# Noether's second theorem and covariant field theory of mechanical stresses in inhomogeneous ionic liquids

## TL;DR

This paper develops a covariant framework using Noether's second theorem to derive symmetric stress tensors in inhomogeneous ionic liquids, linking thermodynamic potentials to mechanical stresses in various models.

## Contribution

It introduces a novel covariant approach to derive symmetric stress tensors from thermodynamic functionals in ionic liquids, unifying several existing models.

## Key findings

- Derived analytical symmetric stress tensors for multiple ionic liquid models.
- Confirmed consistency of stress expressions with self-consistent field equations.
- Provided a covariant methodology applicable to inhomogeneous ionic systems.

## Abstract

In this paper, we present a covariant approach that utilizes Noether's second theorem to derive a symmetric stress tensor from the grand thermodynamic potential functional. We focus on the practical case where the density of the grand thermodynamic potential is dependent on the first and second coordinate derivatives of the scalar order parameters. Our approach is applied to several models of inhomogeneous ionic liquids that consider electrostatic correlations of ions or short-range correlations related to packing effects. Specifically, we derive analytical expressions for the symmetric stress tensors of the Cahn-Hilliard-like model, Bazant-Storey-Kornyshev model, and Maggs-Podgornik-Blossey model. All of these expressions are found to be consistent with respective self-consistent field equations.