标题： 界面在Glauber-Kawasaki动力学中的线性涨落 显示英文标题

标题： Linear fluctuation of interfaces in Glauber-Kawasaki dynamics

摘要： 在本文中，我们找到了Glauber + Kawasaki粒子动力学在流体动力学平均曲率界面极限附近的时空质量波动场的标度极限。 在这里，Glauber速率被缩放为$K=K_N$，Kawasaki速率被缩放为$N^2$，空间被缩放为$1/N$。 我们启动该过程，使得形成的界面$\Gamma_t$是平稳的，即，$\Gamma_t$是“平坦”的。 当Glauber速率在$T^d$上平衡时，$\Gamma_t=\Gamma=\{x: x_1=0\}$是不可移动的，流体动力学极限由$\rho(t,v) = \rho_+$表示，对于$v_1\in (0,1/2)$和$\rho(t,v)= \rho_-$对于$v_1\in (-1/2,0)$对于所有$t\ge 0$，其中$v=(v_1,\ldots,v_d)\in T^d$与$[-1/2,1/2)^d$相一致。 由于在形成过程中，界面附近的边界区域具有宽度$O(1/\sqrt{K_N})$，我们将波动场中的$v_1$坐标按$\sqrt{K_N}$进行缩放，以便缩放极限能够捕捉到界面“附近”的信息。我们当$K_N\uparrow \infty$和$K_N= O(\sqrt{\log(N)})$在$d\leq 2$时将波动极限识别为高斯场。 在一维情况下，场极限由${\bf e}(v_1) B_t$给出，其中$B_t$是一个布朗运动，${\bf e}$是一个递减的“驻波”解$\phi$的归一化导数，该解满足$\partial^2_{v_1} \phi - V'(\phi)=0$在$R$上，其中$V'$是 Glauber 率的均质化。 在二维情况下，极限是${\bf e}(v_1)Z_t(v_2)$，其中$Z_t$是一维随机热方程的解。 极限场中函数${\bf e}(\cdot)$的出现表明界面波动保持了过渡层$\phi$的形状。 显示英文摘要

摘要： 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 $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface $\Gamma_t$ formed is stationary that is, $\Gamma_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $\Gamma_t=\Gamma=\{x: x_1=0\}$ 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 $v=(v_1,\ldots,v_d)\in T^d$ identified with $[-1/2,1/2)^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N= O(\sqrt{\log(N)})$ in $d\leq 2$. In the one dimensional case, the field limit is given by ${\bf e}(v_1) B_t$ where $B_t$ is a Brownian motion and ${\bf e}$ is the normalized derivative of a decreasing `standing wave' solution $\phi$ of $\partial^2_{v_1} \phi - V'(\phi)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is ${\bf e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function ${\bf e}(\cdot)$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $\phi$.