Exact fluctuation relation for open systems beyond the Zwanzig FEP equation

Abstract

We develop a fluctuation framework to quantify the free energy difference between two equilibrium states connected by nonequilibrium processes under arbitrary dynamics and system-environment coupling. 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 asymptotic equilibration postulate, admit trajectory representations for general driving and coupling protocols. A decomposition of the HMF increment along each trajectory separates the work-like contributions associated with changes in $\lambda(t)$ and $C(t)$, the heat-like exchange with the environment, and a feedback-like functional defined with respect to the initial protocol. In the frozen-driving regime with a noninteracting reference, the equalities reduce to new FEP-like expressions involving an environment functional and an explicit overlap correction, 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, while the present trajectory equality closely matches the exact free energy difference over a broad range of coupling strengths.