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Nambu Non-equilibrium Thermodynamics: Axiomatic Formulation and Foundation

So Katagiri, Yoshiki Matsuoka, Akio Sugamoto

TL;DR

The paper addresses the challenge of describing systems that simultaneously exhibit reversible and dissipative dynamics far from equilibrium, beyond the reach of near-equilibrium linear theories. It introduces Nambu Non-equilibrium Thermodynamics (NNET), which uses a Nambu bracket to govern reversible flow along multiple Hamiltonians and an entropy-gradient gradient flow to drive irreversibility. Key contributions include an axiomatic formulation on a thermodynamic state space, a non-equilibrium steady-state criterion, and a triangular reaction demonstration revealing geometric conserved quantities and higher-order transport without assuming detailed balance or linearity. The framework offers a covariant, flexible platform for describing cyclic, oscillatory, and dissipative behavior in open systems, with future work on nonlinear extensions and stochastic fluctuations.

Abstract

We present a theoretical framework for non-equilibrium thermodynamics, termed Nambu Non-equilibrium Thermodynamics (NNET), which unifies reversible dynamics described by the Nambu bracket and irreversible processes driven by entropy gradients. The formulation provides a covariant description of systems far from equilibrium, where entropy may transiently decrease as a result of reversible circulations or exchanges with the surroundings, extending the applicability of conventional thermodynamic formalisms. As an illustrative example, a triangular chemical reaction system is analyzed. It is shown that, without assuming detailed balance or linearity, two geometric structures that behave as conserved quantities in the reversible limit naturally emerge: one associated with cyclic symmetry in the reaction space, and another that vanishes under symmetric reaction rates. These results demonstrate that NNET provides a unified and covariant formulation for describing both cyclic dynamics and dissipative processes within a single theoretical structure.

Nambu Non-equilibrium Thermodynamics: Axiomatic Formulation and Foundation

TL;DR

The paper addresses the challenge of describing systems that simultaneously exhibit reversible and dissipative dynamics far from equilibrium, beyond the reach of near-equilibrium linear theories. It introduces Nambu Non-equilibrium Thermodynamics (NNET), which uses a Nambu bracket to govern reversible flow along multiple Hamiltonians and an entropy-gradient gradient flow to drive irreversibility. Key contributions include an axiomatic formulation on a thermodynamic state space, a non-equilibrium steady-state criterion, and a triangular reaction demonstration revealing geometric conserved quantities and higher-order transport without assuming detailed balance or linearity. The framework offers a covariant, flexible platform for describing cyclic, oscillatory, and dissipative behavior in open systems, with future work on nonlinear extensions and stochastic fluctuations.

Abstract

We present a theoretical framework for non-equilibrium thermodynamics, termed Nambu Non-equilibrium Thermodynamics (NNET), which unifies reversible dynamics described by the Nambu bracket and irreversible processes driven by entropy gradients. The formulation provides a covariant description of systems far from equilibrium, where entropy may transiently decrease as a result of reversible circulations or exchanges with the surroundings, extending the applicability of conventional thermodynamic formalisms. As an illustrative example, a triangular chemical reaction system is analyzed. It is shown that, without assuming detailed balance or linearity, two geometric structures that behave as conserved quantities in the reversible limit naturally emerge: one associated with cyclic symmetry in the reaction space, and another that vanishes under symmetric reaction rates. These results demonstrate that NNET provides a unified and covariant formulation for describing both cyclic dynamics and dissipative processes within a single theoretical structure.

Paper Structure

This paper contains 6 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Axiomatic Formulation of Non-equilibrium Thermodynamics Based on the Nambu Bracket
  3. Relationship Between the Proposed Framework and GENERIC
  4. Demonstration of the Formulation Using a Triangular Reaction System
  5. Triangular Reaction
  6. Conclusion and Discussion

Key Result

Lemma 5

Conservation of Hamiltonians In the absence of the irreversible part, the Hamiltonians $H_{1},\dots,H_{n-1}$ are conserved.

Theorems & Definitions (6)

  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10