自组织临界性

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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.

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， 自组织临界性Self-organized criticality (SOC)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。

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[1] published in 1987 in Physical Review Letters, and is considered to be one of the mechanisms by which complexity[2] arises in nature. Its concepts have been applied across fields as diverse as geophysics,[3] physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology[4][5][6] and others.


The concept was put forward by Per Bak, Chao Tang and Kurt Wiesenfeld ("BTW") in a paper[1] published in 1987 in Physical Review Letters, and is considered to be one of the mechanisms by which complexity[2]


这个概念是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld (“ BTW”)在一篇名为 bak1987的论文中提出的。1987年发表在《物理评论快报》上，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。 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 SOC。


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 Conway's Game of Life and the extensive work of Stephen Wolfram, made it clear that complexity could be generated as an 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 transitions carried out in the 1960s and 1970s showed how scale invariant phenomena such as fractals and power laws emerged at the critical point between phases.

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 Conway's Game of Life and the extensive work of Stephen Wolfram, made it clear that complexity could be generated as an 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 transitions carried out in the 1960s and 1970s showed how scale invariant phenomena such as fractals and power laws emerged at the critical point between phases.

自组织临界性Self-organized criticality（SOC)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在相变的临界点上的。



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.

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.

自组织临界性Self-organized criticality（SOC)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 "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]. Other experiments include those carried out on magnetic-domain patterns, the Barkhausen effect and vortices in superconductors.

Due to BTW's metaphorical visualization of their model as a "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. 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 exponents[7][8]), 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.

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) ，以及研究出现 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确定任意算法是否显示 SOC 的一般规则的问题。

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-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 Omori law describing the frequency of aftershocks); 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; 1/f noise in the amplitude of electrophysiological signals; and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "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.

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-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 Omori law describing the frequency of aftershocks[9][3]); 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;[5][10] 1/f noise in the amplitude of electrophysiological signals;[4] and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "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 SOC 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，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. 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 annealing scheme, as suggested by previous work on extremal optimization.


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.[11] 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 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。 图着色就是这种最佳化问题的一个例子。 SOC 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.[13] These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.[14]

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.[12] These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.[13]

无标度网络Scale-free networks最近引起的兴奋为 SOC相关研究提出了一些有趣的新问题: 许多不同的 SOC模型已经被证明是作为一种涌现现象产生这样的网络，而不是网络研究人员提出的更简单的模型，其中网络往往被假定独立于任何物理空间或动力学存在。虽然许多单一现象已被证明在狭窄的范围内表现出无标度特性，但是到目前为止提供了大量数据的现象是球状蛋白质中溶剂可及的表面区域。这些研究量化了蛋白质的微分几何，并解决了许多关于生物复杂性出现的进化之谜


Despite the considerable interest and research output generated from the SOC hypothesis, there remains no general agreement with regards to its mechanisms in abstract mathematical form. Bak Tang and Wiesenfeld based their hypothesis on the behavior of their sandpile model.[1] However, it has been argued that this model would actually generate 1/f2 noise rather than 1/f noise.[15] This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models generally produce 1/fa spectra, with a<2.[16] Other simulation models were proposed later that could produce true 1/f noise,[17] and experimental sandpile models were observed to yield 1/f noise.[18] In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory,[19] mean field theory,[20] the convergence of random variables,[21] and cluster formation.[22] A continuous model of self-organised criticality is proposed by using tropical geometry.[23]

Despite the considerable interest and research output generated from the SOC hypothesis, there remains no general agreement with regards to its mechanisms in abstract mathematical form. Bak Tang and Wiesenfeld based their hypothesis on the behavior of their sandpile model.[1] However, it has been argued that this model would actually generate 1/f2 noise rather than 1/f noise.[14] This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models generally produce 1/fa spectra, with a<2.[15] Other simulation models were proposed later that could produce true 1/f noise,[16] and experimental sandpile models were observed to yield 1/f noise.[17] In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory,[18] mean field theory,[19] the convergence of random variables,[20] and cluster formation.[21] A continuous model of self-organised criticality is proposed by using tropical geometry.[22]

尽管 SOC 假说引起了相当大的兴趣和研究成果，但是关于其抽象数学形式的机制仍然没有普遍的一致性。Bak Tang 和 Wiesenfeld 基于他们的沙堆模型的行为建立了他们的假设。然而,有人认为，这种模型实际上会产生1 / f sup 2 / sup 噪声而不是1 / f 噪声。这种说法是基于未经测试的比例假设，更严格的分析表明沙堆模型一般产生1 / f sup a / sup 光谱，其中 a < 2。 除了上面提到的非保守理论模型之外，其他关于 SOC 的理论模型都是基于信息论，例如 一个 自组织临界Self-organised criticality的连续模型是通过使用热带几何来提出的。

Examples of self-organized critical dynamics自组织临界动力学的例子

In chronological order of development:

In chronological order of development:

按发展时间顺序排列:



  • Stick-slip model of fault failure[9][3]
  • 断层破坏的粘滑模型[9][3]

See also 参见

  • Ilya Prigogine, a systems scientist who helped formalize dissipative system behavior in general terms.

References参考资料

  1. ↑ 1.0 1.1 1.2 Bak, P., Tang, C. and Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of 1/f noise". Physical Review Letters. 59 (4): 381–384. Bibcode:1987PhRvL..59..381B. doi:10.1103/PhysRevLett.59.381. PMID 10035754.{{cite journal}}: CS1 maint: multiple names: authors list (link) Papercore summary: http://papercore.org/Bak1987.
  2. ↑ 2.0 2.1 Bak, P., and Paczuski, M. (1995). "Complexity, contingency, and criticality". Proc Natl Acad Sci U S A. 92 (15): 6689–6696. Bibcode:1995PNAS...92.6689B. doi:10.1073/pnas.92.15.6689. PMC 41396. PMID 11607561.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. ↑ 3.0 3.1 3.2 3.3 Smalley, R. F., Jr.; Turcotte, D. L.; Solla, S. A. (1985). "A renormalization group approach to the stick-slip behavior of faults". Journal of Geophysical Research. 90 (B2): 1894. Bibcode:1985JGR....90.1894S. doi:10.1029/JB090iB02p01894.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  4. ↑ 4.0 4.1 K. Linkenkaer-Hansen; V. V. Nikouline; J. M. Palva & R. J. Ilmoniemi. (2001). "Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations". J. Neurosci. 21 (4): 1370–1377. doi:10.1523/JNEUROSCI.21-04-01370.2001. PMC 6762238. PMID 11160408.
  5. ↑ 5.0 5.1 J. M. Beggs & D. Plenz (2006). "Neuronal Avalanches in Neocortical Circuits". J. Neurosci. 23 (35): 11167–77. doi:10.1523/JNEUROSCI.23-35-11167.2003. PMC 6741045. PMID 14657176.
  6. ↑ Chialvo, D. R. (2004). "Critical brain networks". Physica A. 340 (4): 756–765. arXiv:cond-mat/0402538. Bibcode:2004PhyA..340..756R. doi:10.1016/j.physa.2004.05.064.
  7. ↑ Tang, C. and Bak, P. (1988). "Critical exponents and scaling relations for self-organized critical phenomena". Physical Review Letters. 60 (23): 2347–2350. Bibcode:1988PhRvL..60.2347T. doi:10.1103/PhysRevLett.60.2347. PMID 10038328.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  8. ↑ Tang, C. and Bak, P. (1988). "Mean field theory of self-organized critical phenomena". Journal of Statistical Physics (Submitted manuscript). 51 (5–6): 797–802. Bibcode:1988JSP....51..797T. doi:10.1007/BF01014884.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  9. ↑ 9.0 9.1 9.2 Turcotte, D. L.; Smalley, R. F., Jr.; Solla, S. A. (1985). "Collapse of loaded fractal trees". Nature. 313 (6004): 671–672. Bibcode:1985Natur.313..671T. doi:10.1038/313671a0.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  10. ↑ Poil, SS; Hardstone, R; Mansvelder, HD; Linkenkaer-Hansen, K (Jul 2012). "Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks". Journal of Neuroscience. 32 (29): 9817–23. doi:10.1523/JNEUROSCI.5990-11.2012. PMC 3553543. PMID 22815496.
  11. ↑ Hoffmann, H. and Payton, D. W. (2018). "Optimization by Self-Organized Criticality". Scientific Reports. 8 (1): 2358. Bibcode:2018NatSR...8.2358H. doi:10.1038/s41598-018-20275-7. PMC 5799203. PMID 29402956.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  12. ↑ Moret, M. A. and Zebende, G. (2007). "Amino acid hydrophobicity and accessible surface area". Phys. Rev. E. 75 (1): 011920. Bibcode:2007PhRvE..75a1920M. doi:10.1103/PhysRevE.75.011920. PMID 17358197.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  13. ↑ Phillips, J. C. (2014). "Fractals and self-organized criticality in proteins". Physica A. 415: 440–448. Bibcode:2014PhyA..415..440P. doi:10.1016/j.physa.2014.08.034.
  14. ↑ Jensen, H. J., Christensen, K. and Fogedby, H. C. (1989). "1/f noise, distribution of lifetimes, and a pile of sand". Phys. Rev. B. 40 (10): 7425–7427. Bibcode:1989PhRvB..40.7425J. doi:10.1103/physrevb.40.7425. PMID 9991162.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  15. ↑ Laurson, Lasse; Alava, Mikko J.; Zapperi, Stefano (15 September 2005). "Letter: Power spectra of self-organized critical sand piles". Journal of Statistical Mechanics: Theory and Experiment. 0511. L001.
  16. ↑ Maslov, S., Tang, C. and Zhang, Y. - C. (1999). "1/f noise in Bak-Tang-Wiesenfeld models on narrow stripes". Phys. Rev. Lett. 83 (12): 2449–2452. arXiv:cond-mat/9902074. Bibcode:1999PhRvL..83.2449M. doi:10.1103/physrevlett.83.2449.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  17. ↑ Frette, V., Christinasen, K., Malthe-Sørenssen, A., Feder, J, Jøssang, T and Meaken, P (1996). "Avalanche dynamics in a pile of rice". Nature. 379 (6560): 49–52. Bibcode:1996Natur.379...49F. doi:10.1038/379049a0.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  18. ↑ Dewar, R. (2003). "Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states". J. Phys. A: Math. Gen. 36 (3): 631–641. arXiv:cond-mat/0005382. Bibcode:2003JPhA...36..631D. doi:10.1088/0305-4470/36/3/303.
  19. ↑ Vespignani, A., and Zapperi, S. (1998). "How self-organized criticality works: a unified mean-field picture". Phys. Rev. E. 57 (6): 6345–6362. arXiv:cond-mat/9709192. Bibcode:1998PhRvE..57.6345V. doi:10.1103/physreve.57.6345. hdl:2047/d20002173.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  20. ↑ Kendal, WS (2015). "Self-organized criticality attributed to a central limit-like convergence effect". Physica A. 421: 141–150. Bibcode:2015PhyA..421..141K. doi:10.1016/j.physa.2014.11.035.
  21. ↑ Hoffmann, H. (2018). "Impact of Network Topology on Self-Organized Criticality". Phys. Rev. E. 97 (2): 022313. Bibcode:2018PhRvE..97b2313H. doi:10.1103/PhysRevE.97.022313. PMID 29548239.
  22. ↑ Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (2018-08-15). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences (in English). 115 (35): E8135–E8142. arXiv:1806.09153. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.




Further reading延伸阅读

1995年). "Self-organized criticality in living systems". Physics Letters A 物理学快报. 203

第203卷 (1

第一期): 29–32

29-- 32页. arXiv:adap-org/9401001. Bibcode:1995PhLA..203...29A. CiteSeerX [//citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.456.9543%0A%0A10.1.1.456.9543 10.1.1.456.9543 10.1.1.456.9543]. doi:10.1016/0375-9601(95)00372-A. {{cite journal}}: Check |citeseerx= value (help); Check date values in: |year= (help); Text "doi 10.1016 / 0375-9601(95)00372-A" ignored (help); Text "作者链接阿达米，c" ignored (help); Text "生命系统中的自组织临界性" ignored (help); line feed character in |author= at position 10 (help); line feed character in |citeseerx= at position 16 (help); line feed character in |issue= at position 2 (help); line feed character in |journal= at position 18 (help); line feed character in |pages= at position 12 (help); line feed character in |volume= at position 4 (help); line feed character in |year= at position 5 (help) 

}}

}}



1996年). How Nature Works: The Science of Self-Organized Criticality

自然如何运作: 自组织临界性的科学. New York: Copernicus

出版商哥白尼. ISBN [[Special:BookSources/978-0-387-94791-4

[国际标准图书编号978-0-387-94791-4]|978-0-387-94791-4

[国际标准图书编号978-0-387-94791-4]]].  

}}

}}



  • Bak, P. and Paczuski, M.

作者 Bak，p. and Paczuski，m。 (1995

1995年). [http://pnas.org/cgi/content/abstract/92/15/6689

Http://pnas.org/cgi/content/abstract/92/15/6689 "Complexity, contingency, and criticality"]. Proceedings of the National Academy of Sciences of the USA 美国美国国家科学院院刊杂志. 92

第92卷 (15

第15期): 6689–6696

6689-- 6696. Bibcode:1995PNAS...92.6689B. doi:[//doi.org/10.1073%2Fpnas.92.15.6689%0A%0A10.1073%20%2F%20pnas.%2092.15.6689 10.1073/pnas.92.15.6689 10.1073 / pnas. 92.15.6689]. PMC [//www.ncbi.nlm.nih.gov/pmc/articles/PMC41396%0A%0A41396 41396 41396]. PMID [//pubmed.ncbi.nlm.nih.gov/11607561

11607561 11607561 11607561]. {{cite journal}}: Check |doi= value (help); Check |pmc= value (help); Check |pmid= value (help); Check |url= value (help); Check date values in: |year= (help); Text "标题复杂性、偶然性和临界性" ignored (help); line feed character in |author= at position 25 (help); line feed character in |doi= at position 24 (help); line feed character in |issue= at position 3 (help); line feed character in |journal= at position 59 (help); line feed character in |pages= at position 16 (help); line feed character in |pmc= at position 6 (help); line feed character in |pmid= at position 9 (help); line feed character in |url= at position 48 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help)CS1 maint: multiple names: authors list (link) 

  |bibcode = 1995PNAS...92.6689B }}

92.6689 b }



  • Bak, P. and Sneppen, K.

作者 Bak，p. and Sneppen，k。 (1993

1993年). "Punctuated equilibrium and criticality in a simple model of evolution". Physical Review Letters 物理评论快报. 71

第71卷 (24

第24期): 4083–4086

4083-- 4086. Bibcode:1993PhRvL..71.4083B. doi:[//doi.org/10.1103%2FPhysRevLett.71.4083%0A%0A10.1103%20%2F%20physrvlett.%2071.4083 10.1103/PhysRevLett.71.4083 10.1103 / physrvlett. 71.4083]. PMID [//pubmed.ncbi.nlm.nih.gov/10055149

10055149 10055149 10055149]. {{cite journal}}: Check |doi= value (help); Check |pmid= value (help); Check date values in: |year= (help); Text "简单演化模型中的间断平衡和临界性" ignored (help); line feed character in |author= at position 24 (help); line feed character in |doi= at position 28 (help); line feed character in |issue= at position 3 (help); line feed character in |journal= at position 24 (help); line feed character in |pages= at position 16 (help); line feed character in |pmid= at position 9 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help)CS1 maint: multiple names: authors list (link)

| bibcode=1993PhRvL..71.4083B}}

1993 phrvl. . 71.4083 b }



  • Bak, P., Tang, C. and Wiesenfeld, K.

作者 Bak，p. ，Tang，c. and Wiesenfeld，k。 (1987

1987年). "Self-organized criticality: an explanation of [math]\displaystyle{ 1/f }[/math] noise". Physical Review Letters 物理评论快报. 59

第59卷 (4

第四期): 381–384

381-- 384. Bibcode:1987PhRvL..59..381B. doi:[//doi.org/10.1103%2FPhysRevLett.59.381%0A%0A10.1103%20%2F%20physrvlett.%2059.381 10.1103/PhysRevLett.59.381 10.1103 / physrvlett. 59.381]. PMID [//pubmed.ncbi.nlm.nih.gov/10035754

10035754 10035754 10035754]. {{cite journal}}: Check |doi= value (help); Check |pmid= value (help); Check date values in: |year= (help); Text "题目自组织临界性: 数学噪音的解释" ignored (help); line feed character in |author= at position 37 (help); line feed character in |doi= at position 27 (help); line feed character in |issue= at position 2 (help); line feed character in |journal= at position 24 (help); line feed character in |pages= at position 14 (help); line feed character in |pmid= at position 9 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help)CS1 maint: multiple names: authors list (link) 

| bibcode=1987PhRvL..59..381B}}

1987 / phrvl. . 59. . 381 b }



  • Bak, P., Tang, C. and Wiesenfeld, K.

作者 Bak，p. ，Tang，c. and Wiesenfeld，k。 (1988

1988年). "Self-organized criticality

标题自组织临界性". Physical Review A 物理评论 a 期刊. 38

第38卷 (1

第一期): 364–374

364-- 374. Bibcode:1988PhRvA..38..364B. doi:[//doi.org/10.1103%2FPhysRevA.38.364%0A%0A10.1103%20%2F%20PhysRevA.%2038.364 10.1103/PhysRevA.38.364 10.1103 / PhysRevA. 38.364]. PMID [//pubmed.ncbi.nlm.nih.gov/9900174

9900174 9900174 9900174]. {{cite journal}}: Check |doi= value (help); Check |pmid= value (help); Check date values in: |year= (help); line feed character in |author= at position 37 (help); line feed character in |doi= at position 24 (help); line feed character in |issue= at position 2 (help); line feed character in |journal= at position 18 (help); line feed character in |pages= at position 14 (help); line feed character in |pmid= at position 8 (help); line feed character in |title= at position 27 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help)CS1 maint: multiple names: authors list (link) Papercore summary. 

|bibcode = 1988PhRvA..38..364B }} Papercore summary.

| bibcode 1988PhRvA. . 38. . 364 b }[ https://archive.is/20130415140421/http://www.Papercore.org/perbak1987文件核心摘要]。



  • [[Mark Buchanan

马克 · 布坎南 |Buchanan, M.

作者: 布坎南。]] (2000

2000年). Ubiquity

标题: Ubiquity. London

地点: 伦敦: Weidenfeld & Nicolson. ISBN [[Special:BookSources/978-0-7538-1297-6

[国际标准图书编号978-0-7538-1297-6]|978-0-7538-1297-6

[国际标准图书编号978-0-7538-1297-6]]].  

}}

}}



  • [[Henrik Jeldtoft Jensen

作者: 亨里克 · 耶尔德托夫特 · 詹森 |Jensen, H. J.

作者 Jensen，h. j。]] (1998

1998年). Self-Organized Criticality

标题自组织临界性. Cambridge: Cambridge University Press

出版商剑桥大学出版社. ISBN [[Special:BookSources/978-0-521-48371-1

[国际标准图书编号978-0-521-48371-1]|978-0-521-48371-1

[国际标准图书编号978-0-521-48371-1]]].  

}}

}}



  • Katz, J. I.

作者 Katz，j. i。 (1986

1986年). "A model of propagating brittle failure in heterogeneous media

在非均匀介质中传播脆性破坏的模型". Journal of Geophysical Research 地球物理研究期刊. 91

第91卷 (B10

第10期): 10412

10412页. Bibcode:1986JGR....9110412K. doi:10.1029/JB091iB10p10412. {{cite journal}}: Check date values in: |year= (help); Text "bibcode 1986JGR... 9110412K" ignored (help); Text "doi 10.1029 / JB091iB10p10412" ignored (help); line feed character in |author= at position 12 (help); line feed character in |issue= at position 4 (help); line feed character in |journal= at position 32 (help); line feed character in |pages= at position 6 (help); line feed character in |title= at position 62 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help)

}}

}}



  • Kron, T./Grund, T.

作者: Kron t. / Grund t。 (2009

2009年). "Society as a Selforganized Critical System

作为一个自组织的批判系统的社会". Cybernetics and Human Knowing 控制论与人类认知. 16

第16卷: 65–82

第65-82页. {{cite journal}}: Check date values in: |year= (help); line feed character in |author= at position 19 (help); line feed character in |journal= at position 30 (help); line feed character in |pages= at position 6 (help); line feed character in |title= at position 43 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help)CS1 maint: multiple names: authors list (link)

}}

}}



2005年). Networks as renormalized models for emergent behavior in physical systems

作为物理系统中突发行为的重整化模型的网络. The Science and Culture Series – Physics. pp. 363–374

第363-374页. arXiv:physics/0502028. Bibcode 2005cmn..conf..363P. doi:10.1142/9789812701558_0042 2005 / cmn. conf. 363 p. ISBN 978-981-256-525-9.  

}}

}}



  • [[Donald L. Turcotte

作者: Donald l. Turcotte |Turcotte, D. L.

作者: Turcotte，D.l。]] (1997

1997年). Fractals and Chaos in Geology and Geophysics

地质学与地球物理学中的分形与混沌. Cambridge: Cambridge University Press

出版商剑桥大学出版社. ISBN [[Special:BookSources/978-0-521-56733-6

[国际标准图书馆编号978-0-521-56733-6]|978-0-521-56733-6

[国际标准图书馆编号978-0-521-56733-6]]].  

}}

}}



1999年). "Self-organized criticality

标题自组织临界性". Reports on Progress in Physics 物理学进展报告. 62

第62卷 (10

第10期): 1377–1429

13771429页. Bibcode:1999RPPh...62.1377T. doi:[//doi.org/10.1088%2F0034-4885%2F62%2F10%2F201%0A%0A10.1088%20%2F%200034-4885%20%2F%2062%20%2F%2010%20%2F%20201 10.1088/0034-4885/62/10/201 10.1088 / 0034-4885 / 62 / 10 / 201]. {{cite journal}}: Check |doi= value (help); Check date values in: |year= (help); Text "bibcode 1999RPPh... 62.1377 t" ignored (help); Text "作者链接 Donald l. Turcotte" ignored (help); line feed character in |author= at position 16 (help); line feed character in |doi= at position 28 (help); line feed character in |issue= at position 3 (help); line feed character in |journal= at position 31 (help); line feed character in |pages= at position 16 (help); line feed character in |title= at position 27 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help) 

}}

}}

2007年). "Realization of {SOC} behavior in a dc glow discharge plasma

直流辉光放电等离子体{ SOC }行为的实现". Physics Letters A 物理学快报. 360

第360卷 (6

第六期): 717–721

717-- 721页. arXiv:physics/0611069. Bibcode:2007PhLA..360..717N. doi:10.1016/j.physleta.2006.09.005. {{cite journal}}: Check date values in: |year= (help); Text "arxiv physics / 0611069" ignored (help); Text "author-link md.Nurujjaman / a.N. Sekar Iyengar" ignored (help); Text "bibcode 2007 phla. . 360. . 717 n" ignored (help); Text "doi 10.1016 / j.physleta. 2006.09.005" ignored (help); line feed character in |author= at position 35 (help); line feed character in |issue= at position 2 (help); line feed character in |journal= at position 18 (help); line feed character in |pages= at position 14 (help); line feed character in |title= at position 60 (help); line feed character in |volume= at position 4 (help); line feed character in |year= at position 5 (help) 

}}

}}

Category:Critical phenomena

范畴: 关键现象

Category:Applied and interdisciplinary physics

类别: 应用和跨学科物理学

Category:Chaos theory

范畴: 混沌理论

Category:Self-organization

类别: 自我组织


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