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'''<font color="#ff8000"> 阿贝尔沙堆模型Abelian sandpile model</font>'''，也被称为 Bak-Tang-Wiesenfeld 模型，是第一个发现的动力系统展示自组织临界性的例子。它是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld 在1987年的一篇论文<ref name=Bak1987>

'''<font color="#ff8000"> 阿贝尔沙堆模型Abelian sandpile model</font>'''，也被称为 Bak-Tang-Wiesenfeld 模型，是第一个发现的动力系统展现自组织临界性的例子。它是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld 在1987年的一篇论文<ref name=Bak1987>

{{cite journal

{{cite journal

  | author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.

  | author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.

The model is a [[cellular automaton]].  In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile.  This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope.  Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.

The model is a [[cellular automaton]].  In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile.  This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope.  Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.

这个模型是一种'''<font color="#ff8000"> 细胞自动机模型Cellular automaton</font>'''。在最初的公式中，有限网格上的每个位置都有一个与沙堆的坡度相对应的关联值。当“沙粒”(或“碎片”)被随机放置在沙堆上时，放置位置的斜坡就会堆积起来，直到倾斜程度超过一个特定的阈值，这个位置倒塌，沙子会转移到邻近的位置，增加它们的斜坡。这个斜坡以“沙粒”(或“碎片”)随机放置的方式逐渐形成，直到斜坡超过一个特定的阈值，在这个阈值的时候，这个位置倒塌，将沙子转移到邻近的位置，增加它们的斜坡。Bak，Tang和 Wiesenfeld考虑了在网格上连续随机放置沙粒的过程; 每次这样在特定位置放置沙粒有可能没有产生影响，也有可能会引起级联反应，影响到周围的其他位置。

这个模型是一种'''<font color="#ff8000"> 细胞自动机模型Cellular automaton</font>'''。在最初的公式中，有限网格上的每个位置都有一个与沙堆的坡度相对应的关联值。当“沙粒”(或“碎片”)被随机放置在沙堆上时，放置位置的斜坡就会堆积起来，直到倾斜程度超过一个特定的阈值，这个位置倒塌，沙子会转移到邻近的位置，增加它们的斜坡。Bak，Tang和 Wiesenfeld考虑了在网格上连续随机放置沙粒的过程; 每次这样在特定位置放置沙粒有可能不会产生影响，也有可能会引起级联反应，影响到周围的其他位置。

#向这个顶点添加一粒沙子，同时让其他顶点的沙粒数保持不变，也就是对于所有的<math>(x,y)\neq(x_i,y_i)</math>，设定<br /><math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> 和<br /><math>z_i(x,y)=z_{i-1}(x,y)</math>。

#向这个顶点添加一粒沙子，同时让其他顶点的沙粒数保持不变，也就是对于所有的<math>(x,y)\neq(x_i,y_i)</math>，设定<br /><math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> 和<br /><math>z_i(x,y)=z_{i-1}(x,y)</math>。

#如果所有的顶点都是稳定的，即如果对于<math>(x,y)\in\Gamma</math>，<math>z_i(x,y)<4</math>，那么构型<math>z_i</math>被认为是稳定的。在这种情况下，继续下一轮迭代。

#如果所有的顶点都是稳定的，即如果对于<math>(x,y)\in\Gamma</math>，<math>z_i(x,y)<4</math>，那么构型<math>z_i</math>被认为是稳定的。在这种情况下，继续下一轮迭代。

#如果至少有一个顶点是不稳定的，即对于一些<math>(x_u,y_u)\in\Gamma</math>，<math>z_i(x_u,y_u)\geq 4</math>，<math>z_i</math>被认为是不稳定的。在这种情况下，随机选择任意不稳定顶点<math> (x_u,y_u)\in\Gamma</math>。将该顶点的沙粒数减少4个，清空这个顶点，并将其每个(最多4个)直接邻居的沙粒数增加1个。即：<br /><math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>，<br /><math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> 如果 <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>.<br />。如果一个在边界的顶点产生崩塌，这将导致沙粒的净损失（两粒在网格的角落，一粒在其他地方）。

#如果至少有一个顶点是不稳定的，即对于一些<math>(x_u,y_u)\in\Gamma</math>，<math>z_i(x_u,y_u)\geq 4</math>，<math>z_i</math>被认为是不稳定的。在这种情况下，随机选择任意不稳定顶点<math> (x_u,y_u)\in\Gamma</math>。将该顶点的沙粒数减少4个，清空这个顶点，并将其每个(最多4个)直接邻居的沙粒数增加1个。即：<br /><math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>， 如果 <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>.<br />，<br /><math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math>。如果一个在边界的顶点产生崩塌，这将导致沙粒的净损失（两粒在网格的角落，一粒在其他地方）。

#由于沙粒的重新分布，一个顶点的崩塌会使其他顶点不稳定。这样，重复崩塌的过程，直到<math>z_i</math>状态下的所有顶点最终稳定下来，继续下一轮迭代。

#由于沙粒的重新分布，一个顶点的崩塌会使其他顶点不稳定。这样，重复崩塌的过程，直到<math>z_i</math>状态下的所有顶点最终稳定下来，继续下一轮迭代。

 

The toppling of several vertices during one iteration is referred to as an ''avalanche''. Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final stable configuration does not depend on the chosen order; this is one sense in which the sandpile is [[Abelian group|''abelian'']]. Similarly, the number of times each vertex topples during each iteration is also independent of the choice of toppling order.

The toppling of several vertices during one iteration is referred to as an ''avalanche''. Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final stable configuration does not depend on the chosen order; this is one sense in which the sandpile is [[Abelian group|''abelian'']]. Similarly, the number of times each vertex topples during each iteration is also independent of the choice of toppling order.

在一次迭代中多个顶点的崩塌被称为雪崩。每一次雪崩最终都会停止，也就是说，经过有限数量的顶点崩塌，会达到某种稳定的配置，这样自动机就得到了很好的定义。此外，尽管顶点崩塌的顺序常常有许多可能的选择，但最终的稳定状态并不依赖于所选择的顺序; 这是沙堆模型具有的可交换性质。类似地，在每次迭代过程中，每个顶点的崩塌次数也与崩塌顺序的选择是无关。

在一次迭代中多个顶点的崩塌被称为雪崩。每一次雪崩最终都会停止，也就是说，经过有限数量的顶点崩塌，会达到某种稳定的配置，这样自动机就得到了很好的定义。此外，尽管顶点崩塌的顺序常常有许多可能的选择，但最终的稳定状态并不依赖于所选择的顺序; 这是沙堆模型具有的可交换性质。类似地，在每次迭代过程中，每个顶点的崩塌次数也与崩塌顺序的选择是无关的。

==Definition (undirected finite multigraphs)定义（无向有限多图）==

==Definition (undirected finite multigraphs)定义（无向有限多图）==

:<math>z(u) \to z(u) + 1</math> for all <math>u\sim v</math>, <math>u\neq s</math>.

:<math>z(u) \to z(u) + 1</math> for all <math>u\sim v</math>, <math>u\neq s</math>.

为了将'''<font color="#ff8000"> 沙堆模型</font>'''从标准方格的矩形网格推广到任意无向有限多重图 <math> G=(V,E)</math> ，在 <math> V</math> 中指定了一个不允许崩塌的特殊沉没顶点<math> s</math>。模型的配置(状态)服从函数<math> z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math>，计算每个非沉没顶点上的非负沙粒数。非沉没顶点<math> v\in V\setminus\{s\} </math>当满足<math> z(v)\geq \deg(v) </math>时是不稳定的，它会产生崩塌，将给它的每个(非沉没)邻居分发一颗沙粒：

为了将'''<font color="#ff8000"> 沙堆模型</font>'''从标准方格的矩形网格推广到任意无向有限多重图 <math> G=(V,E)</math> ，在 <math> V</math> 中指定了一个不允许崩塌的特殊沉没顶点<math> s</math>。模型的构型(状态)服从函数<math> z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math>，计算每个非沉没顶点上的非负沙粒数。非沉没顶点<math> v\in V\setminus\{s\} </math>当满足<math> z(v)\geq \deg(v) </math>时是不稳定的，它会产生崩塌，向给它的每个(非沉没)邻居分发一颗沙粒：

:<math>z(v) \to z(v) - \deg(v)</math>

:<math>z(v) \to z(v) - \deg(v)</math>

:<math>z(u) \to z(u) + 1</math>对于所有的<math>u\sim v</math>, <math>u\neq s</math>.

:<math>z(u) \to z(u) + 1</math>对于所有的<math>u\sim v</math>, <math>u\neq s</math>.

The definition of the sandpile model given above for finite rectangular grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math> can then be seen as a special case of this definition: consider the graph <math>G=(V,E)</math> which is obtained from <math>\Gamma</math> by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of <math>\Gamma</math> such that the [[Degree (graph theory)|degree]] of every non-sink vertex of <math>G</math> is four. In this manner, also sandpile models on non-rectangular grids of the standard square lattice (or of any other lattice) can be defined: Intersect some bounded subset <math>S</math> of <math>\mathbb{R}^2</math> with <math>\mathbb{Z}^2</math>. [[Edge contraction|Contract every edge]] of <math>\mathbb{Z}^2</math> whose two endpoints are not in <math>S\cap\mathbb{Z}^2</math>. The single remaining vertex outside of <math>S\cap\mathbb{Z}^2</math> then constitutes the sink of the resulting sandpile graph.

The definition of the sandpile model given above for finite rectangular grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math> can then be seen as a special case of this definition: consider the graph <math>G=(V,E)</math> which is obtained from <math>\Gamma</math> by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of <math>\Gamma</math> such that the [[Degree (graph theory)|degree]] of every non-sink vertex of <math>G</math> is four. In this manner, also sandpile models on non-rectangular grids of the standard square lattice (or of any other lattice) can be defined: Intersect some bounded subset <math>S</math> of <math>\mathbb{R}^2</math> with <math>\mathbb{Z}^2</math>. [[Edge contraction|Contract every edge]] of <math>\mathbb{Z}^2</math> whose two endpoints are not in <math>S\cap\mathbb{Z}^2</math>. The single remaining vertex outside of <math>S\cap\mathbb{Z}^2</math> then constitutes the sink of the resulting sandpile graph.

上面给出的沙堆模型的定义，是在标准正方形网格<math>\mathbb{Z}^2</math>上的有限矩形网格<math>\Gamma\subset\mathbb{Z}^2</math>，它可以看作是下面定义的一个特例：考虑图<math>G=(V,E)</math>，从<math>\Gamma</math>添加一个沉没顶点，并添加从沉没顶点到每个边界顶点的边，使得<math>G</math>的每个非沉没顶点的度数为4。以这种方式，也可以定义标准正方形网格(或任何其他类型网格)的非矩形格上的沙堆模型: 将<math>\mathbb{R}^2</math>的一些有界子集<math>S</math>与<math>\mathbb{R}^2</math>相交。收缩<math>\mathbb{Z}^2</math>的每条边，其两个端点不在<math>S\cap\mathbb{Z}^2</math>中。<math>S\cap\mathbb{Z}^2</math>之外的一个单独剩余顶点构成了最终沙堆图的沉没顶点。

上面给出的沙堆模型的定义，是在标准正方形网格<math>\mathbb{Z}^2</math>上的有限矩形网格<math>\Gamma\subset\mathbb{Z}^2</math>上，它可以看作是下面定义的一个特例：考虑图<math>G=(V,E)</math>，从<math>\Gamma</math>添加一个沉没顶点，并添加从沉没顶点到每个边界顶点的边，使得<math>G</math>的每个非沉没顶点的度数为4。以这种方式，也可以定义标准正方形网格(或任何其他类型网格)的非矩形格上的沙堆模型: 将<math>\mathbb{R}^2</math>的一些有界子集<math>S</math>与<math>\mathbb{R}^2</math>相交。收缩<math>\mathbb{Z}^2</math>的每条边，其两个端点不在<math>S\cap\mathbb{Z}^2</math>中。<math>S\cap\mathbb{Z}^2</math>之外的一个单独剩余顶点构成了最终沙堆图的沉没顶点。

==Transient and recurrent configurations瞬态和循环构型==

==Transient and recurrent configurations瞬态和循环构型==

In the dynamics of the sandpile automaton defined above, some stable configurations (<math>0\leq z(v)<4</math> for all <math>v\in G\setminus\{s\}</math>) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as ''recurrent configurations'', while the latter are referred to as ''transient configurations''. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the ''minimally stable configuration'' <math>z_m</math>, where every vertex carries <math>z_m(v)=deg(v)-1</math> grains of sand, is reachable from any other stable configuration (add <math>deg(v)-z(v)-1\geq 0</math> grains to every vertex). Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing.

In the dynamics of the sandpile automaton defined above, some stable configurations (<math>0\leq z(v)<4</math> for all <math>v\in G\setminus\{s\}</math>) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as ''recurrent configurations'', while the latter are referred to as ''transient configurations''. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the ''minimally stable configuration'' <math>z_m</math>, where every vertex carries <math>z_m(v)=deg(v)-1</math> grains of sand, is reachable from any other stable configuration (add <math>deg(v)-z(v)-1\geq 0</math> grains to every vertex). Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing.

在上面定义的沙堆自动机的动力学过程中，一些稳定状态的构型（对于所有<math>v\in G\setminus\{s\}</math>，<math>0\leq z(v)<4</math>）经常无限次出现，而另一些则只能出现有限次（如果真的发生的话）。前者被称为“循环构型”，而后者被称为“瞬态构型”。因此，周期性构形由所有稳定的非负构形组成，这些构形可以从任何其他稳定构形中，通过反复向顶点添加沙粒，产生崩塌而得到。很容易看出，“最小稳定配置”<math>zum</math>，其中每个顶点放置<math>z_m(v)=deg(v)-1</math>颗沙粒，可从任何其他稳定构型得到（通过向每个顶点添加<math>deg(v)-z(v)-1\geq 0</math>颗沙粒）。因此，也就是说，周期性构型可以从最小稳定构型开始，通过添加沙粒，再稳定化得到。

在上面定义的沙堆自动机的动力学过程中，一些稳定状态的构型（对于所有<math>v\in G\setminus\{s\}</math>，<math>0\leq z(v)<4</math>）经常无限次出现，而另一些则只能出现有限次（如果真的发生的话）。前者被称为“循环构型”，而后者被称为“瞬态构型”。因此，周期性构形由所有稳定的非负构形组成，这些构形可以从任何其他稳定构形中，通过反复向顶点添加沙粒，产生崩塌而得到。很容易看出，“最小稳定构型”<math>zum</math>，其中每个顶点放置<math>z_m(v)=deg(v)-1</math>颗沙粒，可从任何其他稳定构型得到（通过向每个顶点添加<math>deg(v)-z(v)-1\geq 0</math>颗沙粒）。因此，也就是说，周期性构型可以从最小稳定构型开始，通过添加沙粒，再稳定化得到。

Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together. To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.

Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together. To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.

在这种情况下，<math>\mathbf{x}</math>被称为崩塌或<math>z</math>的稳定过程的里程计函数。

在这种情况下，<math>\mathbf{x}</math>被称为崩塌或<math>z</math>的稳定过程的里程计函数。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）odometer function翻译存疑

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）odometer function翻译存疑

给定上述同构，沙堆群的顺序是<math>\Delta'</math>的行列式，根据矩阵树定理，它是图的生成树数目。

给定上述同构，沙堆群的顺序是<math>\Delta'</math>的行列式，根据矩阵树定理，它是图的生成树数目。

==Self-organized criticality自组织临界性==

==Self-organized criticality自组织临界性==

一旦沙堆模型达到其临界状态，系统对扰动的响应和扰动细节之间就没有关联。一般来说，这意味着再往沙堆形成的斜坡上撒一粒沙子可能不会导致任何事情发生，或者可能导致整个沙堆形成的斜坡在大规模滑坡中崩塌。该模型还显示了[[1/f noise|1/''&fnof;'' noise]]，这是自然界中许多复杂系统的共同特征。

一旦沙堆模型达到其临界状态，系统对扰动的响应和扰动细节之间就没有关联。一般来说，这意味着再往沙堆形成的斜坡上撒一粒沙子可能不会导致任何事情发生，或者可能导致整个沙堆形成的斜坡在大规模滑坡中崩塌。该模型还显示了[[1/f noise|1/''&fnof;'' noise]]，这是自然界中许多复杂系统的共同特征。

This model only displays critical behaviour in two or more dimensions.  The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.

This model only displays critical behaviour in two or more dimensions.  The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.

此模型仅在两个或多个维度中显示关键行为。沙堆模型可以用一维来表示; 然而，一维沙堆模型不是演化到临界状态，而是达到最小稳定状态，其中每个格点都趋向临界坡度。

此模型仅在两个或多个维度中显示临界现象。沙堆模型可以用一维来表示; 然而，一维沙堆模型不是演化到临界状态，而是达到最小稳定状态，其中每个格点都趋向临界坡度。

For two dimensions, it has been hypothesized that the associated [[conformal field theory]] is consists of [[symplectic fermion]]s with a [[central charge]] ''c''&nbsp;=&nbsp;−2.<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

For two dimensions, it has been hypothesized that the associated [[conformal field theory]] is consists of [[symplectic fermion]]s with a [[central charge]] ''c''&nbsp;=&nbsp;−2.<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref>  In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref>  In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复配置，从而重新缩放配置以始终具有相同的物理维度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017  | ref=Pegden2017|class=math.AP }}</ref>

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复构型，从而重新缩放构型，使其最终具有相同的物理尺度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017  | ref=Pegden2017|class=math.AP }}</ref>

.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref>  In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref>  In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

 

There exist several generalizations of the sandpile model to infinite grids. A challenge in such generalizations is that, in general, it is not guaranteed anymore that every avalanche will eventually stop. Several of the generalization thus only consider the stabilization of configurations for which this can be guaranteed.

There exist several generalizations of the sandpile model to infinite grids. A challenge in such generalizations is that, in general, it is not guaranteed anymore that every avalanche will eventually stop. Several of the generalization thus only consider the stabilization of configurations for which this can be guaranteed.

A rather popular model on the (infinite) square lattice with sites <math>(x,y)\in\mathbb{Z}^2</math> is defined as follows:

A rather popular model on the (infinite) square lattice with sites <math>(x,y)\in\mathbb{Z}^2</math> is defined as follows:

扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />

扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）a volume given by the order of the usual sandpile group.翻译存疑。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid翻译存疑。a volume given by the order of the usual sandpile group.翻译存疑。

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

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

=== The divisible sandpile 可分割的沙堆===

=== The divisible sandpile 可分割的沙堆===

A strongly related model is the so called '''divisible sandpile model''', introduced by Levine and Peres in 2008,<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> in which, instead of a discrete number of particles in each site <math>x</math>, there is a real number <math>s(x)</math> representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time <math>t</math>, it will be full for all later times.

A strongly related model is the so called '''divisible sandpile model''', introduced by Levine and Peres in 2008,<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> in which, instead of a discrete number of particles in each site <math>x</math>, there is a real number <math>s(x)</math> representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time <math>t</math>, it will be full for all later times.

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>的时刻是满的，它在以后的所有时间都是满的。

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。