Two-dimensional Coulomb systems on a surface of constant negative curvature

TL;DR

This paper investigates the equilibrium statistical mechanics of classical two-dimensional Coulomb systems on a surface of constant negative curvature, exploring their thermodynamic properties, correlation functions, and exact solvable models.

Contribution

It provides analysis of Coulomb systems on a pseudosphere, including virial expansion, thermodynamic limits, and validation through exactly solvable models.

Findings

Correlation functions have a well-defined thermodynamic limit.

The Coulomb potential on a pseudosphere goes to zero at infinity.

Sum rules for perfect screening are established.

Abstract

We study the equilibrium statistical mechanics of classical two-dimensional Coulomb systems living on a pseudosphere (an infinite surface of constant negative curvature). The Coulomb potential created by one point charge exists and goes to zero at infinity. The pressure can be expanded as a series in integer powers of the density (the virial expansion). The correlation functions have a thermodynamic limit, and remarkably that limit is the same one for the Coulomb interaction and some other interaction law. However, special care is needed for defining a thermodynamic limit of the free energy density. There are sum rules expressing the property of perfect screening. These generic properties can be checked on the Debye-H\"uckel approximation, and on two exactly solvable models~: the one-component plasma and the two-component plasma, at some special temperature.