A Principled Basis for Nonequilibrium Network Flows

The great power of EQuilibrium (EQ) statistical physics comes from its principled foundations: its First Law (conservation), Second Law (variational tendency principle), and its Legendre Transforms from observables $(U, V, N)$ to their driving forces $(T, p, \mu)$. Here, we generalize this structure to Non-EQuilibria (NEQ) in \textit{Caliber Force Theory} (CFT), replacing state entropies with path entropies; and $(U, V, N)$ with dynamic observables (node probabilities, edge traffics, and cycle fluxes). CFT derives dynamical forces and a complete set of conjugate relations: (i) It yields generalized Maxwell-Onsager relations, applicable far from equilibrium; (ii) It constructs dynamical models from mixed force-observable constraints; and (iii) It reveals new relationships -- including an ``equal-traffic'' rule for optimizing molecular motors, and a ``third Kirchhoff's law'' of stochastic transport -- and can resolve some dynamical paradoxes.