Thermodynamics as the Coarse-Grained Limit of SDEF Hydrodynamics

This work derives thermodynamic behavior as the coarse-grained limit of the hydrodynamic regime of the Scalar Drag Emergence Framework (SDEF), a primitive dynamical system defined by a compatibility field and an ancestry (memory) field. No thermodynamic variables are introduced as primitives. Instead, temperature, internal energy, pressure, and entropy arise from coarse-grained statistical behavior of compatibility redistribution and ancestry accumulation. In this limit, resolved transport becomes mixing, ancestry acts as a reservoir, and saturation produces a pressure-like response. Temperature emerges as a measure of transport fluctuation intensity, internal energy corresponds to accumulated ancestry, pressure corresponds to saturation, and entropy corresponds to the dispersion of ancestry under unresolved transport. A First-Law–like balance relation and Second-Law–like entropy increase arise from coarse-grained dynamics rather than being imposed as axioms. This work establishes thermodynamics as a statistical projection of hydrodynamic transport–ancestry dynamics, completing a unified description in which primitive dynamics give rise to hydrodynamic behavior, and hydrodynamics under coarse-graining gives rise to thermodynamic behavior.