论文标题
在川崎动力学下具有强各向异性相互作用的晶格气体中的亚竞争力
Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamics
论文作者
论文摘要
在本文中，我们分析了在非常低温下强烈的各向异性ISING晶格气体的川崎动力学本地版本的背景下的亚抗性和成核。令$λ\ subset \ mathbb {z}^2 $为有限盒。粒子对$λ$执行简单的排除，但是当它们占据相邻站点时，他们会感觉到具有绑定的能量$ -U_1 <0 $在水平方向上，而垂直方向则感觉到$ -U_2 <0 $。因此，川崎动态在卷$λ$内是保守的。沿着每个债券接触$λ$从外部到内部的边界，用速率$ρ= e^{ - δβ} $创建粒子，而沿着从内部到外部的每个债券沿着每个债券，粒子用速率$ 1 $ an灭，其中$β$是倒数温度，$ uex udeversevermations unvevertion undevers the Aversevertion Wemptorce and $δ> 0 $是活动参数。因此，$λ$的边界扮演着密度$ρ$的无限气体储层的角色。我们考虑参数制度$ U_1> 2U_2 $也称为强烈各向异性制度。我们在{（U_1，U_1+U_2）} $中采用$δ\，我们证明了（分别为完整）配置是一种可稳态（分别为稳定）配置。我们认为对应于有限体积的渐近状态在较大的反温度$β$的极限下。我们研究了从空到满的过渡是如何进行的。特别是，我们在概率，期望和分布中估算了从亚稳态构型到稳定配置的渐近过渡时间。此外，我们确定\ emph {crigical液滴}的大小及其某些属性。我们观察到弱和强烈各向异性方案的行为截然不同。我们发现\ emph {wulff shape}，即，将固定体积的液滴的能量最小化的形状与成核模式无关。
In this paper we analyze metastability and nucleation in the context of a local version of the Kawasaki dynamics for the two-dimensional strongly anisotropic Ising lattice gas at very low temperature. Let $Λ\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $Λ$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $Λ$. Along each bond touching the boundary of $Λ$ from the outside to the inside, particles are created with rate $ρ=e^{-Δβ}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $β$ is the inverse temperature and $Δ>0$ is an activity parameter. Thus, the boundary of $Λ$ plays the role of an infinite gas reservoir with density $ρ$. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $Δ\in{(U_1,U_1+U_2)}$ and we prove that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit of large inverse temperature $β$. We investigate how the transition from empty to full takes place. In particular, we estimate in probability, expectation and distribution the asymptotic transition time from the metastable configuration to the stable configuration. Moreover, we identify the size of the \emph{critical droplets}, as well as some of their properties. We observe very different behavior in the weakly and strongly anisotropic regimes. We find that the \emph{Wulff shape}, i.e., the shape minimizing the energy of a droplet at fixed volume, is not relevant for the nucleation pattern.
