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

Abstract: This paper investigates the scaling limit of one--dimensional lattice Ising--Kac--Kawasaki dynamics. Starting from a martingale formulation for the Kac coarse-grained field $X_γ$, we decompose the dynamics into a discrete conservative drift and a Dynkin martingale. The nonlinear drift is analyzed via a conservative multiscale replacement scheme based on one--block and two--block estimates, which yields a cubic conservative term in the macroscopic limit. For the stochastic component, we characterize the predictable quadratic variation to obtain a divergence-form Gaussian noise. By establishing uniform $H<sup>{-1}$ energy estimates, we prove that $X_γ$ converges to a one--dimensional stochastic Cahn--Hilliard equation with conserved noise. Furthermore, we show that the associated canonical equilibrium measure $μ_γ$ converges weakly to the $φ<sup>4_1$ measure on the conserved-mass hyperplane.

• The paper derives the stochastic Cahn–Hilliard equation as the hydrodynamic limit of 1D Ising–Kac–Kawasaki dynamics with conserved magnetization.
• It employs multiscale replacement techniques and one-/two-block estimates to rigorously establish a cubic drift and divergence-form Gaussian noise from microscopic dynamics.
• The analysis connects spin-exchange processes to macroscopic SPDE behavior by proving equilibrium convergence, tight energy estimates, and ergodic properties.

Rigorous Derivation of Stochastic Cahn–Hilliard Dynamics from 1D Ising–Kac–Kawasaki Models

Introduction and Motivation

This paper establishes a hydrodynamic scaling limit of Kawasaki dynamics on the one-dimensional Ising–Kac lattice, with long-range interactions, connecting microscopic particle exchange processes to stochastic Cahn–Hilliard SPDEs (Model B). The approach directly analyzes lattice processes with conserved magnetization and derives a macroscopic stochastic evolution featuring mass conservation, cubic nonlinearity, and divergence-form noise. The main result is the identification of the macroscopic stochastic Cahn–Hilliard equation — including both its drift and noise structure — as the continuum limit of the Ising–Kac–Kawasaki lattice system.

Microscopic Model and Scaling

The system is defined on a ring $\Lambda_N = \mathbb{Z}/(2N+1)\mathbb{Z}$ with spin configurations $\sigma \in \{-1,1\}^{\Lambda_N}$, subject to a nonlocal Kac-type Ising interaction characterized by kernel $\kappa_\gamma$. Kawasaki exchange dynamics drives conservative nearest-neighbor spin swaps with rates depending on energy differences, ensuring detailed balance with respect to a canonical Gibbs measure and conservation of total magnetization.

For scaling, the coarse-grained field $X_\gamma$ is constructed by spatial smoothing via the Kac kernel, followed by time and amplitude rescaling. The scaling parameters $(\varepsilon_\gamma, \alpha_\gamma, \delta_\gamma)$ are chosen so that spatial mesh shrinks ($N\to\infty$), time accelerates, and field amplitudes are appropriately normalized, targeting spatiotemporal scales where mesoscopic fluctuations emerge.

Martingale Decomposition and Conservative Structure

The stochastic evolution of $X_\gamma$ admits a canonical martingale decomposition:

• The drift term arises from conservative bond currents, structurally expressing spin updates as discrete divergence.
• The martingale term tracks microscopic noise, whose quadratic variation is computed explicitly.

This splitting is critical in identifying the limiting SPDE structure: the drift encodes the deterministic evolution, while the martingale yields the stochastic forcing.

Expansion and Closure: Cubic Drift via Multiscale Replacement

A central challenge is to analyze the nonlinear conservative drift. This requires expressing local bond observables (such as $(\sigma_i - \sigma_{i+1})^2$) in terms of mesoscopic fields. The paper executes a multiscale closure argument using:

• One-block and two-block estimates to replace local lattice observables by their canonical equilibrium averages on mesoscopic blocks.
• Identification of the second-order Boltzmann–Gibbs principle, showing that $d_i^2$ is asymptotically equivalent to $2 (1 - m^2)$ where $\sigma \in \{-1,1\}^{\Lambda_N}$0 is the block magnetization.
• Kac-scale matching to pass block averages to the smoothed field $\sigma \in \{-1,1\}^{\Lambda_N}$1.

This yields the cubic drift $\sigma \in \{-1,1\}^{\Lambda_N}$2 in the macroscopic limit, with explicit coefficients determined by the microscopic parameters.

Noise Characterization: Divergence-Type Gaussian Forcing

The quadratic variation of the martingale term is analyzed using similar replacement methods, identifying the noise as divergence-form Gaussian:

• The local mobility observable is replaced by its equilibrium value.
• The covariance structure converges to a divergence-type operator, ensuring mass conservation in the macroscopic noise.

The limiting noise is thus $\sigma \in \{-1,1\}^{\Lambda_N}$3, where $\sigma \in \{-1,1\}^{\Lambda_N}$4 is space–time white noise.

Hydrodynamic Limit and Main Theorem

Under precise kernel and scaling assumptions, and for initial data with suitable entropy bounds, the main result proves that $\sigma \in \{-1,1\}^{\Lambda_N}$5 converges in law to $\sigma \in \{-1,1\}^{\Lambda_N}$6 solving the stochastic Cahn–Hilliard equation:

$\sigma \in \{-1,1\}^{\Lambda_N}$7

Explicit expressions for $\sigma \in \{-1,1\}^{\Lambda_N}$8 are obtained from limits of the lattice parameters. The convergence holds in appropriate function spaces capturing both spatial and temporal regularity.

Canonical Gibbs Measures and Equilibrium

The equilibrium measure induced by the lattice Gibbs measure $\sigma \in \{-1,1\}^{\Lambda_N}$9 under the map $\kappa_\gamma$0 converges to the $\kappa_\gamma$1 canonical measure on the conserved-mass hyperplane $\kappa_\gamma$2. This is characterized by:

$\kappa_\gamma$3

where $\kappa_\gamma$4 is a Ginzburg–Landau-type free energy functional.

No Wick renormalization is required in 1D; the measure is ergodic and supports spatial regularity strictly below $\kappa_\gamma$5.

Tightness, Moment Bounds, and Energy Estimates

Uniform moment bounds and energy estimates for the shifted field $\kappa_\gamma$6 (via Da Prato–Debussche decomposition) are established. This includes:

• Tightness and compactness arguments (Aubin–Lions–Simon) for passage to the limit.
• Uniform $\kappa_\gamma$7 and $\kappa_\gamma$8 estimates, leveraging energy inequalities and Sobolev embeddings.

The argument demonstrates convergence not only in law but also in equilibrium measure, confirming emergence of reversibility and ergodicity at the macroscopic level.

Implications, Contrasts, and Future Directions

This work advances rigorous hydrodynamic theory for conservative interacting particle systems, linking microscopic structure to nonlinear stochastic PDEs. Unlike non-conservative Ising–Kac–Glauber settings (where renormalized $\kappa_\gamma$9 equations arise), the conservative case demands closure at the current level — a more complex multiscale analysis. The methodology confirms that both drift and noise in the SPDE are emergent properties of the underlying lattice dynamics, including full identification of coefficients.

Potential future directions include:

• Higher-dimensional extensions, where singularities and renormalization become relevant.
• Broader classes of conservative dynamics with different interaction kernels and boundary conditions.
• Quantitative rates of convergence or large deviations for fluctuations.
• Applications to invariant measure structure, ergodicity, and long-time behavior in stochastic hydrodynamics.

Conclusion

This paper provides a comprehensive, rigorous derivation of the one-dimensional stochastic Cahn–Hilliard equation (Model B) with divergence-form noise from the Ising–Kac–Kawasaki lattice dynamics. The analysis encompasses multiscale closure of nonlinear drift and noise terms, moment estimates, equilibrium measure convergence, and tightness in function space. This substantiates physical and mathematical connections between microscopic conservative dynamics and macroscopic stochastic PDEs, offering a template for similar derivations in more complex settings (2603.29923).