更改

[[File:Sandpile identity 300x205.png|upright=1.25|thumb|沙堆在矩形网格上的标识。黄色像素对应三颗沙粒的顶点，淡紫色代表两颗沙粒，绿色表示一颗沙粒，黑色表示零颗沙粒。]]

[[File:Sandpile identity 300x205.png|upright=1.25|thumb|沙堆在矩形网格上的标识。黄色像素对应三颗沙粒的顶点，淡紫色代表两颗沙粒，绿色表示一颗沙粒，黑色表示零颗沙粒。]]

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />  

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />  

For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the

For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the

smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid.< Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" />

smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid.<font color="#ff8000"> Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" /></font>

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在标度限制。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复配置映射到子网格上的重复配置。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在标度限制。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复配置映射到子网格上的重复配置。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）上面的整个长句翻译需要重新审校Informaly, this renormalization simply maps

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）上面的整个长句翻译需要重新审校Informaly, this renormalization simply maps==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

=== 可分割的沙堆===

=== 可分割的沙堆===

Levine和Peres在2008年提出了一个与之密切相关的模型，即所谓的“可分割的沙堆模型”。<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref>与每个位置<math>x</math>上的沙粒数量为离散数不同，有一个实数<math>s(x)</math>代表位置的总质量。如果这个质量是负的，我们就可以把它理解为一个空洞。当一个位置上的质量大于1时，就会发生崩塌; 它将多余的部分均匀地分发给它的邻居，这就导致了如果一个位置在<math>t</math>的时刻质量是1，它在以后的所有时间质量都是1。

Levine和Peres在2008年提出了一个与之密切相关的模型，即所谓的“可分割的沙堆模型”。<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref>与每个位置<math>x</math>上的沙粒数量为离散数不同，有一个实数<math>s(x)</math>代表位置的总质量。如果这个质量是负的，我们就可以把它理解为一个空洞。当一个位置上的质量大于1时，就会发生崩塌; 它将多余的部分均匀地分发给它的邻居，这就导致了如果一个位置在<math>t</math>的时刻质量是1，它在以后的所有时间质量都是1。

== References引用 ==

== 参考文献 ==

== Further reading ==

== 进一步阅读 ==

* {{cite book

* {{cite book

       | author = Per Bak

       | author = Per Bak

| location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-1-4704-1021-6 | year=2013 | citeseerx=10.1.1.760.283 | s2cid=7851577 }}

| location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-1-4704-1021-6 | year=2013 | citeseerx=10.1.1.760.283 | s2cid=7851577 }}

==External links==

== 拓展链接 ==

*{{cite web|last1=Garcia-Puente|first1=Luis David|title=Sandpiles|url=https://www.youtube.com/watch?v=1MtEUErz7Gg|website=YouTube|publisher=[[Brady Haran]]|accessdate=15 January 2017|format=YouTube video}}

*{{cite web|last1=Garcia-Puente|first1=Luis David|title=Sandpiles|url=https://www.youtube.com/watch?v=1MtEUErz7Gg|website=YouTube|publisher=[[Brady Haran]]|accessdate=15 January 2017|format=YouTube video}}

[[Category:Nonlinear systems]]

[[Category:Nonlinear systems]]

[[Category:Cellular automaton rules]]

[[Category:Cellular automaton rules]]