The scaling limit of one-dimensional Ising-Kac-Kawasaki dynamics is the stochastic Cahn-Hilliard equation with conserved Gaussian noise, and the equilibrium measures converge weakly to the phi^4_1 measure on the conserved-mass hyperplane.
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Stochastic Cahn--Hilliard Equations from One-Dimensional Ising--Kac--Kawasaki Dynamics
The scaling limit of one-dimensional Ising-Kac-Kawasaki dynamics is the stochastic Cahn-Hilliard equation with conserved Gaussian noise, and the equilibrium measures converge weakly to the phi^4_1 measure on the conserved-mass hyperplane.