沙堆模型

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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.

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。

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.

阿贝尔沙堆模型Abelian sandpile model，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年的一篇论文中提出的。

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 细胞自动机模型Cellular automaton。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。

The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs). It is closely related to the dollar game, a variant of the chip-firing game introduced by Biggs. 该模型已经在无限栅格、其他(非方形)栅格和任意图(包括有向多重图)上进行了研究。它与美元游戏密切相关，美元游戏是Biggs引入的一种chip-firing游戏的变体。

Definition (rectangular grids)定义（矩形网格）

The sandpile model is a cellular automaton originally defined on a [math]\displaystyle{ N\times M }[/math] rectangular grid (checkerboard) [math]\displaystyle{ \Gamma\subset\mathbb{Z}^2 }[/math] of the standard square lattice [math]\displaystyle{ \mathbb{Z}^2 }[/math].

沙堆模型是一个元胞自动机cellular automaton ，它最初定义在[math]\displaystyle{ N\times M }[/math]矩形网格(棋盘格)上，其顶点在标准的正方形格子[math]\displaystyle{ \mathbb{Z}^2 }[/math]上[math]\displaystyle{ \Gamma\subset\mathbb{Z}^2 }[/math]。

To each vertex (side, field) [math]\displaystyle{ (x,y)\in\Gamma }[/math] of the grid, we associate a value (grains of sand, slope, particles) [math]\displaystyle{ z_0(x,y)\in\{0,1,2,3\} }[/math], with [math]\displaystyle{ z_0\in\{0,1,2,3\}^\Gamma }[/math] referred to as the (initial) configuration of the sandpile.

对于栅格中的每个顶点[math]\displaystyle{ (x,y)\in\Gamma }[/math]，我们关联一个值(沙粒、坡度、颗粒) [math]\displaystyle{ z_0(x,y)\in\{0,1,2,3\} }[/math]，这样所有顶点的初始状态[math]\displaystyle{ z_0\in\{0,1,2,3\}^\Gamma }[/math]被称为沙堆的(初始)构型。

The dynamics of the automaton at iteration [math]\displaystyle{ i\in\mathbb{N} }[/math] are then defined as follows: 自动机的动力学过程在第[math]\displaystyle{ i\in\mathbb{N} }[/math]次迭代时的定义如下:

1. Choose a random vertex [math]\displaystyle{ (x_i,y_i)\in\Gamma }[/math] according to some probability distribution (usually uniform).
2. Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set
   [math]\displaystyle{ z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 }[/math] and
   [math]\displaystyle{ z_i(x,y)=z_{i-1}(x,y) }[/math] for all [math]\displaystyle{ (x,y)\neq(x_i,y_i) }[/math].
3. If all vertices are stable, i.e. [math]\displaystyle{ z_i(x,y)\lt 4 }[/math] for all [math]\displaystyle{ (x,y)\in\Gamma }[/math], also the configuration [math]\displaystyle{ z_i }[/math] is said to be stable. In this case, continue with the next iteration.
4. If at least one vertex is unstable, i.e. [math]\displaystyle{ z_i(x_u,y_u)\geq 4 }[/math] for some [math]\displaystyle{ (x_u,y_u)\in\Gamma }[/math], the whole configuration [math]\displaystyle{ z_i }[/math] is said to be unstable. In this case, choose any unstable vertex [math]\displaystyle{ (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
   [math]\displaystyle{ z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4, }[/math], and
   [math]\displaystyle{ 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]\displaystyle{ ( x_u \pm 1, y_u\pm 1)\in\Gamma }[/math].
   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).
5. 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]\displaystyle{ z_i }[/math] eventually become stable and continue with the next iteration.

1. 根据某种概率分布(通常为均匀分布)随机选择一个顶点。根据一些概率分布(通常是均匀的)选择一个随机顶点[math]\displaystyle{ (x_i,y_i)\in\Gamma }[/math]。
2. 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，也就是设定[math]\displaystyle{ \lt br /\gt z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 }[/math]，对于所有的[math]\displaystyle{ (x,y)\neq(x_i,y_i) }[/math]，[math]\displaystyle{ \lt br /\gt z_i(x,y)=z_{i-1}(x,y) }[/math]。
3. 如果所有的顶点都是稳定的，即如果[math]\displaystyle{ (x,y)\in\Gamma }[/math]中[math]\displaystyle{ z_i(x,y)\lt 4 }[/math]，那么[math]\displaystyle{ z_i }[/math]被认为是稳定的。在这种情况下，继续下一轮迭代。
4. 如果至少有一个顶点是不稳定的，即对于一些[math]\displaystyle{ (x_u,y_u)\in\Gamma }[/math]，[math]\displaystyle{ z_i(x_u,y_u)\geq 4 }[/math]，[math]\displaystyle{ 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]\displaystyle{ (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：[math]\displaystyle{ \lt br /\gt z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) – 4 }[/math]，[math]\displaystyle{ \lt br /\gt 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.\lt br /\gt }[/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）。
5. 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]\displaystyle{ 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. 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。

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

To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph [math]\displaystyle{ G=(V,E) }[/math], a special vertex [math]\displaystyle{ s\in V }[/math] called the sink is specified that is not allowed to topple. A configuration (state) of the model is then a function [math]\displaystyle{ z:V\setminus\{s\}\rightarrow\mathbb{N}_0 }[/math] counting the non-negative number of grains on each non-sink vertex. A non-sink vertex [math]\displaystyle{ v\in V\setminus\{s\} }[/math] with

[math]\displaystyle{ z(v)\geq \deg(v) }[/math]

is unstable; it can be toppled, which sends one of its grains to each of its (non-sink) neighbors:

[math]\displaystyle{ z(v) \to z(v) - \deg(v) }[/math]

[math]\displaystyle{ z(u) \to z(u) + 1 }[/math] for all [math]\displaystyle{ u\sim v }[/math], [math]\displaystyle{ u\neq s }[/math].

为了将 沙堆模型从标准方格的矩形网格推广到任意无向有限多重图 [math]\displaystyle{ G=(V,E) }[/math] ，在 [math]\displaystyle{ V }[/math] 中指定了一个不允许崩塌的特殊沉没顶点[math]\displaystyle{ s }[/math]。模型的配置(状态)服从函数[math]\displaystyle{ 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]\displaystyle{ v\in V\setminus\{s\} }[/math]当满足[math]\displaystyle{ 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：

[math]\displaystyle{ z(v) \to z(v) - \deg(v) }[/math]

[math]\displaystyle{ z(u) \to z(u) + 1 }[/math]对于所有的[math]\displaystyle{ u\sim v }[/math], [math]\displaystyle{ u\neq s }[/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. 元胞自动机像之前一样进行，即在每次迭代中，向随机选择的非沉没顶点添加一个沙粒，不断进行崩塌过程，直到所有顶点都稳定。

The definition of the sandpile model given above for finite rectangular grids [math]\displaystyle{ \Gamma\subset\mathbb{Z}^2 }[/math] of the standard square lattice [math]\displaystyle{ \mathbb{Z}^2 }[/math] can then be seen as a special case of this definition: consider the graph [math]\displaystyle{ G=(V,E) }[/math] which is obtained from [math]\displaystyle{ \Gamma }[/math] by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of [math]\displaystyle{ \Gamma }[/math] such that the degree of every non-sink vertex of [math]\displaystyle{ 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]\displaystyle{ S }[/math] of [math]\displaystyle{ \mathbb{R}^2 }[/math] with [math]\displaystyle{ \mathbb{Z}^2 }[/math]. Contract every edge of [math]\displaystyle{ \mathbb{Z}^2 }[/math] whose two endpoints are not in [math]\displaystyle{ S\cap\mathbb{Z}^2 }[/math]. The single remaining vertex outside of [math]\displaystyle{ 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\mathbb{Z}^2上的有限矩形网格\Gamma\subset\mathbb{Z}^2，它可以看作是下面定义的一个特例：考虑图G=(V,E)，从\Gamma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G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\mathbb{R}^2的一些有界子集S与\mathbb{Z}^2相交。收缩\mathbb{Z}^2的每条边，其两个端点不在S\cap\mathbb{Z}^2中。S\cap\mathbb{Z}^2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

In the dynamics of the sandpile automaton defined above, some stable configurations ([math]\displaystyle{ 0\leq z(v)\lt 4 }[/math] for all [math]\displaystyle{ 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]\displaystyle{ z_m }[/math], where every vertex carries [math]\displaystyle{ z_m(v)=deg(v)-1 }[/math] grains of sand, is reachable from any other stable configuration (add [math]\displaystyle{ 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]\displaystyle{ v\In G\setminus\{s\} }[/math]，[math]\displaystyle{ 0\leq z（v）\lt 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]\displaystyle{ zum }[/math]，其中每个顶点放置[math]\displaystyle{ zum（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]\displaystyle{ 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. 并非所有非负稳定构型都是循环的。例如，在一个至少包含由两个连通的非沉没顶点的图结构的沙堆模型中，如果这两个顶点没有放置沙粒，那么这个稳定结构是非循环的。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放置的沙粒的总数。为了达到两个顶点都不放置沙粒的构型，从一个不是这种情况的构型出发，必然涉及到两个顶点中至少有一个崩塌的步骤。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崩塌。由于崩塌会将对每个相邻的顶点转移一颗沙粒，这意味着两个顶点共同放置的沙粒总数不能低于一颗，因而得证。

Sandpile group沙堆群

Given a configuration [math]\displaystyle{ z }[/math], [math]\displaystyle{ z(v)\in\mathbb{N}_0 }[/math] for all [math]\displaystyle{ v\in G\setminus\{s\} }[/math], toppling unstable non-sink vertices on a finite connected graph until no unstable non-sink vertex remains leads to a unique stable configuration [math]\displaystyle{ z^\circ }[/math], which is called the stabilization of [math]\displaystyle{ z }[/math]. Given two stable configurations [math]\displaystyle{ z }[/math] and [math]\displaystyle{ w }[/math], we can define the operation [math]\displaystyle{ z*w \to (z+w)^\circ }[/math], corresponding to the vertex-wise addition of grains followed by the stabilization of the resulting sandpile. 给定一个构型[math]\displaystyle{ z }[/math]，[math]\displaystyle{ z(v)\in\mathbb{N}_0 }[/math]对于所有[math]\displaystyle{ v\in G\setminus\{s\} }[/math]，在有限连通图上使不稳定的非沉没顶点崩塌，直到没有不稳定的非汇顶点保留，这将导致唯一的“稳定”配置math>z^\circ</math>，这就是[math]\displaystyle{ z }[/math]的“稳定化”。给定两个稳定构型[math]\displaystyle{ z }[/math]和[math]\displaystyle{ w }[/math]，我们可以定义运算[math]\displaystyle{ z*w \to (z+w)^\circ }[/math]，对应于沙粒的顶点方向相加，然后稳定得到的沙堆。

==Zcy（讨论）corresponding to the vertex-wise addition of grains翻译存疑

Given an arbitrary but fixed ordering of the non-sink vertices, multiple toppling operations, which can e.g. occur during the stabilization of an unstable configuration, can be efficiently encoded by using the graph Laplacian [math]\displaystyle{ \Delta=D-A }[/math], where [math]\displaystyle{ D }[/math] is the degree matrix and [math]\displaystyle{ A }[/math] is the adjacency matrix of the graph.

给定一个任意但固定的非沉没顶点的顺序，进行多个崩塌操作，可以通过使用图拉普拉斯[math]\displaystyle{ \Delta=D-A }[/math]高效地编码多个崩塌操作（例如，在不稳定构型的稳定过程中可能发生），其中[math]\displaystyle{ D }[/math]是图的度矩阵，[math]\displaystyle{ A }[/math]是图的邻接矩阵。

Deleting the row and column of [math]\displaystyle{ \Delta }[/math] corresponding with the sink yields the reduced graph Laplacian [math]\displaystyle{ \Delta' }[/math]. Then, when starting with a configuration [math]\displaystyle{ z }[/math] and toppling each vertex [math]\displaystyle{ v }[/math] a total of [math]\displaystyle{ \mathbf{x}(v)\in\mathbb{N}_0 }[/math] times yields the configuration [math]\displaystyle{ z-\Delta'\boldsymbol{\cdot}~\mathbf{x} }[/math], where [math]\displaystyle{ \boldsymbol{\cdot} }[/math] is the contraction product. Furthermore, if [math]\displaystyle{ \mathbf{x} }[/math] corresponds to the number of times each vertex is toppled during the stabilization of a given configuration [math]\displaystyle{ z }[/math], then

[math]\displaystyle{ z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x} }[/math]

删除与沉没顶点相对应的 [math]\displaystyle{ \Delta }[/math]的行和列，得到简化图拉普拉斯矩阵 [math]\displaystyle{ \Delta' }[/math]。然后，当以一个构型[math]\displaystyle{ z }[/math] 开始，并将每个顶点[math]\displaystyle{ v }[/math]进行总共[math]\displaystyle{ \mathbf{x}(v)\in\mathbb{N}_0 }[/math]次的崩塌操作时，产生[math]\displaystyle{ z-\Delta'\boldsymbol{\cdot}~\mathbf{x} }[/math]构型，其中[math]\displaystyle{ \boldsymbol{\cdot} }[/math]是收缩积。此外，如果 [math]\displaystyle{ \mathbf{x} }[/math] 对应于在给定构型[math]\displaystyle{ z }[/math]的稳定过程中每个顶点产生崩塌的次数，则

[math]\displaystyle{ z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x} }[/math]

In this case, [math]\displaystyle{ \mathbf{x} }[/math] is referred to as the toppling or odometer function (of the stabilization of [math]\displaystyle{ z }[/math]). 在这种情况下，[math]\displaystyle{ \mathbf{x} }[/math]被称为崩塌或[math]\displaystyle{ z }[/math]的稳定过程的里程计函数。 ==Zcy（讨论）odometer function翻译存疑

Under the operation [math]\displaystyle{ * }[/math], the set of recurrent configurations forms an abelian group isomorphic to the cokernel of the reduced graph Laplacian [math]\displaystyle{ \Delta' }[/math], i.e. to [math]\displaystyle{ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta' }[/math], whereby [math]\displaystyle{ n }[/math] denotes the number of vertices (including the sink). More generally, the set of stable configurations (transient and recurrent) forms a commutative monoid under the operation [math]\displaystyle{ * }[/math]. The minimal ideal of this monoid is then isomorphic to the group of recurrent configurations. 在[math]\displaystyle{ * }[/math]运算下，递归构形的集合构成一个与约化图拉普拉斯矩阵[math]\displaystyle{ \Delta' }[/math]的核同构的阿贝尔群。对于 [math]\displaystyle{ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta' }[/math]，其中[math]\displaystyle{ n }[/math] 表示顶点数（包括沉没顶点）。更一般地说，稳定构型集（瞬态和循环）在[math]\displaystyle{ * }[/math].运算下形成 交换幺半群Commutative monoid。这个幺半群的最小理想同构于循环构型群。 ==Zcy（讨论）ideal翻译存疑

The group formed by the recurrent configurations, as well as the group [math]\displaystyle{ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta' }[/math] to which the former is isomorphic, is most commonly referred to as the sandpile group. Other common names for the same group are critical group, Jacobian group or (less often) Picard group. Note, however, that some authors only denote the group formed by the recurrent configurations as the sandpile group, while reserving the name Jacobian group or critical group for the (isomorphic) group defined by [math]\displaystyle{ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta' }[/math] (or for related isomorphic definitions). Finally, some authors use the name Picard group to refer to the direct product of the sandpile group and [math]\displaystyle{ \mathbb{Z} }[/math], which naturally appears in a cellular automaton closely related to the sandpile model, referred to as the chip firing or dollar game.

由循环构形形成的群，以及与之同构的群[math]\displaystyle{ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta' }[/math]，通常称为 沙堆群Sandpile group。相同群的其它常用名称是“临界群”、“Jacobian群”或(不常见的)“Picard群”。然而，要注意的是，有些作者只把循环构型形成的组称为沙堆组，而为由[math]\displaystyle{ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta' }[/math]（或相关的同构定义）定义的同构群保留Jacobian群或临界群的称呼。最后，一些作者使用Picard群来指代沙堆群和[math]\displaystyle{ \mathbb{Z} }[/math]的直接产物，后者出现在与沙堆模型密切相关的元胞自动机中，被称为 碎片点火或美元游戏Chip firing or Dollar game。

Given the isomorphisms stated above, the order of the sandpile group is the determinant of [math]\displaystyle{ \Delta' }[/math], which by the matrix tree theorem is the number of spanning trees of the graph.

给定上述同构，沙堆群的顺序是[math]\displaystyle{ \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

Main article: Self-organized criticality

The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its critical state, at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the phase transitions between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is self-organized. 模型最初源于这样一个事实，即在晶格上的模拟中，它被吸引到了它的临界状态，此时系统的关联长度和关联时间趋于无穷大，不需要对系统参数进行任何微调。这与早期临界现象的例子形成了对比，例如固体和液体之间，或液体和气体之间的相变，其中临界点只能通过精确调节（例如，温度）来达到。因此，在沙堆模型中，我们可以说临界性是自组织的。

Once the sandpile model reaches its critical state there is no correlation between the system's response to a perturbation and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays 1/ƒ noise, a feature common to many complex systems in nature. 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/ƒ 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. 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。

For two dimensions, it has been hypothesized that the associated conformal field theory is consists of symplectic fermions with a central charge c = −2. 对于二维，相关共形场理论被认为是中心电荷为c = −2的symplectic fermion辛费米子组成的；

Properties属性

Least action principle最小作用原理

The stabilization of chip configurations obeys a form of least action principle: each vertex topples no more than necessary in the course of the stabilization.

芯片结构的稳定遵循一种“最小作用原理”的形式：每个顶点在稳定过程中不超过必要的崩塌量。

This can be formalized as follows. Call a sequence of topples legal if it only topples unstable vertices, and stabilizing if it results in a stable configuration. The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible. Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex [math]\displaystyle{ v }[/math], the number of times [math]\displaystyle{ v }[/math] topples is the same in all legal stabilizing sequences. According to the least action principle, a minimal stabilizing sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence. In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence. 这可以形式化如下。如果一个崩塌序列只推倒不稳定的顶点，则称其为“合法的”，使用一组合法的顶点序列，如果它的结果是一个稳定的构型，则称其为“稳定的”。稳定沙堆的标准方法是找到一个最大的合法崩塌序列，也就是说，让崩塌序列尽可能地长。这种序列具有明显的稳定性，沙堆的可交换性质是所有这些序列都等价于倾斜序列的置换，也就是说，对于任何顶点[math]\displaystyle{ v }[/math]，在所有合法的稳定序列中[math]\displaystyle{ v }[/math]的崩塌次数都是相同的。根据最小作用原理，最小稳定序列等价于合法的（且稳定的）崩塌序列的置换。特别地，由最小稳定序列产生的构型与由最大合法序列产生的构型是相同的。

More formally, if [math]\displaystyle{ \mathbf{u} }[/math] is a vector such that [math]\displaystyle{ \mathbf{u}(v) }[/math] is the number of times the vertex [math]\displaystyle{ v }[/math] topples during the stabilization (via the toppling of unstable vertices) of a chip configuration [math]\displaystyle{ z }[/math], and [math]\displaystyle{ \mathbf{n} }[/math] is an integral vector (not necessarily non-negative) such that [math]\displaystyle{ z-\mathbf{n}\Delta' }[/math] is stable, then [math]\displaystyle{ \mathbf{u}(v) \leq \mathbf{n}(v) }[/math] for all vertices [math]\displaystyle{ v }[/math]. 更正式地说，如果[math]\displaystyle{ \mathbf{u} }[/math]是一个向量， [math]\displaystyle{ \mathbf{u}（v） }[/math]是碎片构型[math]\displaystyle{ z }[/math]在稳定过程中（通过不稳定顶点的崩塌）顶点[math]\displaystyle{ v }[/math]崩塌的次数，并且[math]\displaystyle{ \mathbf{n} }[/math]是一个积分向量（不一定是非负的），使得[math]\displaystyle{ z-\mathbf{n}\Delta' }[/math]是稳定的，那么对于所有顶点[math]\displaystyle{ v }[/math]，[math]\displaystyle{ \mathbf{u}(v) \leq \mathbf{n}(v) }[/math]。 ==Zcy（讨论）[math]\displaystyle{ \mathbf{n} }[/math] is an integral vector (not necessarily non-negative)这一句话的翻译存疑。

Scaling limits缩放限制

Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.

沙堆标识在方形网格不断增加的动画。 黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。

The animation shows the recurrent configuration corresponding to the identity of the sandpile group on different [math]\displaystyle{ N\times N }[/math] square grids of increasing sizes [math]\displaystyle{ N\geq 1 }[/math], whereby the configurations are rescaled to always have the same physical dimension. Visually, the identities on larger grids seem to become more and more detailed and to "converge to a continuous image". Mathematically, this suggests the existence of scaling-limits of the sandpile identity on square grids based on the notion of weak-* convergence (or some other, generalized notion of convergence). Indeed, existence of scaling limits of recurrent sandpile configurations has been proved by Wesley Pegden and Charles Smart . In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids. 动画显示了对应网格尺寸[math]\displaystyle{ N\geq 1 }[/math]不断增大，不同大小的[math]\displaystyle{ 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。 . In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.

Generalizations and related models归纳与相关模型

Sandpile models on infinite grids有向图上的沙堆模型

文件:Sandpile on infinite grid, 3e7 grains.png

30 million grains dropped to a site of the infinite square grid, then toppled according to the rules of the sandpile model. White color denotes sites with 0 grains, green is for 1, purple is for 2, gold is for 3. The bounding box is 3967×3967.

文件:Sandpile on infinite grid, 3e7 grains.png

3000万粒沙粒落在无限方形网格的一个位置上，然后按照沙堆模型的规则产生崩塌。白色表示0颗沙粒位置，绿色表示1颗，紫色表示2颗，金色表示3颗。图中框的大小是3967×3967。

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.

沙堆模型可以推广到无限网格中。这种归纳法的一个挑战是，一般来说，不再保证每次雪崩最终都会停止。因此，一些一般化方法只考虑了构型的稳定性，因为这一点是能保证的。

A rather popular model on the (infinite) square lattice with sites [math]\displaystyle{ (x,y)\in\mathbb{Z}^2 }[/math] is defined as follows: 在（无限）方格上有一个相当流行的模型，其位置[math]\displaystyle{ (x,y)\in\mathbb{Z}^2 }[/math]定义如下：

Begin with some nonnegative configuration of values [math]\displaystyle{ z(x,y)\in \mathbf{Z} }[/math] which is finite, meaning 从\mathbf{z}</math>中有限值[math]\displaystyle{ z(x,y)\in \mathbf{Z} }[/math]的一些非负配置开始，这意味着

[math]\displaystyle{ \sum_{x,y}z(x,y)\lt \infty. }[/math]

Any site [math]\displaystyle{ (x,y) }[/math] with 任何位置[math]\displaystyle{ (x,y) }[/math]有

[math]\displaystyle{ z(x,y)\geq 4 }[/math]

is unstable and can topple (or fire), sending one of its chips to each of its four neighbors: 就是“不稳定的”，并且会产生崩塌，将它位置上的沙粒分发给它的四个邻居：

[math]\displaystyle{ z(x,y) \rightarrow z(x,y) - 4, }[/math]

[math]\displaystyle{ z( x \pm 1, y) \rightarrow z( x \pm 1, y) + 1, }[/math]

[math]\displaystyle{ z(x, y \pm 1) \rightarrow z( x, y \pm 1 ) + 1. }[/math]

Since the initial configuration is finite, the process is guaranteed to terminate, with the grains scattering outward. 由于初始构型是有限的，这一过程必然会随着沙粒向外散布终止。

A popular special case of this model is given when the initial configuration is zero for all vertices except the origin. If the origin carries a huge number of grains of sand, the configuration after relaxation forms fractal patterns (see figure). When letting the initial number of grains at the origin go to infinity, the rescaled stabilized configurations were shown to converge to a unique limit. 给出了该模型的一个常见特例，即除原点外的所有顶点的初始配置都为零。如果原点放置大量沙粒，松弛后的构型形成分形图案(见图)。当初始颗粒数趋于无穷时，重缩放的稳定构型收敛到唯一极限。

Sandpile models on directed graphs有向图上的沙堆模型

The sandpile model can be generalized to arbitrary directed multigraphs. The rules are that any vertex [math]\displaystyle{ v }[/math] with 沙堆模型可以推广到任意有向多重图。规则是任何顶点[math]\displaystyle{ v }[/math]有

[math]\displaystyle{ z(v)\geq \deg^{+}(v) }[/math]

is unstable; toppling again sends chips to each of its neighbors, one along each outgoing edge: 则是不稳定的；崩塌将它的碎片沿着边的输出方向，分发给它的邻居：

[math]\displaystyle{ z(v) \rightarrow z(v) - \deg^{+}(v) + \deg(v,v) }[/math]

and, for each [math]\displaystyle{ u\neq v }[/math]: 并且，对于每个[math]\displaystyle{ u\neq v }[/math]：

[math]\displaystyle{ z(u) \rightarrow z(u) + \deg(v,u) }[/math]

where [math]\displaystyle{ \deg(v,u) }[/math] is the number of edges from [math]\displaystyle{ v }[/math] to [math]\displaystyle{ u }[/math]. 其中，[math]\displaystyle{ \deg(v,u) }[/math]是从[math]\displaystyle{ v }[/math]到[math]\displaystyle{ u }[/math]的边数。

In this case the Laplacian matrix is not symmetric. If we specify a sink [math]\displaystyle{ s }[/math] such that there is a path from every other vertex to [math]\displaystyle{ s }[/math], then the stabilization operation on finite graphs is well-defined and the sandpile group can be written 在这种情况下，拉普拉斯矩阵是不对称的。如果我们指定一个沉没顶点[math]\displaystyle{ s }[/math]，使得每一个顶点都有一条到[math]\displaystyle{ s }[/math]的路径，那么有限图上的稳定操作是定义良好的，并且沙堆群可以像之前一样被写出来

[math]\displaystyle{ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta' }[/math]

as before.

The order of the sandpile group is again the determinant of [math]\displaystyle{ \Delta' }[/math], which by the general version of the matrix tree theorem is the number of oriented spanning trees rooted at the sink. 沙堆群的顺序又是[math]\displaystyle{ \Delta' }[/math]的行列式，根据矩阵树定理的一般版本，它是根在沉没顶点的有向spanning tree生成树的数目。 ==Zcy（讨论）The order of the sandpile group翻译存疑。

The extended sandpile model扩展沙堆模型

Sandpile dynamics induced by the harmonic function H=x*y on a 255x255 square grid.

由谐波函数H=x*y在255x255方格网格上引起的沙堆动力学。

To better understand the structure of the sandpile group for different finite convex grids [math]\displaystyle{ \Gamma\subset\mathbb{Z}^2 }[/math] of the standard square lattice [math]\displaystyle{ \mathbb{Z}^2 }[/math], Lang and Shkolnikov introduced the extended sandpile model in 2019. The extended sandpile model is defined nearly exactly the same as the usual sandpile model (i.e. the original Bak–Tang–Wiesenfeld model ), except that vertices at the boundary [math]\displaystyle{ \partial\Gamma }[/math] of the grid are now allowed to carry a non-negative real number of grains. In contrast, vertices in the interior of the grid are still only allowed to carry integer numbers of grains. The toppling rules remain unchanged, i.e. both interior and boundary vertices are assumed to become unstable and topple if the grain number reaches or exceeds four.

为了更好地理解不同有限凸网格[math]\displaystyle{ \Gamma\subset\mathbb{Z}^2 }[/math]的沙堆群的结构，Lang和Shkolnikov在2019年提出了“扩展沙堆模型”。扩展沙堆模型的定义与“通常的沙堆模型”几乎完全相同（即原始的Bak–Tang–Wiesenfeld模型），除了网格边界[math]\displaystyle{ \partial\Gamma }[/math]的顶点现在允许放置非负实数的沙粒。相比之下，网格内部的顶点仍然只允许放置整数个粒子。崩塌规则保持不变，即假设当沙粒数达到或超过4时，内部顶点和边界顶点都变得不稳定并发生崩塌。

Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the extended sandpile group, of which the usual sandpile group is a discrete subgroup. Different to the usual sandpile group, the extended sandpile group is however a continuous Lie group. Since it is generated by only adding grains of sand to the boundary [math]\displaystyle{ \partial\Gamma }[/math] of the grid, the extended sandpile group furthermore has the topology of a torus of dimension [math]\displaystyle{ |\partial\Gamma| }[/math] and a volume given by the order of the usual sandpile group. 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]\displaystyle{ \partial\Gamma }[/math]上形成的，扩展后的沙堆群还具有维度[math]\displaystyle{ |\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。 ==Zcy（讨论）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 geodesics of this torus passing through the identity. This question leads to the definition of the sandpile dynamics 特别感兴趣的问题是循环构型如何通过恒等式，沿着这个环面的连续测地线动态变化的问题。这个问题引出了沙堆动力学的定义 ==Zcy（讨论）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？

[math]\displaystyle{ D_H(t)=(I-t\Delta H)^\circ }[/math] （扩展沙堆模型）

respectively

[math]\displaystyle{ \tilde{D}_H(t)=(I+\lfloor-t\Delta H\rfloor)^\circ }[/math] （普通沙堆模型）

induced by the integer-valued harmonic function [math]\displaystyle{ H }[/math] at time [math]\displaystyle{ t\in\mathbb{R}\setminus\mathbb{Z} }[/math], with [math]\displaystyle{ I }[/math] the identity of the sandpile group and [math]\displaystyle{ \lfloor.\rfloor }[/math] the floor function. 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]\displaystyle{ 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. 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]\displaystyle{ t }[/math] in the sandpile dynamics induced by some harmonic function [math]\displaystyle{ 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]\displaystyle{ H }[/math] to the respective sub-grid. 由整值调和函数[math]\displaystyle{ H }[/math]在时间[math]\displaystyle{ t\in\mathbb{R}\setminus\mathbb{Z} }[/math]，沙堆群的同一性[math]\displaystyle{ I }[/math]和底函数[math]\displaystyle{ \lfloor.\rfloor }[/math]导出的。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]\displaystyle{ 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。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]\displaystyle{ t }[/math]时的构型，动力学由大型网格上的谐波函数[math]\displaystyle{ H }[/math]导出到相应的构型，这种构型在[math]\displaystyle{ H }[/math]限制到各自子网格的沙堆动力学中时同时出现。 ==Zcy（讨论）上面的长句翻译需要重新审校Informaly, this renormalization simply maps

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, in which, instead of a discrete number of particles in each site [math]\displaystyle{ x }[/math], there is a real number [math]\displaystyle{ 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]\displaystyle{ 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”。与每个位置[math]\displaystyle{ x }[/math]上的沙粒数量为离散数不同，有一个实数[math]\displaystyle{ 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]\displaystyle{ t }[/math]的时刻是满的，它在以后的所有时间都是满的。

Further reading

• Per Bak (1996). How Nature Works: The Science of Self-Organized Criticality. New York: Copernicus. ISBN 978-0-387-94791-4. 
• Per Bak; Chao Tang; Kurt Wiesenfeld (1987). "Self-organized criticality: an explanation of 1/ƒ noise". Physical Review Letters. 59 (4): 381–384. Bibcode:1987PhRvL..59..381B. doi:10.1103/PhysRevLett.59.381. PMID 10035754.
• Per Bak; Chao Tang; Kurt Wiesenfeld (1988). "Self-organized criticality". Physical Review A. 38 (1): 364–374. Bibcode:1988PhRvA..38..364B. doi:10.1103/PhysRevA.38.364. PMID 9900174.
• Cori, Robert; Rossin, Dominique; Salvy, Bruno (2002). "Polynomial ideals for sandpiles and their Gröbner bases" (PDF). Theor. Comput. Sci. 276 (1–2): 1–15. doi:10.1016/S0304-3975(00)00397-2. Zbl 1002.68105.
• Klivans, Caroline (2018). The Mathematics of Chip-Firing. CRC Press. 
• Amini, Omid; Baker, Matthew; Faber, Xander, eds. (2013). "Algebraic geometry of sandpiles". Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011. Contemporary Mathematics. 605. Providence, RI: American Mathematical Society. pp. 211–256. doi:10.1090/conm/605/12117. ISBN 978-1-4704-1021-6. Zbl 1281.14002. 

External links

• Garcia-Puente, Luis David. "Sandpiles" (YouTube video). YouTube. Brady Haran. Retrieved 15 January 2017.