A dynamical classification of the range of pair interactions

Abstract

We formalize a classification of pair interactions based on the convergence properties of the {\it 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) \rightarrow \infty) \sim 1/r^a defining a {\it bounded} pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the {\it pair force} is absolutely integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to this case as {\it 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 {\it dynamically long-range} case, i.e., a \leq d-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 a \leq d-1 (and notably, for the case of gravity, a=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 a > d-2 (or a < d-2), for which the PDF of the {\it 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.

12 pages, 1 figure; significantly shortened and focussed, additional references, version to appear in J. Stat. Phys

References in corpus (9)

• Out-of-equilibrium tricritical point in a system with long-range interactions
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model
• Slow relaxation in long-range interacting systems with stochastic dynamics
• 1-d gravity in infinite point distributions
• Microcanonical quasi-stationarity of long-range interacting systems in contact with a heat bath
• Statistics of the gravitational force in various dimensions of space: from Gaussian to Levy laws
• The field inside a random distribution of parallel dipoles
• Force distribution in a randomly perturbed lattice of identical particles with $1/r^2$ pair interaction
• Gravitational force in an infinite one-dimensional Poisson distribution

Cited by in corpus (11)

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• Light propagation and the average expansion rate in near-FRW universes
• Caloric curves of classical self-gravitating systems in general relativity
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• Formation of disks with long-lived spiral arms from violent gravitational dynamics
• Finite $N$ corrections to Vlasov dynamics and the range of pair interactions
• Projection of the gravitational dynamics on a subspace of probability distributions: curl-free Gaussian ansatz
• Self-similarity and stable clustering in a family of scale-free cosmologies
• Lagrangian-space Gaussian ansatz for the matter redshift-space power spectrum and correlation function
• Existence of shape-dependent thermodynamic limit in spin systems with short- and long-range interactions
• Expansion into the vacuum of stochastic gases with long-range interactions