Nambu Non-equilibrium Thermodynamics II:Reduction of a complex system to a simple one (2508.19455v1)

Abstract: Non-equilibrium thermodynamics far from equilibrium is investigated in terms of a generalized ``Nambu dynamics'' (termed Nambu non-equilibrium thermodynamics (NNET)); NNET consists of Nambu dynamics with multiple Hamiltonians and an entropy causing dissipation. It is shown that a general complex non-linear non-equilibrium system far from equilibrium can be reduced to a simple NNET, on the basis of an existence proposition of NNET. This proposition is, however, proved only formally, so that there may exist various obstacles to destroy it. Non-equilibrium thermodynamics studied in this article is quite general, including the case in which the product of dynamical and affinity forces forms a higher-order mixed tensor.

• The paper demonstrates a reduction approach that maps complex non-linear systems onto simpler NNET models using Nambu dynamics and entropy.
• It employs a novel framework incorporating multiple Hamiltonians and a generalized Carnot cycle to quantify dissipation and entropy production.
• The work bridges theory and practice by offering formal propositions and practical examples from chemical and chaotic systems.

Summary of "Nambu Non-equilibrium Thermodynamics II: Reduction of a complex system to a simple one"

The paper "Nambu Non-equilibrium Thermodynamics II: Reduction of a complex system to a simple one" explores the concept of simplifying complex non-linear systems in non-equilibrium thermodynamics using a generalized framework termed Nambu Non-equilibrium Thermodynamics (NNET). This framework leverages Nambu dynamics involving multiple Hamiltonians and entropy for dissipative processes. The central thesis of the paper is the proposition that any complex autonomous system can potentially be reduced to a simple NNET configuration, albeit with some caveats regarding formal proofs and potential obstacles in application.

Nambu Non-equilibrium Thermodynamics (NNET)

NNET extends classical thermodynamics far from equilibrium by incorporating generalized dynamics characterized by the Nambu bracket. The system is defined in terms of thermodynamic variables $\{x^i\}$, Hamiltonians $\{H_k\}$, and entropy $S$, leading to evolution equations:

$$\dot{x}^i = \{x^i, H_1, \dots, H_{N-1}\}_{NB} + \delta^{ij}\partial_j S$$

where the Nambu bracket denotes a multi-variable extension analogous to the Poisson bracket.

Entropy and Response Theories

The paper highlights the fundamental role of entropy in non-equilibrium behavior, particularly in systems far from equilibrium. Entropy change is decomposed into reversible and irreversible components, emphasizing dissipation and entropy generation in cyclic processes. The paper further introduces a generalized Carnot cycle for complex systems, refining the understanding of thermodynamic efficiency and entropy production.

Linear response theory is revisited in the NNET framework, where transport coefficients are modified by incorporating generalized kinetic constants, capturing the essence of non-linear interactions in far-from-equilibrium scenarios.

Reduction from Complex to Simple Systems

The essence of the paper is the formal proposition that complex non-linear systems can potentially be reduced to simpler NNET models. This reduction relies on the identification of equivalent Hamiltonians and entropies for a given autonomous system, bridging the behavior of these systems with simpler canonical dynamics under the Nambu framework.

Existence Proposition

The paper provides a formal existence proposition for NNET, invoking mathematical principles such as Helmholtz and Darboux theorems. This proposition asserts the feasibility of mapping complex systems onto simpler, structured NNET representations, contingent on the existence of necessary conditions and overcoming obstacles such as chaos, fractals, and singularities.

Practical Examples and Applications

Several practical examples are explored, including chemical reactions and chaotic systems:

• Triangular Chemical Reaction: The classical example of triangular reaction demonstrates explicit construction within the NNET framework using Helmholtz decomposition.
• Belousov-Zhabotinsky (BZ) Reaction: Illustrates complex cyclic dynamics in chemical systems, mapped onto NNET structure, showcasing oscillatory behavior.
• Hindmarsh-Rose and Chaotic Models: Demonstrates the reduction of neuron-like spiking models and chaotic systems using NNET, emphasizing its applicability in diverse dynamic environments.

Conclusion and Implications

The paper offers a robust framework for simplifying complex systems in non-equilibrium thermodynamics. The NNET model potentially serves as a unified approach to understanding a variety of systems, from chemical reactions to chaotic dynamics, enabling clearer insights into the underlying coherent behaviors and cyclic motions that characterizes these systems. Future work is suggested to rigorously analyze necessary conditions and address discussed issues such as global non-existence of first integrals and dealing with dynamic obstacles like chaos and fractals.