A gauge-theoretic formulation of Nambu non-equilibrium thermodynamics

We present a gauge-theoretic formulation of Nambu non-equilibrium thermodynamics (NNET) within a common gauge-fixed worldline framework. In this formulation, the generalized thermodynamic force is described by an Abelian thermodynamic connection, and ordinary Onsager dynamics is recovered by fixing its temporal component through the dissipation function. To extend this construction to NNET, we introduce a closed two-form as additional geometric input. This form determines a divergence-free Nambu vector field and thereby defines a characteristic direction on the extended thermodynamic state space. The Onsager gauge condition is then generalized so that the dissipative gauge fixing is imposed along this Nambu characteristic direction. The resulting constitutive relation reproduces the basic NNET evolution equation, combining nondissipative Nambu circulation with dissipative Onsager relaxation. We further show that the localized BF-type expression is not an independent pre-gauge-fixed action, but a differential-form representation of the Nambu-adapted gauge-fixed worldline theory. The Faddeev–Popov operator associated with the BRST gauge fixing becomes, on shell, the negative of the material derivative along the full NNET flow, and its time-independent zero-mode profiles satisfy the same equation as first integrals of the deterministic dynamics. Finite residual transformations obey a Hamilton–Jacobi equation governed by the same Onsager tensor, implying that the gauge parameter is nonincreasing along the original NNET flow. These results provide a unified geometric interpretation of Onsager dissipation and Nambu dynamics within a single gauge-fixing framework.