Universality in some classical Coulomb systems of restricted dimension

TL;DR

This paper demonstrates that in certain classical Coulomb systems confined in flat manifolds, key thermodynamic properties and charge correlations are universal, independent of microscopic details, especially in two and three dimensions.

Contribution

It establishes the universality of charge correlations and thermodynamic dependencies in Coulomb systems confined in restricted dimensions, extending understanding of boundary effects.

Findings

Charge correlation functions are universal under specified conditions.

Free energy and grand potential depend universally on the confinement width W.

Results are validated on exactly solvable models and specific cases for d=2 and d=3.

Abstract

Coulomb systems in which the particles interact through the $d$-dimensional Coulomb potential but are confined in a flat manifold of dimension $d - 1$ are considered. The Coulomb potential is defined with some boundary condition involving a characteristic macroscopic distance $W$ in the direction perpendicular to the manifold~: either it is periodic of period $W$ in that direction, or it vanishes on one ideal conductor wall parallel to the manifold at a distance $W$ from it, or it vanishes on two parallel walls at a distance $W$ from each other with the manifold equidistant from them. Under the assumptions that classical equilibrium statistical mechanics is applicable and that the system has the macroscopic properties of a conductor, it is shown that the suitably smoothed charge correlation function is universal, and that the free energy and the grand potential have universal dependences on $W$ (universal means independent of the microscopic detail). The cases $d = 2$ are discussed in detail, and the generic results are checked on an exactly solvable model. The case $d = 3$ of a plane parallel to an ideal conductor is also explicitly worked out.