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1 results

  • 11 - Cluster Expansions

  • J. R. Dorfman, University of Maryland, College Park, Henk van Beijeren, Universiteit Utrecht, The Netherlands, T. R. Kirkpatrick, University of Maryland, College Park
  • Book:
    Contemporary Kinetic Theory of Matter
    Published online:
    18 June 2021
    Print publication:
    24 June 2021, pp 437-466

  • Summary

    Symmetric functions of phase variables of N particles, can be expanded, formally, in terms of a series of Ursell cluster functions. They depend successively on one, two, ..., particle variables. For equilibrium systems, cluster expansions are used to obtain virial expansions of the thermodynamic functions, The cluster expansion method can be applied to N-particle time displacement operators and to the initial distribution function for a non-equilibrium system. Assuming a factorization property of the initial distribution function, one obtains expansions of the time dependent two and higher particle distributions in terms of successively higher products of one particle functions. This expansion of the pair distribution function, together with the first hierarchy equation generalizes the Boltzmann equation to higher densities in terms of the dynamics of successively higher number of particles considered in isolation. Contributions from correlated binary collision sequences appear and for hard spheres the Enskog equation is an approximation.