Exact fluctuation relation for open systems beyond the Jarzynski equality

We derive exact fluctuation equalities for open systems that recover free energy differences between two equilibrium endpoints connected by nonequilibrium processes with arbitrary dynamics and coupling. The exponential of the free energy difference is expressed in terms of ensemble averages of the Hamiltonian of mean force (HMF) shift and the chi-squared divergence between the initial and final marginal probability distribution of the open system. A trajectory counterpart of this relation follows from an asymptotic equilibration postulate, which treats relaxation to the final stationary canonical state as a boundary condition rather than as a consequence of constraints on the driven dynamics. In the frozen-coupling regime, the HMF shift reduces to the bare-system Hamiltonian shift, yielding a clear heat-work decomposition. The Jarzynski equality (JE) is recovered under the assumption of Hamiltonian dynamics for the combined system. We validate the theory on a dissipative, phase-space-compressing drive followed by an underdamped Langevin relaxation, where the assumptions underlying the JE break down, whereas our equality reproduces the exact free energy differences.

💡 Research Summary

In this paper the authors address a fundamental limitation of the Jarzynski equality (JE), namely its reliance on microscopic reversibility, Liouville’s theorem, and detailed balance. They develop a set of exact fluctuation relations that remain valid for open systems regardless of the strength of system‑environment coupling and irrespective of whether the underlying dynamics are Hamiltonian, stochastic, or even non‑Markovian. The central object is the Hamiltonian of mean force (HMF), (H^{}{\beta}(X_S,\lambda,C)), which incorporates the effect of the environment by integrating out its degrees of freedom. For two equilibrium end‑points characterized by control parameters ((\lambda(0),C(0))) and ((\lambda(t{\rm eq}),C(t_{\rm eq}))) at the same inverse temperature (\beta), the free‑energy difference of the open system is defined as (\Delta F^{}{S}=F^{*}{S}(\lambda(t_{\rm eq}),C(t_{\rm eq}),\beta)-F^{*}_{S}(\lambda(0),C(0),\beta)).

The first main result (Theorem 1) expresses the exponential of (-\beta\Delta F^{*}_{S}) as an average over the final marginal distribution of the system:

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