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Four Levels of Thermodynamic Convergence of Singularly Perturbed Markov Semigroups

Xinyu Zhang, Liu Hong

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 $CD(-κ,\infty)$, 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.

Four Levels of Thermodynamic Convergence of Singularly Perturbed Markov Semigroups

Abstract

Assuming the dynamical convergence 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 -uniform curvature--dimension bound , Level~II shows convergence of the non-adiabatic entropy production. By further assuming coefficient convergence, Level~III yields sharp 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 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.
Paper Structure (32 sections, 36 theorems, 299 equations, 1 figure, 1 table)

This paper contains 32 sections, 36 theorems, 299 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Setting and Preliminary Thermodynamic Results
  3. Convention (generic notation).
  4. Semigroup assumptions and $\Gamma$-calculus objects
  5. Diffusion-form specialization and dictionary
  6. Initial data.
  7. Preliminary thermodynamic results
  8. Main Results: Four levels of Thermodynamic Upgrade
  9. Assumptions
  10. Convergence results for thermodynamic functionals
  11. Case Studies: Averaging and Stiff-Potential Limits
  12. Averaging limit in a slow--fast regime
  13. Outline.
  14. Linear OU model: verification of assumptions
  15. Nonlinear multiplicative noise
  16. ...and 17 more sections

Key Result

lemma 1

Let $\mathcal{L}$ be given by Eq. eq:backward-eps1 and let $\mathcal{L}^\dagger$ denote its $L^2(\pi)$-adjoint. Define the symmetric and antisymmetric parts Then for all $f,g\in C_c^\infty(X)$, Moreover, writing $\mathrm{id}:X\to\mathbb{R}^d$ for the coordinate map, we have in the sense of componentwise identities on $X$.

Figures (1)

  • Figure 1: Framework overview. Assumptions (left) yield properties (middle) and define the thermodynamic convergence levels (right). Single arrows indicate sufficient implications; double arrows indicate definitions (typically adding the previous level).

Theorems & Definitions (72)

  • lemma 1: Dictionary between $\mathcal{L}$, $\Gamma$ and $(A,\gamma)$
  • definition 1: Thermodynamic terms
  • lemma 2: Entropy identity
  • definition 2: Four levels of thermodynamic convergence
  • theorem 1
  • definition 3: Steady-state Level III$_{\mathrm{ss}}$ thermodynamic convergence
  • corollary 1: Steady-state housekeeping lower semicontinuity
  • definition 4: Recovery sequence
  • theorem 2
  • proof
  • ...and 62 more