Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models

TL;DR

This paper establishes a quantitative hydrodynamic limit for Langevin dynamics in gradient interface models, demonstrating regularity and fluctuation properties using homogenization techniques, with potential extensions to non-uniformly convex potentials.

Contribution

It introduces a novel quantitative homogenization approach to analyze Langevin dynamics for convex interface models, linking hydrodynamic limits with surface tension regularity.

Findings

Proved a quantitative hydrodynamic limit for the dynamics.

Established $C^2$ regularity of the surface tension.

Derived large-scale Lipschitz estimates for trajectories.

Abstract

We study the Langevin dynamics corresponding to the $\nabla\phi$ (or Ginzburg-Landau) interface model with a uniformly convex interaction potential. We interpret these Langevin dynamics as a nonlinear parabolic equation forced by white noise, which turns the problem into a nonlinear homogenization problem. Using quantitative homogenization methods, we prove a quantitative hydrodynamic limit, obtain the $C^2$ regularity of the surface tension, prove a large-scale Lipschitz-type estimate for the trajectories of the dynamics, and show that the fluctuation-dissipation relation can be seen as a commutativity of homogenization and linearization. Finally, we explain why we believe our techniques can be adapted to the setting of degenerate (non-uniformly) convex interaction potentials.