2026/01/20 by Jingxu Wu, Jie Shi · 1 voice
Physics and Astronomy · #physics.gen-ph
We develop a rigorous analytical framework for metastable stochastic transitions in Landau-type gradient systems inspired by QCD phenomenology. The functional F(σ;u)=∫Ω[\fracκ2|∇σ|2+V(σ;u)] dx, depending smoothly on a control parameter u∈\mathcal U, is analyzed through the Euler-Lagrange map E(σ;u)=-κΔσ+V'(σ;u) and its Hessian Lσ,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 Lσ,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 L2(Ω) topology, ensuring compactness, convergence of gradient flows, and spectral continuity of \mathcal Lσ,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.