A Gauge-Theoretic Formulation of Nambu Non-equilibrium Thermodynamics

Abstract

We present a gauge-theoretic formulation of Nambu non-equilibrium thermodynamics (NNET). In this framework, the thermodynamic one-form $A=A_{i}d{a}^i$ is treated as an Abelian thermodynamic connection. The flatness condition $F=dA=0$ expresses the integrability of this connection in the equilibrium thermodynamic sector, while irreversible Onsager relaxation is encoded by the gauge-fixing condition $A_0=\frac{1}{2}{L}^{ij}{A}_i{A}_j$ and by the positivity of the resulting entropy production. By further introducing a BF/Chern--Simons--like coupling $A\wedge B\wedge dt$, with the local Clebsch representation $B=d{H}_1\wedge d{H}_2$, the Nambu bracket naturally emerges. The resulting variational principle yields the basic equation of NNET, in which the Nambu part describes reversible circulation and the Onsager part describes dissipative relaxation. This formulation provides a unified geometric foundation for the integrable thermodynamic sector, the gauge-fixed dissipative sector, and the Nambu-extended non-equilibrium sector.

13 pages