A probabilistic approach to system-environment coupling

We introduce a unified statistical framework for quantifying system-environment coupling by treating the interaction energy $V_{\mathcal{S}\mathcal{E}}$ as a stochastic variable. Using a reference-particle decomposition, we derive exact, closed-form expressions for the mean and variance of $V_{\mathcal{S}\mathcal{E}}$ in terms of the single-particle density and up to four-body correlation functions. When $V_{\mathcal{S}\mathcal{E}}$ is approximately Gaussian, these two moments suffice to compute the free energy shift of the strongly coupled system. To validate our framework, we ran explicit Monte Carlo simulations of the full system-environment configurations across a range of system sizes, generating reference distributions of the interaction energy $V_{\mathcal{S}\mathcal{E}}$. We then applied our derived analytical formulas to predict these distributions and found excellent agreement in both the weak- and strong-coupling regimes.