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

Qi Zhang

Abstract

We study the scaling limit of the one-dimensional lattice Ising--Kac--Kawasaki dynamics under conservative Kawasaki exchange rate. For the Kac coarse-grained field \(X_γ\), we derive a martingale formulation with a discrete conservative drift and a Dynkin martingale. The nonlinear drift is identified by a conservative multiscale replacement scheme based on one-block/two-block estimates, yielding a cubic conservative term in the macroscopic equation. For the stochastic part, we compute the predictable quadratic variation, and obtain a divergence-form Gaussian noise. As a consequence, \(X_γ\) converges to a one-dimensional stochastic Cahn--Hilliard equation with conserved noise.

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

Abstract

We study the scaling limit of the one-dimensional lattice Ising--Kac--Kawasaki dynamics under conservative Kawasaki exchange rate. For the Kac coarse-grained field , we derive a martingale formulation with a discrete conservative drift and a Dynkin martingale. The nonlinear drift is identified by a conservative multiscale replacement scheme based on one-block/two-block estimates, yielding a cubic conservative term in the macroscopic equation. For the stochastic part, we compute the predictable quadratic variation, and obtain a divergence-form Gaussian noise. As a consequence, converges to a one-dimensional stochastic Cahn--Hilliard equation with conserved noise.

Paper Structure

This paper contains 19 sections, 26 theorems, 337 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main result.
  3. Preliminary
  4. Fourier extension from the lattice to the torus
  5. Well-posedness of the $1$d stochastic Cahn--Hilliard equation
  6. Dirichlet forms, entropy inequality, and the Kipnis–Varadhan bound
  7. Expansion for the discrete drift
  8. (i) Linear part.
  9. (ii) Nonlinear part.
  10. (iii) Remainder.
  11. The Second-order Boltzmann--Gibbs Principle
  12. One-block replacement
  13. Two-block replacement for the conserved field
  14. Kac-scale regularity of mesoscopic block averages
  15. Convergence of the noise term
  16. ...and 4 more sections

Key Result

Theorem 1

Assume that the assumption $({\bf A})$-$({\bf C})$ hold. After Fourier extension and letting $\gamma\to0$, the Kac coarse-grained field $X_{\gamma}$ converges in law to $X$ in $L^2(0,T;L^2(\mathbb T))\cap C([0,T],H^{-3}(\mathbb T))$, where $X$ is the unique solution of the one-dimensional stochastic Moreover, $X(\cdot)$ has conservation law:

Theorems & Definitions (49)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Equivalence of discrete and continuum Sobolev norms
  • Lemma 4
  • proof
  • Proposition 2
  • Definition 2.1: Global Dirichlet form on the canonical sector
  • ...and 39 more