Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models

Abstract

We study the Langevin dynamics corresponding to the ∇φ (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.

Upper bounds on the fluctuations for a class of degenerate convex $\nabla ϕ$-interface models

“…It seems for instance reasonable to us that the fluctuation estimate of Proposition 3.3 can be used to prove that the surface tension of the model is strictly convex (i.e., that the eigenvalues of its Hessian are always strictly positive). The strict convexity of the surface tension plays an important role in the proof of the hydrodynamic limit in [50], and we further believe that this result could be combined with the estimate of Theorem 1.2 to prove a quantitative version of the hydrodynamic limit following the techniques of [11]. Once the quantitative hydrodynamic limit has been established, it should be possible to develop a large-scale regularity theory for the model (see [11,Theorem 1.5]).…”

“…The strict convexity of the surface tension plays an important role in the proof of the hydrodynamic limit in [50], and we further believe that this result could be combined with the estimate of Theorem 1.2 to prove a quantitative version of the hydrodynamic limit following the techniques of [11]. Once the quantitative hydrodynamic limit has been established, it should be possible to develop a large-scale regularity theory for the model (see [11,Theorem 1.5]). This result would then be useful to quantify the ergodicity of the environment appearing in the Helffer-Sjöstrand representation formula and would be helpful to establish a quantitative version of the scaling limit of the model (following the insight of [67,51]).…”

“…This result would then be useful to quantify the ergodicity of the environment appearing in the Helffer-Sjöstrand representation formula and would be helpful to establish a quantitative version of the scaling limit of the model (following the insight of [67,51]). We refer to the introduction of [11] for a more detailed description of this line of research. We plan to investigate this in a future work.…”