Sine-Gordon Theory for the Equation of State of Classical Hard-Core Coulomb systems. III Loopwise Expansion

TL;DR

This paper develops an exact field theoretical framework for ionic solutions with charged hard spheres, deriving approximations up to two loops and connecting to known theories like the random phase approximation.

Contribution

It introduces a rigorous field theory representation for Coulomb systems and explores mean field, Gaussian, and two-loop approximations, linking to existing liquid state theories.

Findings

Mean field free energy provides a lower bound.

Mean field pressure is an upper bound.

Two-loop results recover known mode expansion outcomes.

Abstract

We present an exact field theoretical representation of an ionic solution made of charged hard spheres. The action of the field theory is obtained by performing a Hubbard-Stratonovich transform of the configurational Boltzmann factor. It is shown that the Stillinger-Lovett sum rules are satisfied if and only if all the field correlation functions are short range functions. The mean field, Gaussian and two-loops approximations of the theory are derived and discussed. The mean field approximation for the free energy constitutes a rigorous lower bound for the exact free energy, while the mean field pressure is an upper bound. The one-loop order approximation is shown to be identical with the random phase approximation of the theory of liquids. Finally, at the two-loop order and in the pecular case of the restricted primitive model, one recovers results obtained in the framework of the mode expansion theory.