When a fast-slow Markov diffusion is coarse-grained — the fast variables averaged out, leaving an effective equation for the slow ones — the dynamics converge. Trajectories converge, invariant measures converge, free energies converge. This is standard singular perturbation theory.
The paper establishes that convergence comes in four hierarchical levels, each stricter than the last. Level I: free energy converges. Level II: non-adiabatic entropy production converges (requires uniform curvature-dimension bounds). Level III: sharp lower bounds on both adiabatic and total entropy production (requires coefficient convergence, not just dynamical convergence). Level IV: a locking condition eliminates all entropy loss from unresolved microscopic degrees of freedom.
Each level requires a stronger hypothesis than the previous. Free energy convergence is nearly automatic — it follows from dynamical convergence alone. Entropy production convergence requires geometric control over the curvature of the probability landscape. Adiabatic entropy bounds require that the coefficients themselves converge, not just the solutions. And the locking condition — where the coarse-grained dynamics perfectly captures all thermodynamic content — requires a structural alignment between the slow manifold and the fast equilibration.
The hierarchy reveals that “thermodynamic convergence” is not one thing but four nested things. A coarse-grained model can accurately predict free energies while systematically underestimating entropy production. The dynamics converge before the thermodynamics does, and the thermodynamics converges in stages — free energy first, then dissipation, then the full entropy budget. Accuracy at one level is not evidence for accuracy at the next.