The exclusion process, sometimes called Kawasaki dynamics or lattice gas model, describes a system of particles moving on a discrete square lattice with an interaction governed by the exclusion rule under which at most one particle can occupy each site. We mostly discuss the symmetric and reversible case. The weakly asymmetric case recently attracts attention related to KPZ equation; cf. Bertini and Giacomin (Commun Math Phys 183:571–607, 1995) for a simple exclusion case and Gonçalves and Jara (Arch Ration Mech Anal 212:597–644, 2014) for an exclusion process with speed change, see also Gonçalves et al. (Ann Probab 43:286–338, 2015), Gubinelli and Perkowski (J Am Math Soc 31:427–471, 2018). In Sect. 1, as a warm-up, we consider a simple exclusion process and discuss its hydrodynamic limit and the corresponding fluctuation limit in a proper space–time scaling. From this model, one can derive a linear heat equation and a stochastic partial differential equation (SPDE) in the limit, respectively. Section 2 is devoted to the entropy method originally invented by Guo et al. (Commun Math Phys 118:31–59, 1988). We consider the exclusion process with speed change, in which the jump rate of a particle depends on the configuration nearby the particle. This gives a non-trivial interaction among particles. We study only the case that the jump rate satisfies the so-called gradient condition. The hydrodynamic limit, which leads to a nonlinear diffusion equation, follows from the local ergodicity or the local equilibrium of the system, and this is shown by establishing one-block and two-block estimates. We also discuss the fluctuation limit which follows by showing the so-called Boltzmann–Gibbs principle. Section 3 explains the relative entropy method originally due to Yau (Lett Math Phys 22:63–80, 1991). This is a variant of GPV method and gives another proof for the hydrodynamic limit. The difference between these two methods is as follows. Let \(N^d\) be the volume of the domain on which the system is defined (typically, d-dimensional discrete box with side length N) and denote the (relative) entropy by H. Then, H relative to a global equilibrium behaves as \(H=O(N^d)\) (or entropy per volume is O(1)) as \(N\rightarrow \infty .\) GPV method rather relies on the fact that the entropy production I, which is the time derivative of H, behaves as \(O(N^{d-2})\) so that I per volume is o(1), and this characterizes the limit measures. On the other hand, Yau’s method shows \(H=o(N^d)\) for H relative to local equilibria so that the entropy per volume is o(1) and this proves the hydrodynamic limit. In Sect. 4, we consider Kawasaki dynamics perturbed by relatively large Glauber effect, which allows creation and annihilation of particles. This leads to the reaction–diffusion equation in the hydrodynamic limit. We discuss especially the equation with reaction term of bistable type and the problem related to the fast reaction limit or the sharp interface limit leading to the motion by mean curvature. We apply the estimate on the relative entropy due to Jara and Menezes (Non-equilibrium fluctuations of interacting particle systems, 2017; Symmetric exclusion as a random environment: invariance principle, 2018), which is actually obtained as a combination of GPV and Yau’s estimates. This makes possible to study the hydrodynamic limit for microscopic systems with another diverging factors apart from that caused by the space–time scaling.

中文翻译：

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排除过程的水动力极限

排除过程有时称为Kawasaki动力学或晶格气体模型，它描述了一个粒子在离散的方形晶格上移动的系统，其相互作用受排除规则控制，在该规则下，最多一个粒子可以占据每个位置。我们主要讨论对称和可逆的情况。最近，与KPZ方程有关的弱非对称情况引起了人们的关注。cf. Bertini和Giacomin（Commun Math Phys 183：571–607，1995）是一个简单的排除案例，Gonçalves和Jara（Arch Ration Mech Anal 212：597–644，2014）是一个具有速度变化的排除过程，另请参见Gonçalves等。 。（Ann Probab 43：286–338，2015），Gubinelli和Perkowski（J Am Math Soc 31：427–471，2018）。在宗派。1，作为热身，我们考虑一个简单的排除过程，并在适当的时空尺度上讨论其流体动力极限和相应的波动极限。从该模型中，可以分别得出极限处的线性热方程和随机偏微分方程（SPDE）。第2节专门介绍了Guo等人最初发明的熵方法。（Commun Math Phys 118：31–59，1988）。我们考虑具有速度变化的排除过程，其中粒子的跳跃率取决于粒子附近的配置。这在粒子之间产生了非平凡的相互作用。我们仅研究跳跃率满足所谓的梯度条件的情况。导致非线性扩散方程的流体动力极限来自系统的局部遍历性或局部平衡，通过建立一个单块和两个块的估算值可以看出这一点。我们还将通过显示所谓的玻耳兹曼–吉布斯原理来讨论波动极限。第3节解释了最初由Yau引起的相对熵方法（Lett Math Phys 22：63–80，1991）。这是GPV方法的一种变体，为流体动力极限提供了另一种证明。这两种方法之间的区别如下。让\（N ^ d \）是定义系统的域的体积（通常是边长为N的d维离散盒），并用H表示（相对）熵。然后，ħ相对于全球平衡表现为\（H = O（N ^ d）\） （或熵每体积是Ô（1））作为\（N \ RIGHTARROW \ infty。\） GPV方法，而依赖于H的时间导数的熵产生I表现为\（O（N ^ {d-2}）\）的事实，因此每个体积的I为o（1），这是极限措施的特征。另一方面，Yau方法显示H相对于局部平衡的\（H = o（N ^ d）\），因此单位体积的熵为o（1）证明了流体动力极限。在宗派。参照图4，我们认为川崎动力学受到较大的格劳伯效应的干扰，该效应允许粒子的产生和an灭。这导致流体扩散极限中的反应扩散方程。我们特别讨论了具有双稳态类型反应项的方程，以及与快速反应极限或尖锐的界面极限相关的问题，这些极限导致平均曲率运动。我们将估算值应用于因Jara和Menezes引起的相对熵（相互作用粒子系统的非平衡波动，2017年;作为随机环境的对称排除：不变性原理，2018年），它实际上是GPV和Yau估算值的组合而获得的。