Addendum: Thermodynamic quasi-equilibria in high power magnetron discharges: a generalized Poisson–Boltzmann relation (2023 Plasma Sources Sci. Technol. 32 055012)

A recent publication by Köhn et al (2023 Plasma Sources Sci. Technol. 32 055012) studied the quasi-equilibria of high power magnetron discharges through thermodynamic principles. A generalized, magnetic-field aware Poisson–Boltzmann relation for the electric potential and the electron density was established using a non-standard (multi-objective) variational principle. This addendum demonstrates that, assuming slow or quasistatic evolution, the same result can be realized via a standard (single-objective) variational principle, thereby streamlining the theoretical framework while preserving the robustness of the finding.