Generalized free energy and excess/housekeeping decomposition in nonequilibrium systems: From large deviations to thermodynamic speed limits

In genuine nonequilibrium systems under continuous driving, thermodynamic forces are nonconservative and cannot be described by any free energy potential. Nonetheless, we show that such systems can be associated with a derived from a large-deviation variational principle. This variational principle yields a decomposition of fluxes, forces, and entropy production into a conservative part and a nonconservative part, exemplifying an information-geometric Pythagorean theorem. The decomposition is broadly applicable—including to stochastic master equations as well as closed and open deterministic chemical reaction networks—and accessible to thermodynamic inference from short-time trajectory data. We also show that the excess entropy production obeys a thermodynamic speed limit bounding the rate of state evolution and external fluxes. We illustrate the framework on driven Markov jump processes, nonlinear chemical oscillators, and real-world metabolic networks, where we obtain tight dissipation bounds and identify futile metabolic cycles. Connections are drawn to large deviations, Onsager theory, and previous excess/housekeeping decompositions.