Mesoscopic nonequilibrium thermodynamics of single macromolecules and dynamic entropy-energy compensation

We introduce axiomatically a complete thermodynamic formalism for a single macromolecule, either with or without detailed balance, in an isothermal ambient fluid based on its stochastic dynamics. With detailed balance, the theory yields mesoscopic, nonequilibrium for entropy $\left({{\rm Y}}_t\right)$ and free energy $\left({\Psi }_t\right)$ of the macromolecule. ${{\rm Y}}_t$ and ${\Psi }_t$ fluctuate. Expectation $\mathrm{}(d/dt)E\left[{\Psi }_t\right]<~0,“=”$holds if and only if the macromolecule is at thermal equilibrium, in which we show that ${{\rm Y}}_t$ still fluctuates but ${\Psi }_t$ is a constant. The entropy fluctuation of Landau, $E\left[\right(\Delta {{\rm Y}}_t{)}^2],$ precisely matches the fluctuation in the internal energy, which in turn equals the fluctuation in heat dissipation. As a generalization of Clausius’ classic result, the dynamic fluctuations in the entropy and energy of the macromolecule are exactly compensated at thermal equilibrium. For systems with detailed balance, Helmholtz free energy is shown to be the potential of Onsager’s thermodynamic force.