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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.15343 |
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Table of Contents:
- We develop a rigorous analytical framework for metastable stochastic transitions in Landau-type gradient systems inspired by QCD phenomenology. The functional $F(σ;u)=\int_Ω[\fracκ{2}|\nablaσ|^2+V(σ;u)]\,dx$, depending smoothly on a control parameter $u\in\mathcal U$, is analyzed through the Euler-Lagrange map $\mathcal{E}(σ;u)=-κΔσ+V'(σ;u)$ and its Hessian $\mathcal{L}_{σ,u}=-κΔ+V''(σ;u)$. By combining variational methods, $Γ$- 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 $\mathcal D=\{u:\det\mathcal L_{σ,u}=0\}$, whose crossings produce only fold or cusp catastrophes in generic one- and two-parameter slices. The $Γ$-limit is taken with respect to the $L^2(Ω)$ topology, ensuring compactness, convergence of gradient flows, and spectral continuity of $\mathcal L_{σ,u}$. 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.