Nambu Non-equilibrium Thermodynamics I: Foundation

Abstract: We propose a novel framework of non-equilibrium thermodynamics, termed Nambu Non-equilibrium Thermodynamics (NNET), which integrates reversible dynamics governed by the Nambu bracket with irreversible dynamics driven by entropy gradients. This framework is developed axiomatically and enables the description of far-from-equilibrium systems in which entropy can transiently decrease -- a regime difficult to handle within conventional approaches such as Onsager's linear response theory, Prigogine's General Evolution Criterion (GEC), or the GENERIC framework. As an illustrative example, we analyze a triangular chemical reaction system. We demonstrate that, without assuming detailed balance or linearity, two geometric conserved quantities naturally emerge: one associated with cyclic symmetry in the reaction space, and another that vanishes under symmetric reaction rates. This highlights the extended descriptive power of NNET and its capacity to unify cyclic dynamics and dissipation within a covariant structure. The generalization to higher-order nonlinear systems and applications to oscillatory or spiking non-equilibrium behavior will be discussed in subsequent papers.

• The paper introduces Nambu Non-equilibrium Thermodynamics, a framework that combines reversible Nambu brackets with irreversible entropy gradients to describe systems far from equilibrium.
• It develops an axiomatic formulation by defining a thermodynamic state space on a manifold and splitting dynamics into reversible and irreversible components.
• The framework is demonstrated through a triangular chemical reaction system, revealing cyclic conservation laws and nonlinear interaction effects.

"Nambu Non-equilibrium Thermodynamics I: Foundation"

Introduction and Motivation

The paper introduces an innovative framework for analyzing non-equilibrium thermodynamics, named Nambu Non-equilibrium Thermodynamics (NNET). This approach integrates the reversible dynamics characterized by the Nambu bracket with the irreversible mechanisms governed by entropy gradients. Traditional frameworks like Onsager’s linear response theory or the GENERIC formalism focus primarily on systems near equilibrium. They involve assumptions such as detailed balance and linearity, which limit their applicability to systems operating far from equilibrium or exhibiting cyclic dynamics.

Axiomatic Formulation

The core of the paper presents an axiomatic foundation for NNET. It defines a thermodynamic state space represented as a manifold $\mathcal{M}$, where localized thermodynamic variables can be described. Within this space, dynamics are split into reversible and irreversible components:

• Thermodynamic Time Evolution: Expresses the evolution of state variables as $$\dot{x}^i = \partial_{t}^{(H)}x^i + \partial_{t}^{(S)}x^i$$, combining reversible flow via the Nambu bracket with irreversible dissipation via entropy gradients.
• Reversible Dynamics: Governed by an $n$-ary Nambu bracket $\{x^i, H_1, \dots, H_{N-1}\}$, preserving multiple Hamiltonians and suitable for cyclic dynamics.
• Irreversible Dynamics: Defined by $L^{ij} \frac{\partial S}{\partial x^j}$, with $L_{ij}$ convex, ensuring positive entropy production under purely irreversible flows.

The paper introduces conditions under which entropy may decrease due to reversible components, delineating scenarios of open systems where entropy outflows are possible.

Relation to Generic Formalism

NNET contrasts with the GENERIC framework, which requires the entropy to serve as a Casimir within the Poisson structure, guaranteeing compliance with thermodynamic laws. GENERIC rigidly separates reversible and irreversible processes, limiting dynamic description. NNET, however, allows interaction between these components, capturing complex behaviors in oscillatory systems, thus extending the descriptive power beyond GENERIC constraints.

Triangular Reaction System Demonstration

The framework is exemplified through a triangular chemical reaction system. Conventional non-equilibrium thermodynamics assumes detailed balance and linear responses, reducing the description to simple transport coefficients and entropy gradients. NNET expands this by allowing for nonlinear terms, capturing higher-order interactions not evident under traditional constraints.

Three emergent conserved quantities are identified:

1. Geometric Conservation: Independent of detailed balance, representing inherent cyclic symmetry in reactions.
2. Hamiltonian Conservation: Dependent on reaction rate symmetry, linked to dynamics.
3. Entropy as Potential Function: Driving dissipative dynamics beyond the conventional equilibrium concept.

Implications and Conclusion

NNET emerges as a robust tool for describing non-equilibrium systems further from equilibrium, accommodating entropy reductions in periodic dynamics like biological oscillations. The paper outlines plans for expanding these ideas in subsequent installments, potentially transforming the analysis of complex systems resistant to classical thermodynamic methodologies.

Future Directions

Future investigations will explore nonlinear thermodynamic regimes, oscillatory behaviors, and stochastic dynamics, aiming for comprehensive integration within the NNET framework. The implications of redefining entropy in diverse formats, aligning with reversible and dissipative forces, promise new insights into thermodynamic descriptions across domains ranging from fluid dynamics to biological systems.