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Physical Review E 81 (2010) 21102
Gravitational force in an infinite one-dimensional Poisson distribution
A. Gabrielli, M. Joyce1

We consider the statistical properties of the gravitational field F in an infinite one-dimensional homogeneous Poisson distribution of particles using an exponential cutoff of the pair interaction to control and study the divergences which arise. Deriving an exact analytic expression for the probability density function (PDF) P(F) , we show that it is badly defined in the limit in which the well-known Holtzmark distribution is obtained in the analogous three-dimensional case. A well-defined P(F) may, however, be obtained in the infinite range limit by an appropriate renormalization of the coupling strength giving a Gaussian form. Calculating the spatial correlation properties we show that this latter procedure has a trivial physical meaning. Finally we calculate the PDF and correlation properties of differences of forces (at separate spatial points), which are well defined without any renormalization. We explain that the convergence of these quantities is in fact sufficient to allow a physically meaningful infinite system limit to be defined for the clustering dynamics from Poissonian initial conditions.
1 :  LPNHE - Laboratoire de Physique Nucléaire et de Hautes Énergies
Physique/Astrophysique/Cosmologie et astrophysique extra-galactique

Planète et Univers/Astrophysique/Cosmologie et astrophysique extra-galactique

Physique/Matière Condensée/Mécanique statistique
Classical statistical mechanics – Numerical simulations of chaotic systems – Other topics in statistical physics – thermodynamics – and nonlinear dynamical systems
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