Thermodynamic length in stochastic thermodynamics of far-from-equilibrium systems: Unification of fluctuation relation and thermodynamic uncertainty relation.

The Boltzmann distribution for an equilibrium system constrains the statistics of the system by the energetics. Despite the nonequilibrium generalization of the Boltzmann distribution being studied extensively, a unified framework valid for far-from-equilibrium discrete-state systems is lacking. Here, we derive an exact path-integral representation for discrete-state processes and represent it using the exponential of the action for stochastic transition dynamics. Solving the variational problem, the effective action is shown to be equal to the inferred entropy production rate (a thermodynamic quantity) and a nonquadratic dissipation function of the thermodynamic length (TL) defined for microscopic stochastic currents (a dynamic quantity). This formulates a far-from-equilibrium analog of the Boltzmann distribution, namely, the minimum action principle. The nonquadratic dissipation function is physically attributed to incorporating non-Gaussian fluctuations or far-from-equilibrium nonconservative driving. Further, an exact large deviation dynamical rate functional is derived. The equivalence of the variational formulation with the information geometric formulation is proved. The nonquadratic TL recovers the nonquadratic thermodynamic-kinetic uncertainty relation (TKUR) and the speed limits, which are tighter than the close-to-equilibrium quadratic formulations. Moreover, if the transition affinities are known, then the nonquadratic TL recovers the fluctuation relation (FR). The minimum action principle manifests the nonquadratic TKUR and FR as two faces corresponding to the thermodynamic inference and partial control descriptions, respectively. In addition, the validity of these results is extended to coarse-grained observable currents, strengthening the experimental/numerical applicability of them.