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Reduction of Complex Dynamics in Far-from-equilibrium Systems: Nambu Non-equilibrium Thermodynamics

So Katagiri, Yoshiki Matsuoka, Akio Sugamoto

TL;DR

The paper develops a Nambu bracket–based framework (NNET) to reformulate far-from-equilibrium nonlinear dynamics as a combination of Hamiltonian-like flows and entropy-driven dissipation. It shows how, via Helmholtz decomposition and Darboux canonical frames, a general complex system can be locally reduced to a simple NNET with a new entropy $S^C$ and Hamiltonians $H^C_i$, while allowing extension to mixed-tensor geometries for more general nonlinearities. It further proves a local existence proposition for NNETs and discusses global obstacles, including the global non-existence of first integrals and dynamical phenomena such as chaos and fractals, outlining the conditions under which the reduction holds and where it may fail. The work connects to cyclic chemical dynamics, neural spike models, and classical chaotic systems, positioning NNET as a unifying scaffold for oscillatory and chaotic behavior in open, far-from-equilibrium contexts.

Abstract

Far-from-equilibrium thermodynamic systems dominated by strong nonlinearity are reformulated within a dynamical framework based on the Nambu bracket formalism. It is demonstrated that general complex nonlinear non-equilibrium systems can be locally reduced to a simple form of Nambu Non-equilibrium Thermodynamics (NNET). Furthermore, mathematical and dynamical obstacles encountered in extending this reduction globally are discussed, and a generalized formulation that incorporates nonlinear effects through mixed higher-order tensors is proposed.

Reduction of Complex Dynamics in Far-from-equilibrium Systems: Nambu Non-equilibrium Thermodynamics

TL;DR

The paper develops a Nambu bracket–based framework (NNET) to reformulate far-from-equilibrium nonlinear dynamics as a combination of Hamiltonian-like flows and entropy-driven dissipation. It shows how, via Helmholtz decomposition and Darboux canonical frames, a general complex system can be locally reduced to a simple NNET with a new entropy and Hamiltonians , while allowing extension to mixed-tensor geometries for more general nonlinearities. It further proves a local existence proposition for NNETs and discusses global obstacles, including the global non-existence of first integrals and dynamical phenomena such as chaos and fractals, outlining the conditions under which the reduction holds and where it may fail. The work connects to cyclic chemical dynamics, neural spike models, and classical chaotic systems, positioning NNET as a unifying scaffold for oscillatory and chaotic behavior in open, far-from-equilibrium contexts.

Abstract

Far-from-equilibrium thermodynamic systems dominated by strong nonlinearity are reformulated within a dynamical framework based on the Nambu bracket formalism. It is demonstrated that general complex nonlinear non-equilibrium systems can be locally reduced to a simple form of Nambu Non-equilibrium Thermodynamics (NNET). Furthermore, mathematical and dynamical obstacles encountered in extending this reduction globally are discussed, and a generalized formulation that incorporates nonlinear effects through mixed higher-order tensors is proposed.

Paper Structure

This paper contains 24 sections, 1 theorem, 80 equations.

Table of Contents

  1. Introduction
  2. Nambu non-equilibrium thermodynamics (NNET) far from equilibrium
  3. Change of entropy
  4. Generalization of formula in Carnot cycle
  5. Entropy flow
  6. Linear response: non-equilibrium close to equilibrium
  7. Non-linear response: Non-equilibrium far from equilibrium
  8. Reduction of a complex system to a simple NNET
  9. More general non-linear system with mixed tensors
  10. Existence proposition of NNET
  11. Local existence proposition of NNET
  12. Global non-existence of first integrals
  13. Summary
  14. Discussions
  15. (Issue 1) Geometry of the generalized metric of mixed tensor
  16. ...and 9 more sections

Key Result

Proposition 4.1

This proposition is mentioned in the above as the "inverse problem"; it claims that we can find a Nambu Non-equilibrium thermodynamics (NNET) for a given autonomous system, with incompressible $v^{(1)\mu}(x)$ and compressible $v^{(2)\mu}(x)$ flows, acting in the $N$-dimensional space.

Theorems & Definitions (1)

  • Proposition 4.1