Statistical characterization of random electrostatic potentials

In this work we study statistical properties of random electrostatic potentials generated by one dimensional lattices with random charges. We show that the resulting random potentials are correlated Gaussian processes, satisfying the Lindeberg version of the central limit theorem, if certain restrictions are imposed on the individual potentials generated by the particles on the lattice. Since most of the point-particle electrostatic potentials occurring in nature satisfy the Lindeberg condition, the correlation properties of the random potentials are not arbitrary and must comply with the central limit theorem. Based on this theorem we can obtain explicit expressions for these correlations. We thus are able to give a characterization of a broad class of potentials yielding feasible physical scenarios. We illustrate some consequences of our findings by considering dynamical properties of a test particle interacting with the lattice. We show how the long range correlations generate statistical features in these properties, which are best exhibited when considering different length scales.