标题： 凝聚、边界条件以及慢站点对零范围系统的影响 显示英文标题

标题： Condensation, boundary conditions, and effects of slow sites in zero-range systems

摘要： 我们考虑在具有有限缺陷数的$1$维离散环面$\mathbb{Z}/N\mathbb{Z}$上的零范围粒子系统中粒子质量的空间-时间缩放极限。 我们关注两类递增跳跃速率$g$，当$g(n)\sim n^\alpha$时，对于$0<\alpha\leq 1$，以及当$g$是一个有界函数时。 在这种模型中，位于常规位点 $k$ 的粒子以速率 $g(n)$ 等概率跳转到相邻位点，仅取决于位点 $k$ 上的粒子数 $n$。 在缺陷位点$k_{j,N}$处，当$g(n)\sim n^\alpha$时，跳跃速率减慢到$\lambda_j^{-1}N^{-\beta_j}g(n)$，当$g$被限制时，减慢到$\lambda_j^{-1}g(n)$。 这里，$N$是一个缩放参数，其中网格间距被视为$1/N$并且时间被加快了$N^2$。 从初始测量开始，相对于一个不变测度具有$O(N)$相对熵，我们证明了流体动力学极限，并表征了所有缺陷强度下的宏观缺陷点$x_j = \lim_{N\uparrow \infty} k_{j, N}/N$的边界行为。 对于速率$g(n)\sim n^\alpha$，在临界或超临界慢站点（$\beta_j=\alpha$或$\beta_j>\alpha$），由于与演变的原子质量或缺陷处的冷凝相互作用，产生相关的狄利克雷边界条件。 不同的是，当$g$有界时，在任何慢站点（$\lambda_j>1$），我们发现流体动力学密度必须被一个反映缺陷强度的阈值所限制。 此外，由于与存储在慢站点的原子质量的相互作用，相关的边界条件在周期性和狄利克雷之间来回变化。 显示英文摘要

摘要： We consider the space-time scaling limit of the particle mass in zero-range particle systems on a $1$D discrete torus $\mathbb{Z}/N\mathbb{Z}$ with a finite number of defects. We focus on two classes of increasing jump rates $g$, when $g(n)\sim n^\alpha$, for $0<\alpha\leq 1$, and when $g$ is a bounded function. In such a model, a particle at a regular site $k$ jumps equally likely to a neighbor with rate $g(n)$, depending only on the number of particles $n$ at $k$. At a defect site $k_{j,N}$, however, the jump rate is slowed down to $\lambda_j^{-1}N^{-\beta_j}g(n)$ when $g(n)\sim n^\alpha$, and to $\lambda_j^{-1}g(n)$ when $g$ is bounded. Here, $N$ is a scaling parameter where the grid spacing is seen as $1/N$ and time is speeded up by $N^2$. Starting from initial measures with $O(N)$ relative entropy with respect to an invariant measure, we show the hydrodynamic limit and characterize boundary behaviors at the macroscopic defect sites $x_j = \lim_{N\uparrow \infty} k_{j, N}/N$, for all defect strengths. For rates $g(n)\sim n^\alpha$, at critical or super-critical slow sites ($\beta_j=\alpha$ or $\beta_j>\alpha$), associated Dirichlet boundary conditions arise as a result of interactions with evolving atom masses or condensation at the defects. Differently, when $g$ is bounded, at any slow site ($\lambda_j>1$), we find the hydrodynamic density must be bounded above by a threshold value reflecting the strength of the defect. Moreover, due to interactions with masses of atoms stored at the slow sites, the associated boundary conditions bounce between being periodic and Dirichlet.