arXiv:2601.15343v1 Announce Type: new
Abstract: We develop a rigorous analytical framework for metastable stochastic transitions in Landau–type gradient systems inspired by QCD phenomenology. The functional $F(sigma;u)=int_Omega [frac{kappa}{2}|nablasigma|^2+V(sigma;u)],dx$, depending smoothly on a control parameter $uinmathcal U$, is analyzed through the Euler–Lagrange map $mathcal{E}(sigma;u)=-kappaDeltasigma+V'(sigma;u)$ and its Hessian $mathcal{L}_{sigma,u}=-kappaDelta+V”(sigma;u)$. 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 $mathcal D={u:detmathcal L_{sigma,u}=0}$, 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.