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

Zhengyan Wu

TL;DR

The paper rigorously derives the nonlinear fluctuation behavior for the one-dimensional Ising–Kac–Kawasaki model near criticality by proving subsequential convergence of rescaled fluctuations to the stochastic Cahn–Hilliard equation, establishing a multi-scale large deviation principle, and showing Γ-convergence of the IK–K rate function to the CH rate function. The approach relies on regularizing noise, uniform entropy-dissipation and time-regularity estimates, Aubin–Lions compactness, and a skeleton-equation framework to analyze large deviations. Key contributions include a rigorous link between long-range interacting spin systems and nonlocal CH-type fluctuations, as well as a principled description of rare events via multi-scale LDP and variational convergence of rate functionals. These results advance the nonlinear fluctuation theory for conservative spin systems and illustrate how fluctuating hydrodynamics emerges from microscopic long-range dynamics, with potential implications for phase-transition phenomena and non-Gaussian fluctuation statistics near criticality.

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 $Γ$-convergence of the rate function for the rescaled fluctuating Ising-Kac-Kawasaki equation to the rate function of the Cahn-Hilliard equation.

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

TL;DR

The paper rigorously derives the nonlinear fluctuation behavior for the one-dimensional Ising–Kac–Kawasaki model near criticality by proving subsequential convergence of rescaled fluctuations to the stochastic Cahn–Hilliard equation, establishing a multi-scale large deviation principle, and showing Γ-convergence of the IK–K rate function to the CH rate function. The approach relies on regularizing noise, uniform entropy-dissipation and time-regularity estimates, Aubin–Lions compactness, and a skeleton-equation framework to analyze large deviations. Key contributions include a rigorous link between long-range interacting spin systems and nonlocal CH-type fluctuations, as well as a principled description of rare events via multi-scale LDP and variational convergence of rate functionals. These results advance the nonlinear fluctuation theory for conservative spin systems and illustrate how fluctuating hydrodynamics emerges from microscopic long-range dynamics, with potential implications for phase-transition phenomena and non-Gaussian fluctuation statistics near criticality.

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 -convergence of the rate function for the rescaled fluctuating Ising-Kac-Kawasaki equation to the rate function of the Cahn-Hilliard equation.

Paper Structure

This paper contains 25 sections, 29 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Key ideas and technical comments
  4. Nonlinear fluctuations of the dynamical Ising-Kac model
  5. Comments on the literature
  6. Structure of the paper
  7. Preliminaries
  8. Notations
  9. Derivation of the equation
  10. Assumptions of the noise
  11. Well-posedness of the SPDEs
  12. Uniform estimates and compactness
  13. Uniform rescaled entropy dissipation estimates
  14. Uniform time-regularity estimates
  15. Convergence of $\{u_{\gamma}\}_{\gamma\in(0,1]}$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Assume that the Assumption (A1) holds. Suppose that $J\in C^{\infty}(\mathbb{T}^1)$. For every $\gamma\in(0,1]$ and $\delta>0$, let $u_{\gamma,\delta}$ be the weak solution of equgamma-0 with initial data $u_{\gamma,0}$. Suppose that $a<0$, then there exists a subsequence of $(u_{\gamma,\delta})_{\g as $n\rightarrow\infty$ in probability, where $u$ is the weak solution of the Cahn-Hilliard equatio

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • ...and 49 more