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The consistency across scales of a recently developed mathematicalthermodynamic structure, between a continuous stochastic nonlinear dynamicalsystem (diffusion process with Langevin or Fokker-Planck equations) and itsemergent discrete, inter-attractoral Markov jump process, is investigated. Weanalyze how the system's thermodynamic state functions, e.g. free energy $F$,entropy $S$, entropy production $e_p$, and free energy dissipation $\dot{F}$,etc., are related when the continuous system is describe with a coarse-graineddiscrete variable. We show that the thermodynamics derived from the underlyingdetailed continuous dynamics is exact in the Helmholtz free-energyrepresentation. That is, the system thermodynamic structure is the same as ifone only takes a middle-road and starts with the "natural" discretedescription, with the corresponding transition rates empirically determined. By"natural", we mean in the thermodynamic limit of large systems in which thereis an inherent separation of time scales between inter- and intra-attractoraldynamics. This result generalizes a fundamental idea from chemistry and thetheory of Kramers by including thermodynamics: while a mechanical descriptionof a molecule is in terms of continuous bond lengths and angles, chemicalreactions are phenomenologically described by the Law of Mass Action with rateconstants, and a stochastic thermodynamics.

Stochastic Thermodynamics Across Scales: Emergent Inter-attractoral Discrete Markov Jump Process and Its Underlying Continuous Diffusion