Metastable Transitions and $Γ$-Convergent Eyring-Kramers Asymptotics in Landau-QCD Gradient Systems
Jingxu Wu, Jie Shi
TL;DR
The paper develops an analytic framework for metastable transitions in infinite-dimensional Landau-type gradient systems with QCD-inspired potentials, marrying variational methods, $\\Gamma$-convergence, Mosco convergence, spectral theory, and large deviations to derive EK-type asymptotics. Away from a discriminant set where degeneracies occur, minima and index-one saddles persist under parameter changes and discretization, as guaranteed by $\\Gamma$-convergence and $Mosco$ convergence with Cerf continuation guiding critical points. The main result is a field-theoretic Eyring–Kramers law with a prefactor given by a ratio of $\\zeta$-regularized determinants and the unique negative eigenvalue, i.e. $\\mathbb{E}_{\\sigma_0}\\tau_D^\\varepsilon \\sim \\frac{2\\pi}{\\Gamma|\\lambda_-(\\sigma^{\\dagger})|} \\sqrt{\\frac{\\det(\\mathcal{L}_{meta})}{\\det{}'(\\mathcal{L}_{\\sigma^{\\dagger}})}} \\\exp(\\Delta\\mathcal{F}/\\varepsilon^2)$. Near spinodals, universal scalings $\\Delta F \\sim (u-u_c)^{3/2}$ and $|\\lambda_-| \\sim (u-u_c)^{1/2}$ emerge, with the framework remaining robust under discretization and deformations, thereby linking microscopic fluctuations to macroscopic kinetics in QCD-like and related systems. The results provide a mathematically controlled bridge between nucleation-like transitions and field-theoretic metastability, with potential extensions to underdamped dynamics and data-informed potentials.
Abstract
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.