Static Dielectric Permittivity Profiles and Coarse-graining Approaches for Water in Graphene Slit Pores

The dielectric response of nano-confined fluids is crucial across technologies and biological systems, yet its calculation and interpretation from molecular simulations are often muddled by unclear boundary conditions. We re-derive the Green--Kubo relation for the spatially resolved linear dielectric response of fluids in planar confinement, explicitly accounting for boundary conditions and showing that equilibrium-derived profiles agree with those obtained from external fields. We identify common misconceptions in the literature and outline how microscopic dielectric behavior can be coarse-grained to connect with experimental observables. Simulations show that water retains a bulk-like dielectric response down to $\sim 1\,\mathrm{nm}$ confinement. The reduced \emph{effective} dielectric response that governs capacitance arises from the placement of the dielectric interface. Using effective-medium theory, we demonstrate that long-range reductions reported in experiments and theory are consistent with bulk-like behavior beyond about $1\,\mathrm{nm}$ from the surface. The effective response naturally maps onto an interfacial capacitance, and the dielectric properties of simulated water are robust across simulation setups and water models, reflecting universal polarization correlations.