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Metastable Transitions and Γ-Convergent Eyring-Kramers Asymptotics in Landau-QCD Gradient Systemsстатья
- Авторы: Wu Jingxu, Shi Jie
- Журнал: ArXiv e-prints
- Год издания: 2025
- DOI: 10.48550/arXiv.2601.15343
- Аннотация: We develop a rigorous analytical framework for metastable stochastic transitions in Landau--type gradient systems inspired by QCD phenomenology. The functional $F(\sigma;u)=\int_\Omega [\frac{\kappa}{2}|\nabla\sigma|^2+V(\sigma;u)]\,dx$, depending smoothly on a control parameter $u\in\mathcal U$, is analyzed through the Euler--Lagrange map $\mathcal{E}(\sigma;u)=-\kappa\Delta\sigma+V'(\sigma;u)$ and its Hessian $\mathcal{L}_{\sigma,u}=-\kappa\Delta+V''(\sigma;u)$. By combining variational methods, $\Gamma$-- and Mosco convergence, and spectral perturbation theory, we establish the persistence and stability of local minima and index--one saddles under parameter deformations and variational discretizations. The associated mountain--pass solutions form Cerf--continuous branches away from the discriminant set $\mathcal D=\{u:\det\mathcal L_{\sigma,u}=0\}$, whose crossings produce only fold or cusp catastrophes in generic one-- and two--parameter slices. The $\Gamma$--limit is taken with respect to the $L^2(\Omega)$ topology, ensuring compactness, convergence of gradient flows, and spectral continuity of $\mathcal L_{\sigma,u}$. As a consequence, the Eyring--Kramers formula for the mean transition time between metastable wells retains quantitative validity under both parameter deformations and discretization refinement, with convergent free--energy barriers, unstable eigenvalues, and zeta--regularized determinant ratios. This construction unifies the classical intuition of Eyring, Kramers, and Langer with modern variational and spectral analysis, providing a mathematically consistent and physically transparent foundation for metastable decay and phase conversion in Landau--QCD--type systems.
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