Journal of Statistical Physics

, Volume 141, Issue 6, pp 970–989 | Cite as

A Dynamical Classification of the Range of Pair Interactions

Article

Abstract

We formalize a classification of pair interactions based on the convergence properties of the forces acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the “usual” thermodynamic limit. For a pair interaction potential V(r) with V(r→∞)∼1/rγ defining a bounded pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the pair force is absolutely integrable, i.e., for γ>d−1, where d is the spatial dimension. We refer to this case as dynamically short-range, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the dynamically long-range case, i.e., γd−1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for γd−1 (and notably, for the case of gravity, γ=d−2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with γ>d−2 (or γ<d−2), for which the PDF of the difference in forces is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system.

Keywords

Long range interactions Clustering dynamics Thermodynamic limit Classification 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • A. Gabrielli
    • 1
    • 2
  • M. Joyce
    • 3
    • 4
  • B. Marcos
    • 5
  • F. Sicard
    • 3
  1. 1.Istituto dei Sistemi Complessi (ISC)—CNRRomeItaly
  2. 2.ISC-CNR, Physics DepartmentUniversity “Sapienza” of RomeRomeItaly
  3. 3.Laboratoire de Physique Nucléaire et Hautes ÉnergiesUniversité Pierre et Marie Curie—Paris 6, CNRS IN2P3 UMR 7585Paris Cedex 05France
  4. 4.Laboratoire de Physique Théorique de la Matière CondenséeUniversité Pierre et Marie Curie—Paris 6, CNRS UMR 7600Paris Cedex 05France
  5. 5.Laboratoire J.-A. DieudonnéUMR 6621, Université de Nice—Sophia AntipolisNice Cedex 02France

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