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第2行： − In [[physics]], '''self-organized criticality''' ('''SOC''') is a property of [[dynamical system]]s that have a [[critical phenomena|critical point]] as an [[attractor]]. Their macroscopic behavior thus displays the spatial or temporal [[scale invariance|scale-invariance]] characteristic of the [[critical point (physics)|critical point]] of a [[phase transition]], but without the need to tune control parameters to a precise value, because the system, effectively, tunes itself as it evolves towards criticality. − In physics, self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to a precise value, because the system, effectively, tunes itself as it evolves towards criticality.+ − 在物理学中，'''<font color="#ff8000"> 自组织临界性Self-organized criticality (SOC)</font>'''是动力系统的一种特性，动力系统有一个临界点作为'''<font color="#ff8000"> 吸引子Attractor</font>'''。它们在相变临界点的宏观行为因此显示了空间或时间尺度不变特性，但不需要把控制参数调整到一个精确的值，因为系统有效地自我调整趋向于临界状态。 − The concept was put forward by Per Bak, Chao Tang and Kurt Wiesenfeld ("BTW") in a paper published in 1987 in Physical Review Letters, and is considered to be one of the mechanisms by which complexity arises in nature. Its concepts have been applied across fields as diverse as geophysics,physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology and others.+ + + + + + + − − The concept was put forward by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] ("BTW") in a paper<ref name=Bak1987> − {{cite journal − | author = [[Per Bak|Bak, P.]], [[Chao Tang|Tang, C.]] and [[Kurt Wiesenfeld|Wiesenfeld, K.]] − | year = 1987 − | title = Self-organized criticality: an explanation of 1/''f'' noise − | journal = [[Physical Review Letters]] − | volume = 59 − | issue = 4 − | pages = 381–384 − | doi = 10.1103/PhysRevLett.59.381 − | pmid = 10035754 − | bibcode=1987PhRvL..59..381B − }} − Papercore summary: [https://archive.is/20130704122906/http://papercore.org/Bak1987 http://papercore.org/Bak1987].</ref> − published in 1987 in ''[[Physical Review Letters]]'', and is considered to be one of the mechanisms by which [[complexity]]<ref name=Bak1995> − {{cite journal − | author = [[Per Bak|Bak, P.]], and [[Maya Paczuski|Paczuski, M.]] − | year = 1995 − | title = Complexity, contingency, and criticality − | journal =Proc Natl Acad Sci U S A − | volume = 92 − | pages = 6689–6696 − | pmid = 11607561 − | doi = 10.1073/pnas.92.15.6689 − | issue = 15 − | pmc = 41396 − |bibcode = 1995PNAS...92.6689B }}</ref> arises in nature. Its concepts have been applied across fields as diverse as [[geophysics]],<ref name=SmalleyTurcotteSolla85> − {{cite journal − |author1=Smalley, R. F., Jr. |author2=Turcotte, D. L. |author3=Solla, S. A. | year = 1985 − | title = A renormalization group approach to the stick-slip behavior of faults − | journal = Journal of Geophysical Research − | bibcode = 1985JGR....90.1894S − | doi = 10.1029/JB090iB02p01894 − | volume = 90 − | issue = B2 − | pages = 1894 − |url=https://semanticscholar.org/paper/6776d17957204c198e278bda98c935ab1cf8f22b }}</ref> [[physical cosmology]], [[evolutionary biology]] and [[ecology]], [[bio-inspired computing]] and [[optimization (mathematics)]], [[economics]], [[quantum gravity]], [[sociology]], [[solar physics]], [[plasma physics]], [[neurobiology]]<ref name=LinkenkaerHansen2001> − {{cite journal − |author1=K. Linkenkaer-Hansen |author2=V. V. Nikouline |author3=J. M. Palva |author4=R. J. Ilmoniemi. |name-list-style=amp | year = 2001 − | title = Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations − | journal = J. Neurosci. − | volume = 21 − | pages = 1370–1377 − | pmid = 11160408 − | issue = 4 − |doi=10.1523/JNEUROSCI.21-04-01370.2001 |pmc=6762238 }}</ref><ref name=Beggs2003> − {{cite journal − |author1=J. M. Beggs |author2=D. Plenz − |name-list-style=amp | year = 2006 − | title = Neuronal Avalanches in Neocortical Circuits − | journal = J. Neurosci. − | volume = 23 − |issue=35 − |pages=11167–77 − |doi=10.1523/JNEUROSCI.23-35-11167.2003 − |pmid=14657176 − |pmc=6741045 − }}</ref><ref name=Chialvo2004> − {{cite journal − | author =Chialvo, D. R. − | year = 2004 − | title = Critical brain networks − | journal = Physica A − | volume = 340 − | issue =4 − | pages = 756–765 − | doi = 10.1016/j.physa.2004.05.064 − |arxiv = cond-mat/0402538 |bibcode = 2004PhyA..340..756R | author-link = Dante R. Chialvo − }}</ref> and others. − − − 这个概念是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld (“ BTW”)在一篇名为 bak1987的《物理评论快报》上的论文中提出的，其被认为是复杂性在自然界出现的机制之一。它的概念已经被应用于各个领域，比如地球物理学，物理宇宙学，进化生物学和生态学，生物启发计算和优化(数学) ，经济学，量子引力，社会学，太阳物理学，等离子物理学，神经生物学等。'''<font color="#ff8000"> SOC</font>'''通常在多自由度、强非线性动力学的缓慢驱动非平衡系统中被观察到。自从 BTW 的原始论文以来，已经确定了许多单独的例子，但是到目前为止还没有一组已知的一般特征来保证一个系统将显示 '''<font color="#ff8000"> SOC</font>'''。 + − == Overview 概览==+ − Self-organized criticality is one of a number of important discoveries made in [[statistical physics]] and related fields over the latter half of the 20th century, discoveries which relate particularly to the study of [[complexity]] in nature. For example, the study of [[cellular automata]], from the early discoveries of [[Stanislaw Ulam]] and [[John von Neumann]] through to [[John Horton Conway|John Conway]]'s [[Conway's Game of Life|Game of Life]] and the extensive work of [[Stephen Wolfram]], made it clear that complexity could be generated as an [[emergence|emergent]] feature of extended systems with simple local interactions. Over a similar period of time, [[Benoît Mandelbrot]]'s large body of work on [[fractals]] showed that much complexity in nature could be described by certain ubiquitous mathematical laws, while the extensive study of [[phase transition]]s carried out in the 1960s and 1970s showed how [[scale invariance|scale invariant]] phenomena such as [[fractals]] and [[power law]]s emerged at the [[critical point (physics)|critical point]] between phases. − '''<font color="#ff8000"> 自组织临界性Self-organized criticality（SOC)</font>'''是20世纪下半叶统计物理学及相关领域的众多重要发现之一，这些发现尤其与研究自然界的复杂性有关。例如，元胞自动机的研究---- 从 Stanislaw Ulam 和约翰·冯·诺伊曼的早期发现到 John Conway 的生命游戏和 Stephen Wolfram 的大量工作---- 清楚地表明，复杂性可以作为具有简单局部相互作用的扩展系统的一个涌现特征而产生。在相似的时间段内，beno t Mandelbrot 关于分形的大量工作表明，自然界的许多复杂性可以用某些无处不在的数学定律来描述，而在20世纪60年代和70年代对相变的广泛研究表明，诸如分形和幂律等尺度不变现象是如何出现不同相的临界点上的。+ − The term ''self-organized criticality'' was firstly introduced by [[Per Bak|Bak]], [[Chao Tang|Tang]] and [[Kurt Wiesenfeld|Wiesenfeld]]'s 1987 paper, which clearly linked together those factors: a simple [[cellular automaton]] was shown to produce several characteristic features observed in natural complexity ([[fractal]] geometry, [[pink noise|pink (1/f) noise]] and [[power law]]s) in a way that could be linked to [[critical point (physics)|critical-point phenomena]]. Crucially, however, the paper emphasized that the complexity observed emerged in a robust manner that did not depend on finely tuned details of the system: variable parameters in the model could be changed widely without affecting the emergence of critical behavior: hence, ''[[self-organized]]'' criticality. Thus, the key result of BTW's paper was its discovery of a mechanism by which the emergence of complexity from simple local interactions could be ''spontaneous''—and therefore plausible as a source of natural complexity—rather than something that was only possible in artificial situations in which control parameters are tuned to precise critical values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, producing some of the most cited papers in the scientific literature. − The term self-organized criticality was firstly introduced by Bak, Tang and Wiesenfeld's 1987 paper, which clearly linked together those factors: a simple cellular automaton was shown to produce several characteristic features observed in natural complexity (fractal geometry, pink (1/f) noise and power laws) in a way that could be linked to critical-point phenomena. Crucially, however, the paper emphasized that the complexity observed emerged in a robust manner that did not depend on finely tuned details of the system: variable parameters in the model could be changed widely without affecting the emergence of critical behavior: hence, self-organized criticality. Thus, the key result of BTW's paper was its discovery of a mechanism by which the emergence of complexity from simple local interactions could be spontaneous—and therefore plausible as a source of natural complexity—rather than something that was only possible in artificial situations in which control parameters are tuned to precise critical values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, producing some of the most cited papers in the scientific literature.+ − '''<font color="#ff8000"> 自组织临界性Self-organized criticality（SOC)</font>'''这个术语最早由 Bak，Tang 和 Wiesenfeld 在1987年的论文中提出，这篇论文将这些因素清楚地联系在一起: 一个简单的细胞自动机被证明可以产生在自然复杂性中观察到的几个特征(分形几何、粉红噪声和幂律) ，这种方式可以与临界点现象联系起来。然而，关键的是，这篇论文强调，观察到的复杂性是以一种强有力的方式出现的，并不依赖于系统精细调整的细节: 模型中的可变参数可以被广泛改变，而不会影响临界行为的涌现: 因此，具有自组织临界性。因此，BTW 论文的关键结果是发现了一种机制，通过这种机制，从简单的局部相互作用中产生的复杂性可能是自发的---- 因此是合理的自然复杂性的来源---- 而不是只有在控制参数调整到精确的临界值的人工情况下才可能出现的东西。这项研究的发表引起了理论家和实验家的极大兴趣，产生了一些在科学文献中被引用最多的论文。 + + + + − Due to BTW's metaphorical visualization of their model as a "[[Bak–Tang–Wiesenfeld sandpile|sandpile]]" on which new sand grains were being slowly sprinkled to cause "avalanches", much of the initial experimental work tended to focus on examining real avalanches in [[granular matter]], the most famous and extensive such study probably being the Oslo ricepile experiment{{Citation needed|date=March 2018}}. Other experiments include those carried out on magnetic-domain patterns, the [[Barkhausen effect]] and vortices in [[superconductors]].+ + + − 由于 BTW 将他们的模型比喻为一个“沙堆” ，在沙堆上缓慢地喷洒新的沙粒以引起“雪崩” ，所以最初的实验工作主要集中在研究颗粒物质中的真实雪崩，其中最著名和最广泛的研究可能是奥斯陆地震实验。其他实验还包括在磁畴图案、超导体中的巴克豪森效应和涡旋上进行的实验。 − Early theoretical work included the development of a variety of alternative SOC-generating dynamics distinct from the BTW model, attempts to prove model properties analytically (including calculating the critical exponents), and examination of the conditions necessary for SOC to emerge. One of the important issues for the latter investigation was whether conservation of energy was required in the local dynamical exchanges of models: the answer in general is no, but with (minor) reservations, as some exchange dynamics (such as those of BTW) do require local conservation at least on average. In the long term, key theoretical issues yet to be resolved include the calculation of the possible universality classes of SOC behavior and the question of whether it is possible to derive a general rule for determining if an arbitrary algorithm displays SOC.+ − + − Early theoretical work included the development of a variety of alternative SOC-generating dynamics distinct from the BTW model, attempts to prove model properties analytically (including calculating the [[critical exponent]]s<ref name=Tang1988a> − {{cite journal − | author = [[Chao Tang|Tang, C.]] and [[Per Bak|Bak, P.]] − | year = 1988 − | title = Critical exponents and scaling relations for self-organized critical phenomena − | journal = [[Physical Review Letters]] − | volume = 60 − | issue = 23 − | pages = 2347–2350 − | doi = 10.1103/PhysRevLett.60.2347 − | bibcode= 1988PhRvL..60.2347T − | pmid=10038328 − }} − </ref><ref name=Tang1988b> − {{cite journal − | author = [[Chao Tang|Tang, C.]] and [[Per Bak|Bak, P.]] − | year = 1988 − | title = Mean field theory of self-organized critical phenomena − | journal = [[Journal of Statistical Physics]] − | volume = 51 − | issue = 5–6 − | pages = 797–802 − | doi = 10.1007/BF01014884 − | bibcode= 1988JSP....51..797T − | url = https://zenodo.org/record/1232502 − | type = Submitted manuscript − }} − </ref>), and examination of the conditions necessary for SOC to emerge. One of the important issues for the latter investigation was whether [[conservation of energy]] was required in the local dynamical exchanges of models: the answer in general is no, but with (minor) reservations, as some exchange dynamics (such as those of BTW) do require local conservation at least on average. In the long term, key theoretical issues yet to be resolved include the calculation of the possible [[universality class]]es of SOC behavior and the question of whether it is possible to derive a general rule for determining if an arbitrary [[algorithm]] displays SOC. − − 早期的理论工作包括开发各种不同于 BTW 模型的 soc 生成动力学，试图解析证明模型的性质(包括计算临界指数，参见 tang1988a) ，以及检验发生 '''<font color="#ff8000"> SOC</font>'''的必要条件。后一项研究的一个重要问题是，在局部动态交换模型时是否需要能量守恒: 一般的答案是否定的，但有一些保留意见，因为一些交换动力学(如 BTW 的动态)确实需要局部至少平均的能量守恒。从长远来看，有待解决的关键理论问题包括 '''<font color="#ff8000"> SOC</font>''' 行为可能的普适性类的计算，以及是否有可能推导出一个一般规则来确定一个随机算法是否显示 '''<font color="#ff8000"> SOC。</font>''' − − Alongside these largely lab-based approaches, many other investigations have centered around large-scale natural or social systems that are known (or suspected) to display [[scale invariance|scale-invariant]] behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including: [[earthquakes]] (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the [[Gutenberg–Richter law]] describing the statistical distribution of earthquake size, and the [[Aftershock|Omori law]] describing the frequency of aftershocks<ref name=TurcotteSmalleySolla85> − {{cite journal − |author1=Turcotte, D. L. |author2=Smalley, R. F., Jr. |author3=Solla, S. A. | year = 1985 − | title = Collapse of loaded fractal trees − | journal = Nature − | doi= 10.1038/313671a0 − | volume = 313 − | issue = 6004 − | pages = 671–672|bibcode = 1985Natur.313..671T − }}</ref><ref name=SmalleyTurcotteSolla85 />); [[solar flares]]; fluctuations in economic systems such as [[financial markets]] (references to SOC are common in [[econophysics]]); [[landscape formation]]; [[forest fires]]; [[landslides]]; [[epidemics]]; neuronal avalanches in the cortex;<ref name="Beggs2003" /><ref name=Poil2012> − {{cite journal − | pmid = 22815496 − |date=Jul 2012 − |author1=Poil, SS |author2=Hardstone, R |author3=Mansvelder, HD |author4=Linkenkaer-Hansen, K | title = Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks − | volume = 32 − | issue = 29 − | pages = 9817–23 − | doi = 10.1523/JNEUROSCI.5990-11.2012 − | journal = Journal of Neuroscience − | pmc=3553543 − }}</ref> 1/f noise in the amplitude of electrophysiological signals;<ref name=LinkenkaerHansen2001 /> and [[biological evolution]] (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "[[punctuated equilibrium|punctuated equilibria]]" put forward by [[Niles Eldredge]] and [[Stephen Jay Gould]]). These "applied" investigations of SOC have included both modelling (either developing new models or adapting existing ones to the specifics of a given natural system) and extensive data analysis to determine the existence and/or characteristics of natural scaling laws. − − 除了这些大部分基于实验室的方法，许多其他的研究都集中在大规模的自然或社会系统上，这些系统已经知道(或怀疑)表现出尺度不变的行为。虽然这些方法并不总是受到研究对象专家的欢迎(至少最初是这样) ，但 '''<font color="#ff8000"> SOC</font>''' 已经成为解释一些自然现象的强有力的候选者，包括: 地震(早在 '''<font color="#ff8000"> SOC</font>''' 被发现之前，地震就被认为是尺度不变行为的来源，例如描述地震大小统计分布的古腾堡-里克特定律，以及描述余震频率的描述余震的 Omori 定律，命名为 turcottesmalleysolla85太阳耀斑; 经济系统的波动，比如金融市场(经济物理学中经常提到 SOC) ; 景观形成; 森林火灾; 滑坡; 流行病; 大脑皮层的神经雪崩;电生理信号振幅的1 / f 噪声，以及生物进化(其中 SOC 已被调用，例如，作为背后的动力机制的理论“间断平衡”由 Niles Eldredge 和史蒂芬·古尔德提出)。对SOC的这些”应用”研究既包括建模(开发新模型或使现有模型适应特定自然系统的具体情况) ，也包括广泛的数据分析，以确定是否存在和 / 或具有自然幂率的特点。 − − − − In addition, SOC has been applied to computational algorithms. Recently, it has been found that the avalanches from an SOC process, like the BTW model, make effective patterns in a random search for optimal solutions on graphs.<ref name=Hoffmann2018> − − − − − − − − − − − − − − An example of such an optimization problem is [[graph coloring]]. The SOC process apparently helps the optimization from getting stuck in a [[local optimum]] without the use of any [[Simulated annealing|annealing]] scheme, as suggested by previous work on [[extremal optimization]]. − − 此外，SOC 已经应用于计算算法。最近，人们发现来自 SOC 过程的雪崩，如 BTW 模型，在图的最优解的随机搜索中形成有效的模式。 − 第415行：
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无编辑摘要
在物理学中，'''自组织临界性 Self-organized criticality (SOC)</font>'''是动力系统的一种特性，动力系统有一个临界点作为吸引子 Attractor。它们在相变临界点的宏观行为因此显示了空间或时间尺度不变特性，但不需要把控制参数调整到一个精确的值，因为系统有效地自我调整趋向于临界状态。
这个概念是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld (简称“BTW”)在1987年的《物理评论快报 Physical Review Letters》上的论文中提出的，<ref name=Bak1987>{{cite journal | author =Per Bak, Chao Tang and Kurt Wiesenfeld | year = 1987 | title = Self-organized criticality: an explanation of 1/''f'' noise | journal = Physical Review Letters | volume = 59 | issue = 4 | pages = 381–384 | doi = 10.1103/PhysRevLett.59.381| pmid = 10035754 | bibcode=1987PhRvL..59..381B}}
Papercore summary: [https://archive.is/20130704122906/http://papercore.org/Bak1987 http://papercore.org/Bak1987].</ref> 其被认为是复杂性在自然界出现的机制之一。<ref name=Bak1995>
{{cite journal | author =Per Bak, and Maya Paczuski | year = 1995 | title = Complexity, contingency, and criticality | journal =Proc Natl Acad Sci U S A | volume = 92 | pages = 6689–6696 | pmid = 11607561 | doi = 10.1073/pnas.92.15.6689 | issue = 15 | pmc = 41396|bibcode = 1995PNAS...92.6689B }}</ref>它的概念已经被应用于各个领域，比如地球物理学，<ref name=SmalleyTurcotteSolla85>{{cite journal|author1=Smalley, R. F., Jr. |author2=Turcotte, D. L. |author3=Solla, S. A. | year = 1985| title = A renormalization group approach to the stick-slip behavior of faults
| journal = Journal of Geophysical Research| bibcode = 1985JGR....90.1894S| doi = 10.1029/JB090iB02p01894| volume = 90 | issue = B2| pages = 1894|url=https://semanticscholar.org/paper/6776d17957204c198e278bda98c935ab1cf8f22b }}</ref> 物理宇宙学，进化生物学和生态学，生物启发计算和优化(数学) ，经济学，量子引力，社会学，太阳物理学，等离子物理学，神经生物学等。<ref name=LinkenkaerHansen2001>
{{cite journal |author1=K. Linkenkaer-Hansen |author2=V. V. Nikouline |author3=J. M. Palva |author4=R. J. Ilmoniemi. |name-list-style=amp | year = 2001 | title = Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations | journal = J. Neurosci. | volume = 21 | pages = 1370–1377 | pmid = 11160408 | issue = 4|doi=10.1523/JNEUROSCI.21-04-01370.2001 |pmc=6762238 }}</ref><ref name=Beggs2003>{{cite journal |author1=J. M. Beggs |author2=D. Plenz |name-list-style=amp | year = 2006 | title = Neuronal Avalanches in Neocortical Circuits | journal = J. Neurosci. | volume = 23|issue=35 |pages=11167–77 |doi=10.1523/JNEUROSCI.23-35-11167.2003 |pmid=14657176 |pmc=6741045
}}</ref><ref name=Chialvo2004>{{cite journal | author =Chialvo, D. R. | year = 2004 | title = Critical brain networks | journal = Physica A | volume = 340 | issue =4 | pages = 756–765 | doi = 10.1016/j.physa.2004.05.064
|arxiv = cond-mat/0402538 |bibcode = 2004PhyA..340..756R | author-link = Dante R. Chialvo }}</ref>'''SOC'''通常在多自由度、强非线性动力学的缓慢驱动非平衡系统中被观察到。自从 BTW 的原始论文以来，已经确定了许多单独的例子，但是到目前为止还没有一组已知的一般特征来保证一个系统将显示 '''<font color="#ff8000"> SOC</font>'''。
== 概览==
'''自组织临界性'''是20世纪下半叶统计物理学及相关领域的众多重要发现之一，这些发现尤其与研究自然界的复杂性有关。例如，[[元胞自动机]]的研究---- 从 Stanislaw Ulam 和[[约翰·冯·诺伊曼]]的早期发现到 [[John Conway]] 的[[生命游戏]]和 [[Stephen Wolfram]] 的大量工作---- 清楚地表明，复杂性可以作为具有简单局部相互作用的扩展系统的一个[[涌现]]特征而产生。在相似的时间段内，[[Mandelbrot]]关于[[分形]]的大量工作表明，自然界的许多复杂性可以用某些无处不在的数学定律来描述，而在20世纪60年代和70年代对相变的广泛研究表明，诸如分形和[[幂律]]等尺度不变现象是如何出现不同相的[[临界点]]上的。
自组织临界性最早由 Bak，Tang 和 Wiesenfeld 在1987年的论文中提出，这篇论文将这些因素清楚地联系在一起: 一个简单的细胞自动机被证明可以产生在自然复杂性中观察到的几个特征（分形几何、粉红噪声和幂律） ，这种方式可以与临界点现象联系起来。然而，关键的是，这篇论文强调，观察到的复杂性是以一种强有力的方式出现的，并不依赖于系统精细调整的细节：模型中的可变参数可以被广泛改变，而不会影响临界行为的涌现：因此，具有自组织临界性。因此，BTW 论文的关键结果是发现了一种机制，通过这种机制，从简单的局部相互作用中产生的复杂性可能是自发的---- 因此是合理的自然复杂性的来源---- 而不是只有在控制参数调整到精确的临界值的人工情况下才可能出现的东西。这项研究的发表引起了理论家和实验家的极大兴趣，产生了一些在科学文献中被引用最多的论文。
由于 BTW 将他们的模型比喻为一个“沙堆” ，在沙堆上缓慢地喷洒新的沙粒以引起“雪崩” ，所以最初的实验工作主要集中在研究颗粒物质中的真实雪崩，其中最著名和最广泛的研究可能是奥斯陆地震实验。其他实验还包括在磁畴图案、超导体中的巴克豪森效应 Barkhausen effect和涡旋 vortices上进行的实验。
早期的理论工作包括开发各种不同于 BTW 模型的 soc 生成动力学，试图解析证明模型的性质(包括计算临界指数<ref name=Tang1988a>{{cite journal | author = Chao Tang and Per Bak | year = 1988 | title = Critical exponents and scaling relations for self-organized critical phenomena | journal = Physical Review Letters | volume = 60 | issue = 23
| pages = 2347–2350 | doi = 10.1103/PhysRevLett.60.2347| bibcode= 1988PhRvL..60.2347T | pmid=10038328}}
</ref><ref name=Tang1988b>{{cite journal | author = Chao Tang and Per Bak | year = 1988 | title = Mean field theory of self-organized critical phenomena | journal = Journal of Statistical Physics | volume = 51 | issue = 5–6 | pages = 797–802 | doi = 10.1007/BF01014884| bibcode= 1988JSP....51..797T| url = https://zenodo.org/record/1232502 | type = Submitted manuscript }}
</ref>) ，以及检验发生SOC的必要条件。后一项研究的一个重要问题是，在局部动态交换模型时是否需要能量守恒: 一般的答案是否定的，但有一些保留意见，因为一些交换动力学(如 BTW 的动态)确实需要局部至少平均的能量守恒。从长远来看，有待解决的关键理论问题包括SOC行为可能的普适性类的计算，以及是否有可能推导出一个一般规则来确定一个随机算法是否显示SOC。
除了这些大部分基于实验室的方法，许多其他的研究都集中在大规模的自然或社会系统上，这些系统已经知道(或怀疑)表现出尺度不变的行为。虽然这些方法并不总是受到研究对象专家的欢迎(至少最初是这样) ，但 '''<font color="#ff8000"> SOC</font>''' 已经成为解释一些自然现象的强有力的候选者，包括: 地震(早在SOC被发现之前，地震就被认为是尺度不变行为的来源，例如描述地震大小统计分布的古腾堡-里克特定律，以及描述余震频率的描述余震的 Omori 定律<ref name=TurcotteSmalleySolla85>
{{cite journal|author1=Turcotte, D. L. |author2=Smalley, R. F., Jr. |author3=Solla, S. A. | year = 1985| title = Collapse of loaded fractal trees| journal = Nature | doi= 10.1038/313671a0| volume = 313| issue = 6004| pages = 671–672|bibcode = 1985Natur.313..671T }}</ref><ref name=SmalleyTurcotteSolla85 />）；太阳耀斑; 经济系统的波动，比如金融市场(经济物理学中经常提到 SOC) ; 景观形成; 森林火灾; 滑坡; 流行病; 大脑皮层的神经雪崩;<ref name="Beggs2003" /><ref name=Poil2012>{{cite journal | pmid = 22815496 |date=Jul 2012 |author1=Poil, SS |author2=Hardstone, R |author3=Mansvelder, HD |author4=Linkenkaer-Hansen, K | title = Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks | volume = 32 | issue = 29 | pages = 9817–23
| doi = 10.1523/JNEUROSCI.5990-11.2012 | journal = Journal of Neuroscience | pmc=3553543}}</ref>电生理信号振幅的1 / f 噪声，<ref name=LinkenkaerHansen2001 />以及生物进化(其中 SOC 已被调用，例如，作为背后的动力机制的理论“间断平衡”由 Niles Eldredge 和史蒂芬·古尔德提出)。对SOC的这些”应用”研究既包括建模(开发新模型或使现有模型适应特定自然系统的具体情况) ，也包括广泛的数据分析，以确定是否存在和 / 或具有自然幂率的特点。
此外，SOC 已经应用于计算算法。最近，人们发现来自 SOC 过程的雪崩，如 BTW 模型，在图的最优解的随机搜索中形成有效的模式。 <ref name=Hoffmann2018>{{cite journal| author = H. Hoffmann and D. W. Payton| year = 2018| title = Optimization by Self-Organized Criticality| journal = Scientific Reports| volume = 8| issue = 1| pages = 2358| doi=10.1038/s41598-018-20275-7
| pmid = 29402956| pmc = 5799203| bibcode = 2018NatSR...8.2358H}}</ref> 图着色就是这种最佳化问题的一个例子。'''<font color="#ff8000"> SOC</font>''' 过程显然有助于避免优化陷入局部最优，而无需使用任何以前的极值优化工作所建议的退火方案。
{{cite journal
| author = [[H. Hoffmann|Hoffmann, H.]] and [[D. W. Payton|Payton, D. W.]]
| year = 2018
| title = Optimization by Self-Organized Criticality
| journal = [[Scientific Reports]]
| volume = 8
| issue = 1
| pages = 2358
| doi=10.1038/s41598-018-20275-7
| pmid = 29402956
| pmc = 5799203
| bibcode = 2018NatSR...8.2358H
}}</ref>
图着色就是这种最佳化问题的一个例子。'''<font color="#ff8000"> SOC</font>''' 过程显然有助于避免优化陷入局部最优，而无需使用任何以前的极值优化工作所建议的退火方案。
The recent excitement generated by [[scale-free networks]] has raised some interesting new questions for SOC-related research: a number of different SOC models have been shown to generate such networks as an emergent phenomenon, as opposed to the simpler models proposed by network researchers where the network tends to be assumed to exist independently of any physical space or dynamics. While many single phenomena have been shown to exhibit scale-free properties over narrow ranges, a phenomenon offering by far a larger amount of data is solvent-accessible surface areas in globular proteins.<ref name=Moret2007>
The recent excitement generated by [[scale-free networks]] has raised some interesting new questions for SOC-related research: a number of different SOC models have been shown to generate such networks as an emergent phenomenon, as opposed to the simpler models proposed by network researchers where the network tends to be assumed to exist independently of any physical space or dynamics. While many single phenomena have been shown to exhibit scale-free properties over narrow ranges, a phenomenon offering by far a larger amount of data is solvent-accessible surface areas in globular proteins.<ref name=Moret2007>
== Further reading延伸阅读 ==
== 延伸阅读 ==
* {{cite journal
* {{cite journal
| author = Adami, C.
| author = Adami, C.
}}
}}
*[http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=cond-mat&level=1&index1=20771 Self-organized criticality on arxiv.org]
*[http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=cond-mat&level=1&index1=20771 Self-organized criticality on arxiv.org]
[[Category:Critical phenomena]]
[[Category:Critical phenomena]]