Four Levels of Thermodynamic Convergence of Singularly Perturbed Markov Semigroups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911518821449728 |
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| author | Zhang, Xinyu Hong, Liu |
| author_facet | Zhang, Xinyu Hong, Liu |
| contents | 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 $CD(-κ,\infty)$, 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 Itô--Kunita derivative-flow method) and illustrate the theory on slow--fast averaging limits and stiff-potential regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Four Levels of Thermodynamic Convergence of Singularly Perturbed Markov Semigroups Zhang, Xinyu Hong, Liu Probability 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 $CD(-κ,\infty)$, 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 Itô--Kunita derivative-flow method) and illustrate the theory on slow--fast averaging limits and stiff-potential regimes. |
| title | Four Levels of Thermodynamic Convergence of Singularly Perturbed Markov Semigroups |
| topic | Probability |
| url | https://arxiv.org/abs/2603.14906 |