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第1行： − [[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.]] − − 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.<ref name=Bak1987> − {{cite journal − | author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K. − | year = 1987 − | title = Self-organized criticality: an explanation of 1/''ƒ'' noise − | journal = [[Physical Review Letters]] − | volume = 59 − | issue = 4 − | pages = 381–384 − | doi = 10.1103/PhysRevLett.59.381 − | bibcode=1987PhRvL..59..381B − | pmid=10035754 − |author-link=Per Bak |author2-link=Chao Tang }}</ref> 第36行：
第21行： − + + − 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 − | author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B. − | year = 2008 − | title = Chip-Firing and Rotor-Routing on Directed Graphs − | journal = In and Out of Equilibrium 2 − | volume = 60 − | pages = 331–364 − | doi = 10.1007/978-3-7643-8786-0_17 − | bibcode=1987PhRvL..59..381B − |arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref> It is closely related to the [[Chip-firing game#Biggs's Variant|dollar game]], a variant of the [[chip-firing game]] introduced by Biggs.<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>
无编辑摘要
此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。此词条由Zcy初步审校。
此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。此词条由Zcy初步审校。
[[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。]]
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。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。
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。
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。<ref name=Hol2008>{{cite book
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。<ref name=Hol2008>{{cite book