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Applications of Nambu Non-equilibrium Thermodynamics to Specific Phenomena

So Katagiri, Yoshiki Matsuoka, Akio Sugamoto

TL;DR

NNET provides a unified, quantitatively consistent framework for far-from-equilibrium dynamics by decomposing the velocity field into a reversible Nambu flow generated by Hamiltonians $H_1,H_2$ and an irreversible gradient flow from a generalized entropy $S$. The authors construct explicit $H_1$, $H_2$, and $S$ for four paradigmatic systems—the Belousov–Zhabotinsky reaction, Hindmarsh–Rose neuron model, Lorenz, and Chen systems—and show that cycles, spikes, and chaotic attractors emerge from the interplay of the reversible and dissipative components. They demonstrate how regime transitions (steady, periodic, chaotic) imprint distinctive signatures in the $(H_1,S)$ diagnostic space, providing a cross-model, thermodynamically grounded view beyond Onsager or GENERIC formalisms. The work also points to extensions to spatially extended systems and potential implications for pattern formation and neural dynamics, offering a principled route to classify non-equilibrium behaviors.

Abstract

We apply Nambu non-equilibrium thermodynamics (NNET)-a dynamics with multiple Hamiltonians coupled to entropy-induced dissipation-to paradigmatic far-from-equilibrium systems. Concretely, we construct NNET realizations for the Belousov-Zhabotinsky (BZ) reaction (oscillatory), the Hindmarsh-Rose neuron model (spiking), and the Lorenz and Chen systems (chaotic), and analyze their dynamical and thermodynamic signatures. Across all cases the velocity field cleanly decomposes into a reversible Nambu part and an irreversible entropygradient part, anchored by a model-independent quasi-conserved quantity. This construction reproduces cycles, spikes, and strange-attractor behavior and clarifies transitions among steady, periodic, and chaotic regimes via cross-model diagnostics. These results demonstrate that NNET provides a unified, quantitatively consistent framework for oscillatory, spiking, and chaotic non-equilibrium systems, offering a systematic description beyond the scope of linear-response theories such as Onsager's relations or GENERIC.

Applications of Nambu Non-equilibrium Thermodynamics to Specific Phenomena

TL;DR

NNET provides a unified, quantitatively consistent framework for far-from-equilibrium dynamics by decomposing the velocity field into a reversible Nambu flow generated by Hamiltonians and an irreversible gradient flow from a generalized entropy . The authors construct explicit , , and for four paradigmatic systems—the Belousov–Zhabotinsky reaction, Hindmarsh–Rose neuron model, Lorenz, and Chen systems—and show that cycles, spikes, and chaotic attractors emerge from the interplay of the reversible and dissipative components. They demonstrate how regime transitions (steady, periodic, chaotic) imprint distinctive signatures in the diagnostic space, providing a cross-model, thermodynamically grounded view beyond Onsager or GENERIC formalisms. The work also points to extensions to spatially extended systems and potential implications for pattern formation and neural dynamics, offering a principled route to classify non-equilibrium behaviors.

Abstract

We apply Nambu non-equilibrium thermodynamics (NNET)-a dynamics with multiple Hamiltonians coupled to entropy-induced dissipation-to paradigmatic far-from-equilibrium systems. Concretely, we construct NNET realizations for the Belousov-Zhabotinsky (BZ) reaction (oscillatory), the Hindmarsh-Rose neuron model (spiking), and the Lorenz and Chen systems (chaotic), and analyze their dynamical and thermodynamic signatures. Across all cases the velocity field cleanly decomposes into a reversible Nambu part and an irreversible entropygradient part, anchored by a model-independent quasi-conserved quantity. This construction reproduces cycles, spikes, and strange-attractor behavior and clarifies transitions among steady, periodic, and chaotic regimes via cross-model diagnostics. These results demonstrate that NNET provides a unified, quantitatively consistent framework for oscillatory, spiking, and chaotic non-equilibrium systems, offering a systematic description beyond the scope of linear-response theories such as Onsager's relations or GENERIC.

Paper Structure

This paper contains 14 sections, 99 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Approaches for Constructing the Hamiltonians and Entropy
  3. General evolution law of NNET
  4. Constructive procedure for Hamiltonians and entropy
  5. Application
  6. BZ reaction
  7. Hindmarsh-Rose model
  8. Lorenz System
  9. Chen System
  10. Summary and Discussion
  11. NNET in summary (for the present 3D applications)
  12. Analysis of cycles and spikes
  13. Harmonic oscillators
  14. Turning points of spikes

Figures (15)

  • Figure 1: Time variation of concentration X, Y and Z. Horizontal axis represents time, vertical axis represents concentration.
  • Figure 2: Plot of entropy change over time. The horizontal axis represents time and the vertical axis represents entropy.
  • Figure 3:
  • Figure 5: 3D diagram of the limit cycle according to the time variation of the concentration X, Y and Z. The vertical axis represents the concentration of Z, the horizontal axis represents the concentration of X and the remaining axis represents the concentration of Y.
  • Figure 6: Time variation of membrane potential $x$, recovery variable $y$ and bursting variable $z$. Horizontal axis represents time, vertical axis represents $x$, $y$, $z$.
  • ...and 10 more figures