General Considerations on the Finite-Size Corrections for Coulomb Systems in the Debye-Huckel Regime

TL;DR

This paper analyzes finite-size effects in Coulomb systems within the Debye-Huckel regime, deriving general formulas for the grand potential's size-dependent corrections in various geometries.

Contribution

The authors develop a method linking the Laplacian's heat kernel expansion to finite-size corrections of the grand potential, applicable to arbitrary geometries.

Findings

Recovered known bulk grand potential results in 2D and 3D

Derived surface tension for 3D Coulomb systems

Proved universal logarithmic correction in 2D systems

Abstract

We study the statistical mechanics of classical Coulomb systems in a low coupling regime (Debye-Huckel regime) in a confined geometry with Dirichlet boundary conditions. We use a method recently developed by the authors which relates the grand partition function of a Coulomb system in a confined geometry with a certain regularization of the determinant of the Laplacian on that geometry with Dirichlet boundary conditions. We study several examples of confining geometry in two and three dimensions and semi-confined geometries where the system is confined only in one or two directions of the space. We also generalize the method to study systems confined in arbitrary geometries with smooth boundary. We find a relation between the expansion for small argument of the heat kernel of the Laplacian and the large-size expansion of the grand potential of the Coulomb system. This allow us to find the finite-size expansion of the grand potential of the system in general. We recover known results for the bulk grand potential (in two and three dimensions) and the surface tension (for two dimensional systems). We find the surface tension for three dimensional systems. For two dimensional systems our general calculation of the finite-size expansion gives a proof of the existence a universal logarithmic finite-size correction predicted some time ago, at least in the low coupling regime. For three dimensional systems we obtain a prediction for the perimeter correction to the grand potential of a confined system.