Gravitational force in an infinite one-dimensional Poisson distribution

Abstract

We consider the statistical properties of the gravitational field F in an infinite one-dimensional homogeneous Poisson distribution of particles, using an exponential cut-off 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.

9 pages, minor changes, final version published in Phys. Rev. E

References in corpus (5)

• 1-d gravity in infinite point distributions
• Relaxation of a 1-D gravitational system
• The field inside a random distribution of parallel dipoles
• Gravitational Dynamics of an Infinite Shuffled Lattice of Particles
• Force distribution in a randomly perturbed lattice of identical particles with $1/r^2$ pair interaction

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• Entropy production and Vlasov equation for self-gravitating systems
• Finite $N$ corrections to Vlasov dynamics and the range of pair interactions
• One-Dimensional Relativistic Self-Gravitating Systems