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Fluctuation Theorem and Thermodynamic Formalism

Noé Cuneo, Vojkan Jakšić, Claude-Alain Pillet, Armen Shirikyan

TL;DR

The paper provides a rigorous framework proving the Fluctuation Theorem (FT) for entropy production in chaotic discrete-time dynamical systems on compact spaces, extending to empirical measures, all continuous potentials, and weak Gibbs states, including phase transitions. It introduces the Periodic Orbits Fluctuation Principle (POFP) and the Gibbs Fluctuation Principle (GFP), establishing large deviation principles for empirical measures and entropy production under both periodic-orbit and weak Gibbs settings, and extends these results to asymptotically additive potential sequences. The FT is shown to be a structural feature of the dynamical thermodynamic formalism, holding under minimal chaoticity assumptions (expansiveness and specification) without ergodicity, and applying to non-invertible maps and non-time-reversal reversals. The work bridges rigorous large deviation theory with thermodynamic formalism, enabling phase-transition analyses and connections to multifractal analysis, while providing clear criteria and proofs for the LDPs and FR/FT symmetries. Overall, the results significantly broaden the mathematical reach of entropy-production fluctuations in deterministic chaotic systems and offer tools for non-equilibrium statistical mechanics in a wide dynamical-context setting.

Abstract

We study the Fluctuation Theorem (FT) for entropy production in chaotic discrete-time dynamical systems on compact metric spaces, and extend it to empirical measures, all continuous potentials, and all weak Gibbs states. In particular, we establish the FT in the phase transition regime. These results hold under minimal chaoticity assumptions (expansiveness and specification) and require no ergodicity conditions. They are also valid for systems that are not necessarily invertible and involutions other than time reversal. Further extensions involve asymptotically additive potential sequences and the corresponding weak Gibbs measures. The generality of these results allows to view the FT as a structural facet of the thermodynamic formalism of dynamical systems.

Fluctuation Theorem and Thermodynamic Formalism

TL;DR

The paper provides a rigorous framework proving the Fluctuation Theorem (FT) for entropy production in chaotic discrete-time dynamical systems on compact spaces, extending to empirical measures, all continuous potentials, and weak Gibbs states, including phase transitions. It introduces the Periodic Orbits Fluctuation Principle (POFP) and the Gibbs Fluctuation Principle (GFP), establishing large deviation principles for empirical measures and entropy production under both periodic-orbit and weak Gibbs settings, and extends these results to asymptotically additive potential sequences. The FT is shown to be a structural feature of the dynamical thermodynamic formalism, holding under minimal chaoticity assumptions (expansiveness and specification) without ergodicity, and applying to non-invertible maps and non-time-reversal reversals. The work bridges rigorous large deviation theory with thermodynamic formalism, enabling phase-transition analyses and connections to multifractal analysis, while providing clear criteria and proofs for the LDPs and FR/FT symmetries. Overall, the results significantly broaden the mathematical reach of entropy-production fluctuations in deterministic chaotic systems and offer tools for non-equilibrium statistical mechanics in a wide dynamical-context setting.

Abstract

We study the Fluctuation Theorem (FT) for entropy production in chaotic discrete-time dynamical systems on compact metric spaces, and extend it to empirical measures, all continuous potentials, and all weak Gibbs states. In particular, we establish the FT in the phase transition regime. These results hold under minimal chaoticity assumptions (expansiveness and specification) and require no ergodicity conditions. They are also valid for systems that are not necessarily invertible and involutions other than time reversal. Further extensions involve asymptotically additive potential sequences and the corresponding weak Gibbs measures. The generality of these results allows to view the FT as a structural facet of the thermodynamic formalism of dynamical systems.

Paper Structure

This paper contains 24 sections, 33 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Prologue: what is the Fluctuation Theorem?
  3. Transient fluctuation relations
  4. Fluctuation Theorem and Fluctuation Relation
  5. Entropic pressure
  6. What does the Fluctuation Theorem mean?
  7. Interpretation of Theorems A and B
  8. Outlook
  9. Preliminaries
  10. A class of continuous dynamical systems
  11. Asymptotic additivity
  12. Topological pressure
  13. Expansiveness
  14. Periodic Orbits Fluctuation Principle
  15. LDP for empirical measures
  16. ...and 9 more sections

Key Result

Proposition 1.1

In the above setting, the following two relations hold: where $\widehat{P}_n$ is the image of $P_n$ under the reflection $\vartheta(s)=-s$.

Theorems & Definitions (92)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Example 6: Two-sided subshift of finite type
  • Example 7: Uniformly hyperbolic systems
  • Example 8: Anosov diffeomorphisms
  • Example 9: Anosov diffeomorphisms: historical perspective
  • Definition 10
  • ...and 82 more