Statistical field theory of mechanical stresses in Coulomb fluids. Noether's theorem {\sl vs} General covariant approach

In this paper, we introduce a statistical field theory that describes the macroscopic mechanical forces in inhomogeneous Coulomb fluids. Our approach employs the generalization of Noether's first theorem for the case of fluctuating order parameter, to calculate the stress tensor for Coulomb fluids. This tensor encompasses the mean-field stress tensor and the fluctuation corrections derived through the one-loop approximation. The correction for fluctuations includes a term that accounts for the thermal fluctuations of the local electrostatic potential and field in the vicinity of the mean-field configuration. This correlation stress tensor determines how electrostatic correlation affects local stresses in a nonuniform Coulomb fluid. We also use previously formulated general covariant methodology [P.E. Brandyshev and Yu.A. Budkov, J. Chem. Phys. 158, 174114 (2023)] in conjunction with a functional Legendre transformation method and derive within it the same total stress tensor. We would like to emphasize that our general approaches are applicable not only to Coulomb fluids but also to nonionic simple or complex fluids, for which the field-theoretic Hamiltonian is known as a functional of the relevant scalar order parameters.