and C. Toepffer,Phys. Rev. A 40:474 (1989).Google Scholar
  5. 5.
    H. P. McKean, Jr.,Comb. Theory 2:358 (1967).Google Scholar
  6. 6.
    K.-J. Schmitt,J. Stat. Phys. 46:283 (1987).Google Scholar
  7. 7.
    F. Henin,Physica 76:201 (1974).Google Scholar
  8. 8.
    N. N. Bogoliubov, inStudies in Statistical Mechanics, J. de Boer and G. R. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, Part B, Chapter 2.Google Scholar
  9. 9.
    J. Yvon,Les Corrélations et l'Entropie en Mécanique Statistique Classique (Dunod, Paris, 1966), Chapter 2.Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • R. Krieg
    • 1
  • K. -J. Schmitt
    • 1
  • C. Toepffer
    • 1
  1. 1.Institut für Theoretische Physik II der Universität Erlangen-NürnbergErlangenWest Germany

Personalised recommendations