Exact fluctuation relation for open systems beyond the Zwanzig FEP equation

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 $λ(t)$ 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.

💡 Research Summary

This paper presents a groundbreaking fluctuation framework designed to quantify the free energy difference between two equilibrium states connected via non-equilibrium processes, specifically addressing the challenges inherent in open systems with strong system-environment coupling. The fundamental problem addressed is the limitation of the classical Zwanzig Free Energy Perturbation (FEP) method, which suffers from severe convergence issues and numerical instability when the phase-space overlap between the initial and final states is insufficient or when the coupling to the environment is significant.

To overcome these limitations, the authors introduce a framework based on the Hamiltonian of Mean Force (HMF). The core innovation lies in expressing the equilibrium free energy difference as an exponential average of the HMF shift, augmented by an explicit correction factor derived from the $\chi^2$ divergence between the initial and final system marginal distributions. This mathematical refinement allows the framework to remain robust even in regimes where the overlap between states is minimal, effectively correcting for the discrepancy in phase-space coverage.

A significant technical achievement of this work is the decomposition of the HMF increment along individual trajectories into three distinct physical components: a work-like contribution associated with the temporal change in the control parameter $\lambda(t)$, a heat-like contribution representing the energy exchange with the environment $C(t)$, and a feedback-like functional that depends on the initial protocol. This granular decomposition provides a profound physical insight into how energy is transduced and dissipated during non-equilibrium driving.

The researchers further demonstrate that in the frozen-driving regime with a non-interacting reference, the proposed relations reduce to new FEP-like expressions that incorporate an environment functional and an explicit overlap correction, while successfully recovering the traditional Zwanzig formula as a limiting case.

The practical utility of this approach was validated using overdamped Langevin dynamics. The numerical results demonstrate that while the conventional Zwanzig FEP fails due to poor phase-space overlap and agonizingly slow convergence, the newly proposed trajectory equality maintains high precision and rapid convergence across a wide range of coupling strengths. This research provides a powerful and reliable tool for studying the thermodynamics of complex, open systems, offering significant implications for fields such as molecular biology, nanotechnology, and statistical mechanics where environmental interactions are non-negligible.