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[[File:Sandpile identity 300x205.png|upright=1.25|thumb|The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.]]
[[File:Sandpile identity 300x205.png|upright=1.25|thumb|The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.]]
[[文件：沙堆身份300x205.png |直立=1.25 |拇指|矩形网格沙堆群的标识元素。黄色像素对应三个粒子的顶点，淡紫色代表两个粒子，绿色表示一个，黑色表示零。]]
The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.
The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.
The Abelian sandpile model, also known as the Bak–Tang–Wiesenfeld model, was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper.
The Abelian sandpile model, also known as the Bak–Tang–Wiesenfeld model, was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper.
'''<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年的一篇论文中介绍的。
{{cite journal
{{cite journal
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.
这个模型是一个'''<font color="#ff8000"> 细胞自动机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桩的坡度相对应的关联值。这个斜坡以“沙粒”(或“碎片”)随机放置在桩上的方式逐渐形成，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，增加了它们的斜坡。贝克、唐和维森菲尔德考虑了在网格上连续随机放置沙粒的过程; 每次这样放置沙粒在特定地点可能没有效果，或者可能会引起级联反应，影响许多地点。
| year = 1987
| year = 1987
The sandpile model is a cellular automaton originally defined on a N\times M rectangular grid (checkerboard) \Gamma\subset\mathbb{Z}^2 of the standard square lattice \mathbb{Z}^2.
The sandpile model is a cellular automaton originally defined on a N\times M rectangular grid (checkerboard) \Gamma\subset\mathbb{Z}^2 of the standard square lattice \mathbb{Z}^2.
沙堆模型是一个最初定义在 n 乘 m 矩形网格(棋盘格) Gamma 子集 mathbb { z } ^ 2的标准正方形格子数学{ z } ^ 2上的细胞自动机模型。
沙堆模型是一个最初定义在 n 乘 m 矩形网格(棋盘格) Gamma 子集 mathbb { z } ^ 2的标准正方形格子数学{ z } ^ 2上的l细胞自动机模型。
| pages = 381–384
| pages = 381–384
Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set<br />z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 and<br />z_i(x,y)=z_{i-1}(x,y) for all (x,y)\neq(x_i,y_i).
Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set<br />z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 and<br />z_i(x,y)=z_{i-1}(x,y) for all (x,y)\neq(x_i,y_i).
向这个顶点添加一粒沙子，同时让其他顶点的粒子数保持不变，即。对所有(x,y)\neq(x_i,y_i)设置 <br />z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 和<br />z_i(x,y)=z_{i-1}(x,y)。
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.
模型是一个[[元胞自动机]]。在其原始公式中，有限网格上的每个场地都有一个与桩体坡度相对应的相关值。当“砂粒”（或“碎屑”）随机放置在桩上时，坡度逐渐增大，直到坡度超过特定的阈值，此时场地坍塌，将砂土转移到相邻场地，从而增加其坡度。Bak、Tang和Wiesenfeld考虑了在网格上连续随机放置沙粒的过程；在特定位置上的每一个这样的放置都可能没有效果，或者可能引起级联反应，从而影响到许多站点。
If all vertices are stable, i.e. z_i(x,y)<4 for all (x,y)\in\Gamma, also the configuration z_i is said to be stable. In this case, continue with the next iteration.
If all vertices are stable, i.e. z_i(x,y)<4 for all (x,y)\in\Gamma, also the configuration z_i is said to be stable. In this case, continue with the next iteration.
如果所有的顶点都是稳定的，即 z_i(x,y)<4对于 Gamma 中的所有(x,y)\in\Gamma，配置 z_i 也被认为是稳定的。在这种情况下，继续下一个迭代。
If at least one vertex is unstable, i.e. z_i(x_u,y_u)\geq 4 for some (x_u,y_u)\in\Gamma, the whole configuration z_i is said to be unstable. In this case, choose any unstable vertex (x_u,y_u)\in\Gamma at random. Topple this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set<br />z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,, and<br />z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1 if ( x_u \pm 1, y_u\pm 1)\in\Gamma.<br />If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
If at least one vertex is unstable, i.e. z_i(x_u,y_u)\geq 4 for some (x_u,y_u)\in\Gamma, the whole configuration z_i is said to be unstable. In this case, choose any unstable vertex (x_u,y_u)\in\Gamma at random. Topple this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set<br />z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,, and<br />z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1 if ( x_u \pm 1, y_u\pm 1)\in\Gamma.<br />If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
如果至少有一个顶点是不稳定的，即 z_i(x_u,y_u)\geq 4对于 γ 中的某些(x_u,y_u)\in\Gamma，整个构型 z_i 被认为是不稳定的。在这种情况下，随机选择 Gamma 中的任意不稳定顶点(x_u,y_u)\in\Gamma 。通过减少四颗粒数和增加一颗粒数(最多四颗)来推翻这个顶点。设置 <br />z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4，以及 <br />z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1 if ( x_u \pm 1, y_u\pm 1)\in\Gamma.<br /> 如果一个顶点在畴的边界倾斜，这将导致晶粒的净损失(两个晶粒在网格的角落，否则一个晶粒)。
The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs).<ref name=Hol2008>{{cite book
The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs).<ref name=Hol2008>{{cite book
该模型已经在无限格、其他（非正方形）格和任意图（包括有向多重图）上进行了研究。<ref name=Hol2008>{{cite book
Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of z_i eventually become stable and continue with the next iteration.
Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of z_i eventually become stable and continue with the next iteration.
To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph G=(V,E) without loops, a special vertex s\in V called the sink is specified that is not allowed to topple. A configuration (state) of the model is then a function z:V\setminus\{s\}\rightarrow\mathbb{N}_0 counting the non-negative number of grains on each non-sink vertex. A non-sink vertex v\in V\setminus\{s\} with
To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph G=(V,E) without loops, a special vertex s\in V called the sink is specified that is not allowed to topple. A configuration (state) of the model is then a function z:V\setminus\{s\}\rightarrow\mathbb{N}_0 counting the non-negative number of grains on each non-sink vertex. A non-sink vertex v\in V\setminus\{s\} with
为了将'''<font color="#ff8000"> 沙堆模型</font>'''从标准正方格的矩形网格推广到任意无向有限重图 g = (v，e) ，在 v 中指定了一个不允许倒塌的特殊顶点 s。模型的配置(状态)是一个函数 z: v set- { s }-right tarrow mathbb { n } _ 0，计算每个非汇顶点上的非负粒子数。V set- { s }中的非接收点 v
| pages = 331–364
| pages = 331–364
Z (u) to z (u) + 1 for all u sim v，u neq s.
Z (u) to z (u) + 1 for all u sim v，u neq s.
==Definition (rectangular grids)==
==Definition (rectangular grids)定义（矩形网格）==
The sandpile model is a [[cellular automaton]] originally defined on a <math>N\times M</math> rectangular grid (''checkerboard'') <math>\Gamma\subset\mathbb{Z}^2</math> of the [[Square lattice|standard square lattice]] <math>\mathbb{Z}^2</math>.
The sandpile model is a [[cellular automaton]] originally defined on a <math>N\times M</math> rectangular grid (''checkerboard'') <math>\Gamma\subset\mathbb{Z}^2</math> of the [[Square lattice|standard square lattice]] <math>\mathbb{Z}^2</math>.
沙堆模型是一个[[元胞自动机]]最初定义在一个<math>N\times M</math>矩形网格（“棋盘格”）<math>\Gamma\subset\mathbb{Z}^2</math>[[正方形格|标准正方形格]]<math>\mathbb{Z}^2</math>
The cellular automaton then progresses as before, i.e. by adding, in each iteration, one particle to a randomly chosen non-sink vertex and toppling until all vertices are stable.
The cellular automaton then progresses as before, i.e. by adding, in each iteration, one particle to a randomly chosen non-sink vertex and toppling until all vertices are stable.
To each vertex (''side'', ''field'') <math>(x,y)\in\Gamma</math> of the grid, we associate a value (''grains of sand'', ''slope'', ''particles'') <math>z_0(x,y)\in\{0,1,2,3\}</math>, with <math>z_0\in\{0,1,2,3\}^\Gamma</math> referred to as the (initial) configuration of the sandpile.
To each vertex (''side'', ''field'') <math>(x,y)\in\Gamma</math> of the grid, we associate a value (''grains of sand'', ''slope'', ''particles'') <math>z_0(x,y)\in\{0,1,2,3\}</math>, with <math>z_0\in\{0,1,2,3\}^\Gamma</math> referred to as the (initial) configuration of the sandpile.
对于网格的每个顶点（“边”、“场”）<math>（x，y），\Gamma</math>中，我们将0,1,2,3\}</math>中的值（“沙粒”、“坡度”、“粒子”）zu0（x，y）与称为沙堆（初始）配置的<math>z\u联系在一起。
The definition of the sandpile model given above for finite rectangular grids \Gamma\subset\mathbb{Z}^2 of the standard square lattice \mathbb{Z}^2 can then be seen as a special case of this definition: consider the graph G=(V,E) which is obtained from \Gamma by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of \Gamma such that the degree of every non-sink vertex of G 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 S of \mathbb{R}^2 with \mathbb{Z}^2. Contract every edge of \mathbb{Z}^2 whose two endpoints are not in S\cap\mathbb{Z}^2. The single remaining vertex outside of S\cap\mathbb{Z}^2 then constitutes the sink of the resulting sandpile graph.
The definition of the sandpile model given above for finite rectangular grids \Gamma\subset\mathbb{Z}^2 of the standard square lattice \mathbb{Z}^2 can then be seen as a special case of this definition: consider the graph G=(V,E) which is obtained from \Gamma by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of \Gamma such that the degree of every non-sink vertex of G 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 S of \mathbb{R}^2 with \mathbb{Z}^2. Contract every edge of \mathbb{Z}^2 whose two endpoints are not in S\cap\mathbb{Z}^2. The single remaining vertex outside of S\cap\mathbb{Z}^2 then constitutes the sink of the resulting sandpile graph.
The dynamics of the automaton at iteration <math>i\in\mathbb{N}</math> are then defined as follows:
The dynamics of the automaton at iteration <math>i\in\mathbb{N}</math> are then defined as follows:
然后，自动机在迭代<math>i\in\mathbb{N}</math>时的动力学定义如下：
# Choose a random vertex <math>(x_i,y_i)\in\Gamma</math> according to some probability distribution (usually uniform).
# Choose a random vertex <math>(x_i,y_i)\in\Gamma</math> according to some probability distribution (usually uniform).
#根据某种概率分布（通常是均匀的），选择一个随机顶点（x_i，y_i）\in\Gamma</math>。
In the dynamics of the sandpile automaton defined above, some stable configurations (0\leq z(v)<4 for all v\in G\setminus\{s\}) 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 z_m, where every vertex carries z_m(v)=deg(v)-1 grains of sand, is reachable from any other stable configuration (add deg(v)-z(v)-1\geq 0 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 (0\leq z(v)<4 for all v\in G\setminus\{s\}) 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 z_m, where every vertex carries z_m(v)=deg(v)-1 grains of sand, is reachable from any other stable configuration (add deg(v)-z(v)-1\geq 0 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中，一些稳定的组态（0\leq z（v）<4，对于所有v\In G\setminus\{s}）经常出现，而另一些则只能出现有限次（如果有的话）。前者被称为重复配置，而后者被称为瞬态配置。因此，周期性构形由所有稳定的非负构形组成，这些构形可以通过反复向顶点添加砂粒和倾倒而达到。很容易看出，最小稳定配置z峎m，其中每个顶点携带z峎m（v）=deg（v）-1颗粒，可从任何其他稳定配置（每个顶点添加deg（v）-z（v）-1\geq 0颗粒）。因此，等效地说，周期性构型正是通过添加沙粒和稳定化就可以从最小稳定构型得到的构型。
# Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set<br /><math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> and<br /><math>z_i(x,y)=z_{i-1}(x,y)</math> for all <math>(x,y)\neq(x_i,y_i)</math>.
# Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set<br /><math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> and<br /><math>z_i(x,y)=z_{i-1}(x,y)</math> for all <math>(x,y)\neq(x_i,y_i)</math>.
#向该顶点添加一粒沙子，同时让所有其他顶点的粒数保持不变，即为所有<math>（x，y，u i）=z{i-1}（x_i，y_i）+1</math>和<br/><math>z_i（x，y）=z{i-1}（x，y）</math>（x，y）</math>。
# If all vertices are ''stable'', i.e. <math>z_i(x,y)<4</math> for all <math>(x,y)\in\Gamma</math>, also the configuration <math>z_i</math> is said to be stable. In this case, continue with the next iteration.
# If all vertices are ''stable'', i.e. <math>z_i(x,y)<4</math> for all <math>(x,y)\in\Gamma</math>, also the configuration <math>z_i</math> is said to be stable. In this case, continue with the next iteration.
#如果所有顶点都是“稳定”的，即<math>z_i（x，y）<4</math>对于Gamma</math>中的所有<math>（x，y）<4</math>，那么配置<math>z酏i</math>也被称为稳定的。在这种情况下，继续下一个迭代。
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.
并非所有非负稳定构型都是循环的。例如，在一个至少由两个连通的非汇点组成的图上的每个沙堆模型中，每个稳定的结构，其中两个顶点携带零沙粒是非递归的。为了证明这一点，首先要注意的是沙粒的增加只能增加两个顶点共同承载的沙粒的总数。为了达到两个顶点都携带零粒子的配置，而实际情况并非如此，因此必然涉及到两个顶点中至少有一个被推翻的步骤。考虑这些步骤中的最后一个。在这个步骤中，两个顶点中的一个必须最后倒下。由于倾倒把一粒沙子转移到每个相邻的顶点，这意味着两个顶点共同承载的沙粒总数不能低于一粒，因而得证。
# If at least one vertex is ''unstable'', i.e. <math>z_i(x_u,y_u)\geq 4</math> for some <math>(x_u,y_u)\in\Gamma</math>, the whole configuration <math>z_i</math> is said to be unstable. In this case, choose any unstable vertex <math>(x_u,y_u)\in\Gamma</math> at random. ''Topple'' this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set<br /><math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>, and<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> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>.<br />If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
# If at least one vertex is ''unstable'', i.e. <math>z_i(x_u,y_u)\geq 4</math> for some <math>(x_u,y_u)\in\Gamma</math>, the whole configuration <math>z_i</math> is said to be unstable. In this case, choose any unstable vertex <math>(x_u,y_u)\in\Gamma</math> at random. ''Topple'' this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set<br /><math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>, and<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> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>.<br />If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
#如果至少有一个顶点是“不稳定”的，即<math>z_i（x_u，y u u）\geq 4</math>对于某些<math>（x\u u，y_u u）\n在\Gamma</math>中，整个配置<math>zu i</math>称为不稳定。在这种情况下，随机选择\Gamma</math>中的任何不稳定顶点<math>（x\u，y\u）通过将其晶粒度减少4个，并将其（最多4个）直接相邻的每个晶粒数增加1个，即设置<br /><math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>，和<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> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>.<br />如果域边界上的一个顶点倒下，这将导致晶粒净损失（两个晶粒位于网格的一角，否则一个晶粒）。
# Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of <math>z_i</math> eventually become stable and continue with the next iteration.
# Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of <math>z_i</math> eventually become stable and continue with the next iteration.