Topological Irreversible Thermodynamics

This framework integrates conserved fluid helicity into linear irreversible thermodynamics alongside energy and enstrophy, formulating non-equilibrium transport as a generalized gradient flow driven by free energy variations. Because velocity and vorticity both reverse sign under microscopic time reversal, the quadratic densities of energy, enstrophy, and topological linking are strictly even, guaranteeing pure Onsager reciprocity with an identically vanishing Casimir antisymmetric block. Singularities associated with continuous topological changes are resolved by projecting the gradient flow onto the discrete eight-hundred-sixty-one-node manifold, where diffusive transport is governed by the discrete graph Laplacian and bounded by its spectral gap. Macroscopic fluxes proceed through deterministic vortex reconnections that transfer discrete integer units of linking charge, circulation, and enstrophy, following an exact microscopic fluctuation relation. Because enstrophy and linking share the same physical reconnection event, the off-diagonal coupling establishes a Topological Seebeck Effect, where an enstrophy gradient autonomously drives a quantized linking current, and reciprocally, a topological affinity drives enstrophy flow.