Linear Fluctuation of Interfaces in Glauber–Kawasaki Dynamics

Source abstract

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 $$K=K_N$$ K = K N (with $$K_N\uparrow \infty$$ K N ↑ ∞ as $$N\uparrow \infty$$ N ↑ ∞ ), the Kawasaki rates by $$N^2$$ N 2 and space by 1/ N . We start the process so that the continuum interface $$\Gamma _t$$ Γ t formed, between ‘high’ $$\rho _+$$ ρ + and ‘low’ $$\rho _-$$ ρ - density regions, is stationary that is, $$\Gamma _t$$ Γ t is ‘flat’. When the Glauber rates are balanced on $${\mathbb {T}}^d$$ T d , the interface $$\Gamma _t=\Gamma =\{v: v_1=0\}$$ Γ t = Γ = { v : v 1 = 0 } is immobile and the hydrodynamic limit is given by $$\rho (t,v) = \rho _+$$ ρ ( t , v ) = ρ + for $$v_1\in (0,1/2)$$ v 1 ∈ ( 0 , 1 / 2 ) and $$\rho (t,v)= \rho _-$$ ρ ( t , v ) = ρ - for $$v_1\in (-1/2,0)$$ v 1 ∈ ( - 1 / 2 , 0 ) for all $$t\ge 0$$ t ≥ 0 , where $$v=(v_1,\ldots ,v_d)\in {\mathbb {T}}^d$$ v = ( v 1 , … , v d ) ∈ T d identified with $$[-1/2,1/2)^d$$ [ - 1 / 2 , 1 / 2 ) d . In the pre-limit setting, a boundary region of width $$O(1/\sqrt{K_N})$$ O ( 1 / K N ) about the interface arises, where the density is more indeterminate than outside the region. Accordingly, we will scale the $$v_1$$ v 1 coordinate in the fluctuation field by $$\sqrt{K_N}$$ 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 $$K_N= O(\sqrt{\log (N)})$$ K N = O ( log ( N ) ) in $$d\le 2$$ d ≤ 2 . In the one dimensional case, the field limit is given by $$\textbf{e}(v_1) B_t$$ e ( v 1 ) B t where $$B_t$$ B t is a Brownian motion and $$\textbf{e}$$ e is the normalized derivative of a decreasing ‘standing wave’ solution $$\phi$$ ϕ of $$\partial ^2_{v_1} \phi - V'(\phi )=0$$ ∂ v 1 2 ϕ - V ′ ( ϕ ) = 0 on $${\mathbb {R}}$$ R , where $$V'$$ V ′ is the homogenization of the Glauber rates. In two dimensions, the limit is $$\textbf{e}(v_1)Z_t(v_2)$$ e ( v 1 ) Z t ( v 2 ) where $$Z_t$$ Z t is the solution of a one dimensional stochastic heat equation. The appearance of the function $$\textbf{e}(\cdot )$$ e ( · ) in the limit field indicates that the interface fluctuation retains the shape of the transition layer $$\phi$$ ϕ .