Entropy Production from Macroscopic Balances: Bridging Continuum Laws and Stochastic Dynamics

We develop a balance-first framework for nonequilibrium thermodynamics in which entropy production follows directly from macroscopic conservation laws, without assuming the Gibbs relation, local equilibrium, or a predefined temperature field. Entropy flux and production emerge naturally from the flux–force structure, with non-negativity ensured by dissipative admissibility. Substituting microscopic currents from the Fokker–Planck equation recovers the canonical entropy production laws for both overdamped and underdamped Brownian motions, which demonstrates that macroscopic and stochastic descriptions share identical production and extraction identities. The framework further quantifies thermodynamic distances and establishes rigorous bounds linking current, activity, and dissipation, providing a unifying route from continuum balance to stochastic irreversibility. Thus, entropy appears as an emergent, gauge-invariant quantity that connects transport, distance, and control in nonequilibrium systems.