Bivectorial Nonequilibrium Thermodynamics: Cycle Affinity, Vorticity Potential, and Onsager's Principle

We generalize an idea in the works of Landauer and Bennett on computations, and Hill's in chemical kinetics, to emphasize the importance of kinetic cycles in mesoscopic nonequilibrium thermodynamics (NET). For continuous stochastic systems, a NET in phase space is formulated in terms of cycle affinity $\nabla\wedge\big(\mathbf{D}^{-1}\mathbf{b}\big)$ and vorticity potential $\mathbf{A}(\mathbf{x})$ of the stationary flux $\mathbf{J}^{*}=\nabla\times\mathbf{A}$. Each bivectorial cycle couples two transport processes represented by vectors and gives rise to Onsager's reciprocality; the scalar product of the two bivectors $\mathbf{A}\cdot\nabla\wedge\big(\mathbf{D}^{-1}\mathbf{b}\big)$ is the rate of local entropy production in the nonequilibrium steady state. An Onsager operator that maps vorticity to cycle affinity is introduced.