论文标题
在粘性基材上扩展气泡的亚竞争力
Metastability for expanding bubbles on a sticky substrate
论文作者
论文摘要
我们研究了一个维度界面与粘性的不可变元基板或壁相互作用的动力学行为。该界面受到相反方向的两个效果的约束。界面和底物之间的接触获得了能量奖励，而具有恒定强度的外部力将界面从墙壁上拉开。我们的界面是由$ \ mathbb {z} _+$的一维近邻居路径的图表建模的，起点为$ 0 $，在$ 2n $ steps之后以$ 0 $结束，该墙壁对应于水平轴的级别。在平衡处，每条路径$ξ=（ξ_x）_ {x = 0}^{2n} $，给出与$λ^{h（ξ）} \ exp（\fracσ{n} a（ξ）a（ξ）a（ξ））$，其中$ h（n p）是路径$ξ$和$ x $轴之间的区域。然后，我们考虑经典的热浴动力学，该动力学以恒定速率通过角叉平衡了每个$ξ_x$的值。 研究模型的静态，我们以$λ$和该模型的$σ来得出全相图，并确定临界线，该临界线将固定相位的固定相分开，固定力将界面粘贴到墙壁上，并为墙壁的外部力量稳定在墙壁周围的$ξ$，以确定性的形状稳定在壁的距离距离。在动态侧，我们确定了第二个临界线，该线将快速混合阶段（系统在多项式时间中混合）将混合时间呈指数增长的缓慢相位。在这种缓慢的混合体制中，我们获得了$ \ log $比例尺混合时间的尖锐估计，并提供了亚稳态行为的证据。
We study the dynamical behavior of a one dimensional interface interacting with a sticky unpenetrable substrate or wall. The interface is subject to two effects going in opposite directions. Contact between the interface and the substrate are given an energetic bonus while an external force with constant intensity pulls the interface away from the wall. Our interface is modeled by the graph of a one-dimensional nearest-neighbor path on $\mathbb{Z}_+$, starting at $0$ and ending at $0$ after $2N$ steps, the wall corresponding to level-zero the horizontal axis. At equilibrium each path $ξ=(ξ_x)_{x=0}^{2N}$, is given a probability proportional to $λ^{H(ξ)} \exp(\fracσ{N}A(ξ))$, where $H(ξ):=\#\{x \ : ξ_x=0\}$ and $A(ξ)$ is the area enclosed between the path $ξ$ and the $x$-axis. We then consider the classical heat-bath dynamics which equilibrates the value of each $ξ_x$ at a constant rate via corner-flip. Investigating the statics of the model, we derive the full phase diagram in $λ$ and $σ$ of this model, and identify the critical line which separates a localized phase where the pinning force sticks the interface to the wall and a delocalized one, for which the external force stabilizes $ξ$ around a deterministic shape at a macroscopic distance of the wall. On the dynamical side, we identify a second critical line, which separates a rapidly mixing phase (for which the system mixes in polynomial time) to a slow phase where the mixing time grows exponentially. In this slowly mixing regime we obtain a sharp estimate of the mixing time on the $\log$ scale, and provide evidences of a metastable behavior.
