Fluctuating Hydrodynamics of the Ising-Kac-Kawasaki Model and Nonlinear Fluctuations Near Criticality

TL;DR

This paper demonstrates that the rescaled fluctuating Ising-Kac-Kawasaki equation converges to the stochastic Cahn-Hilliard equation, confirming a conjecture about nonlinear fluctuations near criticality and analyzing large deviations and rate functions.

Contribution

It proves the convergence of the fluctuating Ising-Kac-Kawasaki model to the stochastic Cahn-Hilliard equation and establishes related large deviation principles and $\Gamma$-convergence results.

Findings

Rescaled Ising-Kac-Kawasaki equation converges to stochastic Cahn-Hilliard equation.

Multi-scale dynamical large deviations are established.

Rate function convergence via $\Gamma$-convergence is shown.

Abstract

We study the scaling limit behavior of a family of conservative SPDEs as the fluctuating Ising-Kac-Kawasaki dynamics. Precisely, we show that there exists a sequence of the one-dimensional rescaled fluctuating Ising-Kac-Kawasaki equation converges to the solution of the stochastic Cahn-Hilliard equation. This solves a simple version of the conjecture concerning the nonlinear fluctuation phenomenon, proposed by [Giacomin, Lebowitz, Presutti; Math. Surveys Monogr., 1999]. Furthermore, we prove a multi-scale dynamical large deviations in a small noise regime. Finally, we show the $\Gamma$-convergence of the rate function for the rescaled fluctuating Ising-Kac-Kawasaki equation to the rate function of the Cahn-Hilliard equation.