Exact and variational identities for free energy differences in strongly coupled open systems

Abstract

We derive exact identities for open systems connecting two equilibrium endpoints without imposing microscopic reversibility, detailed balance (DB), fluctuation-dissipation structure, or local detailed balance (LDB) on the driven dynamics. The identities express the Hamiltonian of mean force (HMF) free energy differences through exponential moments and an explicit chi-squared overlap between the endpoint marginals. In the frozen-coupling regime, the HMF shift reduces to a bare-system increment and admits a trajectory-level heat-work-reference decomposition. The exact relations then reduce the problem to a scalar-action law. A maximum-entropy construction gives a Bessel-form scalar-action law, independent of the microscopic system, environment, and number of degrees of freedom at the level of the variational reconstruction. This law provides three outputs from the same sampled configurations: the HMF free energy difference, the endpoint-overlap burden, and a Hessian uncertainty estimate. Since many systems in biology, chemistry, physics and engineering violate the underlying assumptions of the standard Jarzynski identity, we validate the framework on a reduced-dimensional model with a non-Liouvillian, phase-space-compressing ramp followed by underdamped Langevin relaxation. The standard Jarzynski work estimator fails for this ramp because phase-space preservation is broken and no compensating Jacobian correction is included, whereas the present endpoint identities recover the exact HMF free energy difference, and the variational construction reproduces it within its local uncertainty.