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Dissipation, Generalized Free Energy, and a Self-consistent Nonequilibrium Thermodynamics of Chemically Driven Open Subsystems

Nonequilibrium thermodynamics of a system situated in a sustained environment with influx and efflux is usually treated as a subsystem in a larger, closed "universe". It remains a question what the minimally required description for the surrounding of such an open driven system is, so that its nonequilibrium thermodynamics can be established solely based on the internal stochastic kinetics. We provide a solution to this problem using insights from studies of molecular motors in a chemical nonequilibrium steady state (NESS) with sustained external drive through a regenerating system, or in a quasi-steady state (QSS) with an excess amount of ATP, ADP, and Pi. We introduce the key notion of {\em minimal work} that is needed, $W_{min}$, for the external regenerating system to sustain a NESS ({\em e.g.}, maintaining constant concentrations of ATP, ADP and Pi for a molecular motor). Using a Markov (master-equation) description of a motor protein, we illustrate that the NESS and QSS have identical kinetics as well as the Second Law in terms of a same positive entropy production rate. The difference between the heat dissipation of a NESS and its corresponding QSS is exactly the $W_{min}$. This provides a justification for introducing an {\em ideal external regenerating system} and yields a {\em free energy balance equation} between the net free energy input $F_{in}$ and total dissipation $F_{dis}$ in an NESS: $F_{in}$ consists of chemical input minus mechanical output; $F_{dis}$ consists of dissipative heat; and the amount of useful energy becoming heat is the NESS entropy production. Furthermore, we show that for non-stationary systems, the $F_{dis}$ and $F_{in}$ correspond to the entropy production rate and housekeeping heat in stochastic thermodynamics, and identify a relative entropy $H$ as a generalized free energy.