Endpoint reweighting identities with overlap diagnostics for free-energy differences in strongly coupled open systems

We develop a reweighting framework to quantify the free energy difference between two equilibrium states of a strongly coupled open system at fixed temperature. For an open system described by the Hamiltonian of mean force (HMF), we show that the equilibrium free energy difference between two canonical endpoints can be written as exponential averages of the HMF shift, divided by an explicit factor built from the chi-squared divergence between the initial and final system marginals. These relations hold at the endpoint level and, under an explicit asymptotic-equilibration postulate, admit trajectory representations for protocols whose evolved system marginal reaches the prescribed final canonical endpoint. In this trajectory representation, the dynamics provide a route for generating samples from the final canonical endpoint. In the frozen-driving regime with a noninteracting reference, the equalities reduce to FEP-like expressions involving an environment functional and an explicit overlap diagnostic, with the Zwanzig formula recovered as a limiting case. We validate the approach on an open system coupled to an environment and evolved under overdamped Langevin dynamics, where conventional Zwanzig FEP suffers from poor phase-space overlap and slow numerical convergence. The finite-sampling tests show that the overlap factor can be reconstructed from the same samples entering the estimator, making the overlap burden directly observable. A multistage coupling ladder then decomposes this burden into local factors and provides direct comparison with recursive FEP, BAR, and MBAR.