Force distribution in a randomly perturbed lattice of identical particles with 1/r² pair interaction

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Andrea Gabrielli
Istituto dei Sistemi Complessi–CNR, Via dei Taurini 19, 00185 Rome, Italy and SMC-INFM, Physics Department, University "La Sapienza" of Rome, 00185 Rome, Italy

Thierry Baertschiger
Dipartimento di Fisica, Università "La Sapienza," 00185 Rome, Italy and Istituto dei Sistemi Complessi–CNR, Via dei Taurini 19, 00185 Rome, Italy

Michael Joyce
Laboratoire de Phyisique Nucléaire et Hautes Energies, Université Pierre et Marie Curie–Paris 6, UMR 7585, F-75005 Paris, France

Bruno Marcos
Dipartimento di Fisica, Università "La Sapienza," 00185 Rome, Italy, and Istituto dei Sistemi Complessi–CNR, Via dei Taurini 19, 00185 Rome, Italy

Francesco Sylos Labini
E. Fermi Center, Via Panisperna 89 A, Compendio del Viminale, 00184 Rome, Italy and Istituto dei Sistemi Complessi–CNR, Via dei Taurini 19, 00185 Rome, Italy

Received 6 March 2006; revised 22 June 2006; published 9 August 2006

Westudy the statistics of the force felt by a particlein the class of a spatially correlated distribution of identicalpointlike particles, interacting via a 1/r² pair force (i.e., gravitationalor Coulomb), and obtained by randomly perturbing an infinite perfectlattice. We specify the conditions under which the force ona particle is a well-defined stochastic quantity. We then studythe small displacements approximation, giving both the limitations of itsvalidity and, when it is valid, an expression for theforce variance. The method introduced by Chandrasekhar to find theforce probability density function for the homogeneous Poisson particle distributionis extended to shuffled lattices of particles. In this way,we can derive an approximate expression for the probability distributionof the force over the full range of perturbations ofthe lattice, i.e., from very small (compared to the latticespacing) to very large where the Poisson limit is recovered.We show in particular the qualitative change in the large-forcetail of the force distribution between these two limits. Excellentaccuracy of our analytic results is found on detailed comparisonwith results from numerical simulations. These results provide basic statisticalinformation about the fluctuations of the interactions (i) of themasses in self-gravitating systems like those encountered in the contextof cosmological N-body simulations, and (ii) of the charges inthe ordered phase of the one-component plasma, the so-called Coulombor Wigner crystal.

URL:	http://link.aps.org/doi/10.1103/PhysRevE.74.021110
DOI:	10.1103/PhysRevE.74.021110
PACS:	02.50.-r; 05.40.-a; 61.43.-j 02.50.-r Probability theory, stochastic processes, and statistics 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 61.43.-j Structure of disordered solids YEAR: 2006
KEYWORDS:	lattice theory, gravitation, electromagnetism, Poisson distribution

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