A pair of flat parallel surfaces, each freely diffusing along the direction of their separation, will eventually come into contact. If the shapes of these surfaces also fluctuate, then contact will occur when their centers-of-mass remain separated by a nonzero distance $\ell$. An example of such a situation is the motion of interfaces between two phases at conditions of thermodynamic coexistence, and in particular the annihilation of domain wall pairs under periodic boundary conditions. Here we present a general approach to calculate the probability distribution of the contact distance $\ell$ and determine how its most likely value ${\ell }^{*}$ depends on the surfaces' lateral size $L$. Using the Edward-Wilkinson equation as a model for interfaces, we demonstrate that ${\ell }^{*}$ scales weakly with system size, i.e., the dependence of ${\ell }^{*}$ on $L$ for both ($1+1$)- and ($2+1$)-dimensional interfaces is such that ${lim}_{L\to \infty }({\ell }^{*}/L)=0$. In particular, for ($2+1$)-dimensional interfaces ${\ell }^{*}$ is an algebraic function of $logL$, a result that is confirmed by computer simulations of slab-shaped domains formed under periodic boundary conditions. This weak scaling implies that such domains remain topologically intact until $\ell$ becomes very small compared to the lateral size of the interface, contradicting expectations from equilibrium thermodynamics.

中文翻译：

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两个波动接口之间的接触距离的缩放比例随系统大小而变小

一对平坦的平行表面最终会相互接触，每个表面沿其分离方向自由扩散。如果这些表面的形状也发生波动，则当它们的质心保持非零距离分开时将发生接触$\ell$。这种情况的一个例子是在热力学共存的条件下，两相之间的界面运动，特别是在周期性边界条件下，畴壁对的an灭。这里我们提出一种通用的方法来计算接触距离的概率分布$\ell$ 并确定其最可能的价值 ${\ell }^{*}$ 取决于表面的横向尺寸 $\mathrm{大号}$。使用Edward-Wilkinson方程作为界面模型，我们证明了${\ell }^{*}$ 随系统规模微弱扩展，即 ${\ell }^{*}$ 上 $\mathrm{大号}$ 对彼此而言 （$\mathrm{1个}+\mathrm{1个}$）-和（$2+\mathrm{1个}$）维接口是这样的 ${林}_{\mathrm{大号}\to \infty }（{\ell }^{*}/\mathrm{大号}）=0$。特别是对于$2+\mathrm{1个}$）维接口 ${\ell }^{*}$ 是...的代数函数 $日志\mathrm{大号}$，该结果已通过计算机模拟，验证了在周期性边界条件下形成的平板状区域。这种弱缩放性意味着此类域在拓扑上保持完整，直到$\ell$ 与界面的横向尺寸相比变得很小，这与平衡热力学的预期相矛盾。