The connection between electromagnetic stress tensors and the dielectric constant in a medium

We perform explicit calculations for microscopic models to confirm that a single, unique electromagnetic stress tensor exists in a dielectric medium: the microscopic tensor $σ^{#}$. We demonstrate that the conventional macroscopic stress tensor of continuum electrodynamics, which contains the derivative of the dielectric coefficient, is merely a representation of $σ^{#}$'s spatial average. This result establishes that, contrary to claims in the literature, both this averaged electromagnetic stress and the hydrodynamic pressure always possess a unique physical meaning.