Four Levels of Thermodynamic Convergence of Singularly Perturbed Markov Semigroups

Abstract

Assuming the dynamical convergence $P_t^\varepsilon\to\bar P_t$ for singular limits of time-homogeneous Markov diffusion semigroups, we develop a semigroup-level framework that upgrades this convergence into four levels of thermodynamic convergence (including non-reversible diffusions and multiplicative noise). Level~I yields convergence of the free energy, and under an $\varepsilon$-uniform curvature--dimension bound $CD(-\kappa,\infty)$, Level~II shows convergence of the non-adiabatic entropy production. By further assuming coefficient convergence, Level~III yields sharp $\liminf$ bounds for the adiabatic and total entropy productions. Moreover, Level~IV holds precisely when a locking condition holds, with no loss on entropy production arising from unresolved microscopic nonequilibrium forcing. We give two verifiable routes to the uniform $CD$ hypothesis (a Ricci-type criterion and an It\^{o}--Kunita derivative-flow method) and illustrate the theory on slow--fast averaging limits and stiff-potential regimes.