Thermodynamic Origin of Degree-Day Scaling in Phase-Change Systems
Phase transitions impose topological constraints on thermodynamic state variables, masking energetic fluctuations at the phase boundary. This constraint is most apparent in melting systems, where temperature remains pinned despite continued energy input. Here we resolve this information loss by introducing a latent temperature-a counterfactual trajectory describing the system's unconstrained thermal evolution. We show that energy conservation alone enforces a rigorous duality between the total latent heat dissipated during phase change and the accumulated exceedance of the latent temperature above the melting point. This duality is mathematically equivalent to the one-dimensional Wasserstein-1 distance between the latent and observed temperature trajectories, with the transport cost set by a characteristic surface dissipation timescale and melting energy. Applied to ice-sheet surface melting, this timescale admits a direct physical interpretation in terms of radiative and turbulent heat loss. The same framework yields a first-principles derivation of the empirical Positive Degree Day law and predicts realistic degree-day factors that emerge from surface energy balance, without ad hoc calibration. More broadly, phase change emerges as an optimal transport process that projects continuous energetic variability onto a constrained thermodynamic boundary.