Nonequilibrium Potential Function of Chemically Driven Single Macromolecules via Jarzynski-Type Log-Mean-Exponential Work

Applying the method from recently developed fluctuation theorems to the stochastic dynamics of single macromolecules in ambient fluid at constant temperature, we establish two Jarzynski-type equalities: (1) between the log-mean-exponential (LME) of the irreversible heat dissiption of a driven molecule in nonequilibrium steady-state (NESS) and $\ln P^{ness}(x)$, and (2) between the LME of the work done by the internal force of the molecule and nonequilibrium chemical potential function $\mu^{ness}(x)$ $\equiv U(x)+k_BT\ln P^{ness}(x)$, where $P^{ness}(x)$ is the NESS probability density in the phase space of the macromolecule and $U(x)$ is its internal potential function. $\Psi$ = $\int\mu^{ness}(x)P^{ness}(x)dx$ is shown to be a nonequilibrium generalization of the Helmholtz free energy and $\Delta\Psi$ = $\Delta U-T\Delta S$ for nonequilibrium processes, where $S$ $=-k_B\int P(x)\ln P(x)dx$ is the Gibbs entropy associated with $P(x)$. LME of heat dissipation generalizes the concept of entropy, and the equalities define thermodynamic potential functions for open systems far from equilibrium.