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

We study the statistics of the force felt by a particle in the class of a spatially correlated distribution of identical pointlike particles, interacting via a 1/r(2) pair force (i.e., gravitational or Coulomb), and obtained by randomly perturbing an infinite perfect lattice. We specify the conditions under which the force on a particle is a well-defined stochastic quantity. We then study the small displacements approximation, giving both the limitations of its validity and, when it is valid, an expression for the force variance. The method introduced by Chandrasekhar to find the force probability density function for the homogeneous Poisson particle distribution is extended to shuffled lattices of particles. In this way, we can derive an approximate expression for the probability distribution of the force over the full range of perturbations of the lattice, i.e., from very small (compared to the lattice spacing) to very large where the Poisson limit is recovered. We show in particular the qualitative change in the large-force tail of the force distribution between these two limits. Excellent accuracy of our analytic results is found on detailed comparison with results from numerical simulations. These results provide basic statistical information about the fluctuations of the interactions (i) of the masses in self-gravitating systems like those encountered in the context of cosmological N-body simulations, and (ii) of the charges in the ordered phase of the one-component plasma, the so-called Coulomb or Wigner crystal.