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

Atul Tanaji Mohite, Heiko Rieger

TL;DR

This work develops a minimum action principle (MinAP) for far-from-equilibrium stochastic thermodynamics of discrete-state systems by deriving an exact path-integral representation (via Doi-Peliti field theory) in which the action equals the inferred entropy production rate (EPR) and introduces a non-quadratic thermodynamic length (TL). It unifies fluctuation relations (FR) and thermodynamic-kinetic uncertainty relations (TKUR) through a non-quadratic, exact large deviation framework, showing TL governs current fluctuations, and the exact rate functional tightens bounds beyond Gaussian approximations. The theory extends to coarse-grained observable currents, enabling practical thermodynamic inference from partial data; it also links to information geometry as an equivalent variational formulation, and provides non-quadratic speed limits and fluctuation descriptions for state-space currents and non-reciprocal systems. Overall, the MinAP formalism offers a cohesive, exact canonical-style framework for analyzing and inferring dissipation in fEQ discrete-state dynamics with broad applicability to experiments and numerical optimization. The results offer a path toward more accurate thermodynamic bounds and inference in finite-size and driven systems, with potential extensions to hypergraphs and optimal control of stochastic networks.

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.

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

TL;DR

This work develops a minimum action principle (MinAP) for far-from-equilibrium stochastic thermodynamics of discrete-state systems by deriving an exact path-integral representation (via Doi-Peliti field theory) in which the action equals the inferred entropy production rate (EPR) and introduces a non-quadratic thermodynamic length (TL). It unifies fluctuation relations (FR) and thermodynamic-kinetic uncertainty relations (TKUR) through a non-quadratic, exact large deviation framework, showing TL governs current fluctuations, and the exact rate functional tightens bounds beyond Gaussian approximations. The theory extends to coarse-grained observable currents, enabling practical thermodynamic inference from partial data; it also links to information geometry as an equivalent variational formulation, and provides non-quadratic speed limits and fluctuation descriptions for state-space currents and non-reciprocal systems. Overall, the MinAP formalism offers a cohesive, exact canonical-style framework for analyzing and inferring dissipation in fEQ discrete-state dynamics with broad applicability to experiments and numerical optimization. The results offer a path toward more accurate thermodynamic bounds and inference in finite-size and driven systems, with potential extensions to hypergraphs and optimal control of stochastic networks.

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.

Paper Structure

This paper contains 32 sections, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Minimum Action Principle
  3. Setup
  4. Variational formulation
  5. Derivation using DPFT
  6. Doi-Peliti Lagrangian in the current-affinity picture
  7. Most likelihood path and the inferred EPR
  8. Thermodynamic length and entropy production
  9. Min-Max principle for the EPR
  10. Mapping to information geometry
  11. Large deviation principle
  12. The exact result
  13. The Gaussian approximation and the Thermodynamic Uncertainty Relation
  14. The dynamical rate functional and the Non-equilibrium Fluctuation-response relation
  15. Physical implications of the exact LDP
  16. ...and 17 more sections

Figures (1)

  • Figure 1: (a) Lagrangian $\mathcal{L}[j, \chi]$ for fixed $J_\gamma=3.5, T_\gamma=4$ (cyan) and $J_\gamma=2.5, T_\gamma=4$ (orange). The corresponding most-likely transition affinity $\chi^* = 2\tanh^{-1}(J_\gamma/T_\gamma)$ is shown as a vertical dotted line. (b) Comparison between the exact large deviation rate functional $I = 2x\tanh^{-1}(x)$, the dynamical rate functional $I_{D}= 2x\sinh^{-1}(x)$, and the close-to-equilibrium quadratic (Gaussian) approximated rate functional $I_{G} = 2x^2$, where $x=J_\gamma/T_\gamma$ is the current precision.