Linear Fluctuation of Interfaces in Glauber–Kawasaki Dynamics

Abstract

In this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by \(K=K_N\) (with \(K_N\uparrow \infty \) as \(N\uparrow \infty \)), the Kawasaki rates by \(N^2\) and space by 1/N. We start the process so that the continuum interface \(\Gamma _t\) formed, between ‘high’ \(\rho _+\) and ‘low’ \(\rho _-\) density regions, is stationary that is, \(\Gamma _t\) is ‘flat’. When the Glauber rates are balanced on \({\mathbb {T}}^d\), the interface \(\Gamma _t=\Gamma =\{v: v_1=0\}\) is immobile and the hydrodynamic limit is given by \(\rho (t,v) = \rho _+\) for \(v_1\in (0,1/2)\) and \(\rho (t,v)= \rho _-\) for \(v_1\in (-1/2,0)\) for all \(t\ge 0\), where \(v=(v_1,\ldots ,v_d)\in {\mathbb {T}}^d\) identified with \([-1/2,1/2)^d\). In the pre-limit setting, a boundary region of width \(O(1/\sqrt{K_N})\) about the interface arises, where the density is more indeterminate than outside the region. Accordingly, we will scale the \(v_1\) coordinate in the fluctuation field by \(\sqrt{K_N}\) so that the fluctuation field scaling limit will capture information ‘near’ the interface. We identify the fluctuation limit as a Gaussian field when \(K_N= O(\sqrt{\log (N)})\) in \(d\le 2\). In the one dimensional case, the field limit is given by \(\textbf{e}(v_1) B_t\) where \(B_t\) is a Brownian motion and \(\textbf{e}\) is the normalized derivative of a decreasing ‘standing wave’ solution \(\phi \) of \(\partial ^2_{v_1} \phi - V'(\phi )=0\) on \({\mathbb {R}}\), where \(V'\) is the homogenization of the Glauber rates. In two dimensions, the limit is \(\textbf{e}(v_1)Z_t(v_2)\) where \(Z_t\) is the solution of a one dimensional stochastic heat equation. The appearance of the function \(\textbf{e}(\cdot )\) in the limit field indicates that the interface fluctuation retains the shape of the transition layer \(\phi \).