Stochastic Cahn--Hilliard Equations from One-Dimensional Ising--Kac--Kawasaki Dynamics

TL;DR

This paper rigorously derives a stochastic Cahn--Hilliard equation as the scaling limit of one-dimensional Ising--Kac--Kawasaki dynamics, connecting microscopic lattice models to macroscopic stochastic PDEs.

Contribution

It provides a detailed analysis of the microscopic dynamics and proves convergence to a stochastic Cahn--Hilliard equation with conserved noise, including equilibrium measure convergence.

Findings

Convergence of the lattice dynamics to the stochastic Cahn--Hilliard equation.

Characterization of the limiting Gaussian noise in divergence form.

Weak convergence of the equilibrium measure to the $\,\phi^4_1$ measure.

Abstract

This paper investigates the scaling limit of one--dimensional lattice Ising--Kac--Kawasaki dynamics. Starting from a martingale formulation for the Kac coarse-grained field $X_\gamma$, we decompose the dynamics into a discrete conservative drift and a Dynkin martingale. The nonlinear drift is analyzed via a conservative multiscale replacement scheme based on one--block and two--block estimates, which yields a cubic conservative term in the macroscopic limit. For the stochastic component, we characterize the predictable quadratic variation to obtain a divergence-form Gaussian noise. By establishing uniform $H^{-1}$ energy estimates, we prove that $X_\gamma$ converges to a one--dimensional stochastic Cahn--Hilliard equation with conserved noise. Furthermore, we show that the associated canonical equilibrium measure $\mu_\gamma$ converges weakly to the $\phi^4_1$ measure on the conserved-mass hyperplane.