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

Abstract

This paper investigates the scaling limit of the one-dimensional lattice Ising--Kac--Kawasaki dynamics near the critical temperature. Starting from a martingale formulation for the Kac coarse-grained field, we decompose the microscopic dynamics into a discrete conservative drift and a pure-jump Dynkin martingale. Under a critical scaling regime, the nonlinear drift is identified via a conservative multiscale replacement scheme based on the second-order Boltzmann--Gibbs Principle, yielding the cubic conservative term. For the fluctuation component, a martingale central limit theorem characterizes the predictable quadratic variation as a divergence-form Gaussian noise. By establishing uniform $H^{-1}$ energy estimates and utilizing a compactness argument, we prove that the Kac coarse-grained field converges to the unique solution of a one-dimensional conservative stochastic Cahn--Hilliard equation. Furthermore, we demonstrate that the associated canonical equilibrium measure induced by the microscopic dynamics converges weakly to the standard $\phi^4_1$ measure on the conserved-mass hyperplane.