Amplitude Function of Asymptotic Correlations Along Charged Wall in Coulomb Fluids

TL;DR

This paper investigates the amplitude function of asymptotic correlations in Coulomb fluids near charged walls, establishing new relations with density profiles and proving factorization properties across various models and dimensions.

Contribution

It introduces a new relation between the amplitude function and the density profile using conformal transformations, extending previous results to more general Coulomb fluid models.

Findings

Proves the factorization property of the amplitude function at any temperature.

Derives a relation between the amplitude function and the charge profile for many-component Coulomb fluids.

Extends the theoretical framework to arbitrary dimensions and geometries.

Abstract

In classical semi-infinite Coulomb fluids, two-point correlation functions exhibit a slow inverse-power law decay along a uniformly charged wall. In this work, we concentrate on the corresponding amplitude function which depends on the distances of the two points from the wall. Recently [L. \v{S}amaj, J. Stat. Phys. {\bf 161}, 227 (2015)], applying a technique of anticommuting variables to a 2D system of charged rectilinear wall with "counter-ions only", we derived a relation between the amplitude function and the density profile which holds for any temperature. In this paper, using the M\"obius conformal transformation of particle coordinates in a disc, a new relation between the amplitude function and the density profile is found for that model. This enables us to prove, at any temperature, the factorization property of the amplitude function in point distances from the wall and to express it in terms of the density profile. Presupposing the factorization property of the amplitude function and using specific sum rules for semi-infinite geometries, a relation between the amplitude function of the charge-charge structure function and the charge profile is derived for many-component Coulomb fluids in any dimension.