Thermodynamic Length in Stochastic Thermodynamics of Far-From-Equilibrium Systems: Unification of Fluctuation Relation and Thermodynamic Uncertainty Relation

TL;DR

This paper develops a unified theoretical framework for far-from-equilibrium stochastic systems, linking thermodynamic length, fluctuation relations, and uncertainty relations through a path-integral and variational approach.

Contribution

It introduces a non-quadratic thermodynamic length that unifies fluctuation relations and thermodynamic uncertainty relations for far-from-equilibrium systems.

Findings

Derives an exact path-integral representation for discrete stochastic processes.

Shows the non-quadratic thermodynamic length recovers known uncertainty and fluctuation relations.

Extends the framework to coarse-grained observable currents.

Abstract

The Boltzmann distribution for an equilibrium system constrains the statistics of the system by the energetics. Despite the non-equilibrium 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 non-quadratic 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 non-quadratic dissipation function is physically attributed to incorporating non-Gaussian fluctuations or far-from-equilibrium non-conservative 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 non-quadratic TL recovers the non-quadratic 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, the non-quadratic TL recovers the fluctuation relation (FR). The minimum action principle manifests the non-quadratic 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.