Thermodynamic potentials from a probabilistic view on the system-environment interaction energy

In open systems with strong coupling, the interaction energy between the system and the environment is significant, so thermodynamic quantities cannot be reliably obtained by traditional statistical mechanics methods. The Hamiltonian of mean force $\mathcal{H}^{*}_{\beta}$ offers an in principle accurate theoretical basis by explicitly accounting for the interaction energy. However, calculating the Hamiltonian of mean force is challenging both theoretically and computationally. We demonstrate that when the condition $\text{Var}_{\mathcal{E}_0} (e^{-\beta {V}_{\mathcal{SE}}}) = 0$ is met, the dependence of thermodynamic variables can be shifted from $\{P_{\beta}(x_{\mathcal{S}}), \mathcal{H}^{*}_{\beta}(x_{\mathcal{S}})\}$ to $\{P_{\beta}(x_{\mathcal{S}}), P(V_{\mathcal{SE}})\}$. This change simplifies thermodynamic measurements. As a central result, we derive a general equality that holds for arbitrary coupling strengths and from which an inequality follows - aligned with Jensen's inequality applied to the Gibbs-Bogoliubov-Feynman bound. This equality, analogous in importance to the Jarzynski equality, offers deeper insight into free energy differences in strongly coupled systems. Finally, by combining our result with said Jarzynski equality, we derive additional relations that further clarify thermodynamic behavior in strongly coupled open systems.