Equilibrium fluctuations of a one-dimensional nongradient Ginzburg-Landau model
We study the equilibrium fluctuations of a one-dimensional nongradient Ginzburg-Landau model. The dynamics in this model is governed by a conservation law together with random noise which also conserves the total charge. The resulting process is nongradient and reversible with respect to a family of time-independent Ginzburg-Landau Gibbs measures. Equilibrium fluctuations for the gradient Ginzburg-Landau model have been studied by several authors. The gradient condition on dynamics means the regular summation by parts can be performed. With this property for the dynamics, the original contributions from the oscillations of fluctuation fields can be largely reduced. For the nongradient system, we also need to control these large oscillations. The usual idea to replace local microscopic quantities by averaging with respect to the Gibbs state does not suffice. \textit{S. R. S. Varadhan} [in: Asymptotic problems in probability theory: stochastic models and diffusions on fractals. Pitman Res. Notes Math. Ser. 283, 75-128 (1993; Zbl 0793.60105)] introduced a perturbation technique to estimate the contributions of large oscillations to lower-order perturbation terms. The hydrodynamic scaling limits for the nongradient Ginzburg-Landau model are then derived with the bulk diffusion coefficient as a thermodynamic quantity. \textit{J. Quastel} [Commun. Pure Appl. Math. 45, No. 6, 623-679 (1992; Zbl 0769.60097)] extended this approach to the diffusion of color in a symmetric exclusion process. In considering fluctuations, we will also use this perturbation method. We identify the correct diffusion and drift terms for the limit fluctuation process by using estimates on the central limit theorem variances.
scientific article; zbMATH DE number 731982