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。它们的宏观行为因此显示了相变临界点的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,因为系统在趋向于临界状态时有效地自我调整。



The concept was put forward by Per Bak, Chao Tang and Kurt Wiesenfeld ("BTW") in a paper引用错误:没有找到与</ref>对应的<ref>标签

Papercore summary: http://papercore.org/Bak1987.</ref>

论文摘要: [ https://archive.is/20130704122906/http://Papercore.org/bak1987 http://Papercore.org/bak1987] / 参考

published in 1987 in Physical Review Letters, and is considered to be one of the mechanisms by which complexity引用错误:没有找到与</ref>对应的<ref>标签 arises in nature. Its concepts have been applied across fields as diverse as geophysics,[1] arises in nature. Its concepts have been applied across fields as diverse as geophysics,[1] physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology[2] physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology[2][3][3][4][4] and others.

}}</ref> and others.

} / ref and others.



SOC is typically observed in slowly driven non-equilibrium systems with many degrees of freedom and strongly nonlinear dynamics. Many individual examples have been identified since BTW's original paper, but to date there is no known set of general characteristics that guarantee a system will display SOC.

SOC is typically observed in slowly driven non-equilibrium systems with many degrees of freedom and strongly nonlinear dynamics. Many individual examples have been identified since BTW's original paper, but to date there is no known set of general characteristics that guarantee a system will display SOC.

SOC是典型的多自由度、强非线性动力学的缓慢驱动非平衡系统。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,细胞自动机的研究---- 从 Stanislaw Ulam 和约翰·冯·诺伊曼的早期发现到 John Conway 的《生命的游戏》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,诸如分形和幂定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、粉红噪声和幂定律) ,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,而不会影响关键行为的出现: 因此,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引用错误:没有找到与</ref>对应的<ref>标签[5][5]), 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.

</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 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.

/ ref) ,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,因为一些交换动态(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引用错误:没有找到与</ref>对应的<ref>标签[1]); 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;[3][6]); 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;[6] 1/f noise in the amplitude of electrophysiological signals;[2] 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.

}}</ref> 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.

{} / ref 1 / f 噪声在电生理信号的振幅,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)。对土壤有机碳的这些”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 / 或具有自然定标法的特点。



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.引用错误:没有找到与</ref>对应的<ref>标签

}}</ref>

{} / ref

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.

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.

图着色就是这种最佳化问题的一个例子。 SOC 过程显然有助于优化陷入局部最优,而无需使用任何退火方案,正如以前的极值优化工作所建议的。



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.引用错误:没有找到与</ref>对应的<ref>标签

}}</ref>

{} / ref

These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.引用错误:没有找到与</ref>对应的<ref>标签

}}</ref>

{} / ref



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.[7] However,

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. However,

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。和 Wiesenfeld 基于他们的沙堆模型的行为建立了他们的假设。然而,

it has been argued that this model would actually generate 1/f2 noise rather than 1/f noise.引用错误:没有找到与</ref>对应的<ref>标签

</ref>

/ 参考

This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models

This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models

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

generally produce 1/fa spectra, with a<2. 引用错误:没有找到与</ref>对应的<ref>标签

</ref>

/ 参考

Other simulation models were proposed later that could produce true 1/f noise,引用错误:没有找到与</ref>对应的<ref>标签 and experimental sandpile models were observed to yield 1/f noise.[8] and experimental sandpile models were observed to yield 1/f noise.[8] In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory[9] In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory[9],

}}</ref>, 

} / ref,

mean field theory引用错误:没有找到与</ref>对应的<ref>标签,

}}</ref>,

} / ref,

the convergence of random variables引用错误:没有找到与</ref>对应的<ref>标签,

}}</ref>,

} / ref,

and cluster formation.引用错误:没有找到与</ref>对应的<ref>标签 A continuous model of self-organised criticality is proposed by using tropical geometry.[10]

}}</ref> A continuous model of self-organised criticality is proposed by using tropical geometry.

{} / ref 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[11][1]

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 1.3 1.4 |bibcode = 1995PNAS...92.6689B }} 引用错误:无效<ref>标签;name属性“SmalleyTurcotteSolla85”使用不同内容定义了多次
  2. 2.0 2.1 2.2 |url=https://semanticscholar.org/paper/6776d17957204c198e278bda98c935ab1cf8f22b }} 引用错误:无效<ref>标签;name属性“LinkenkaerHansen2001”使用不同内容定义了多次
  3. 3.0 3.1 3.2 |doi=10.1523/JNEUROSCI.21-04-01370.2001 |pmc=6762238 }} 引用错误:无效<ref>标签;name属性“Beggs2003”使用不同内容定义了多次
  4. 4.0 4.1 }} 引用错误:无效<ref>标签;name属性“Chialvo2004”使用不同内容定义了多次
  5. 5.0 5.1 / ref / name tang1988b {{cite journal {{cite journal {引用期刊 | author = Tang, C. and Bak, P. | author = Tang, C. and Bak, P. 作者 Tang,c. and Bak,p。 | year = 1988 | year = 1988 1988年 | title = Mean field theory of self-organized critical phenomena | title = Mean field theory of self-organized critical phenomena 自组织临界现象的平均场理论 | journal = Journal of Statistical Physics | journal = Journal of Statistical Physics 统计物理学杂志 | volume = 51 | volume = 51 第51卷 | issue = 5–6 | issue = 5–6 第5-6期 | pages = 797–802 | pages = 797–802 797802页 | doi = 10.1007/BF01014884 | doi = 10.1007/BF01014884 10.1007 / BF01014884 | bibcode= 1988JSP....51..797T | bibcode= 1988JSP....51..797T 1988JSP... 51. . 797 t | url = https://zenodo.org/record/1232502 | url = https://zenodo.org/record/1232502 Https://zenodo.org/record/1232502 | type = Submitted manuscript | type = Submitted manuscript | 打印提交的手稿 }} }} }}
  6. 6.0 6.1 }} 引用错误:无效<ref>标签;name属性“Poil2012”使用不同内容定义了多次
  7. 引用错误:无效<ref>标签;未给name属性为Bak1987的引用提供文字
  8. 8.0 8.1 / ref 和实验沙堆模型被观察到产生1 / f 噪音。参考名称 frette1996 {{cite journal {{cite journal {引用期刊 | author = Frette, V., Christinasen, K., Malthe-Sørenssen, A., Feder, J, Jøssang, T and Meaken, P | author = Frette, V., Christinasen, K., Malthe-Sørenssen, A., Feder, J, Jøssang, T and Meaken, P | author = Frette, V., Christinasen, K., Malthe-Sørenssen, A., Feder, J, Jøssang, T and Meaken, P | year = 1996 | year = 1996 1996年 | title = Avalanche dynamics in a pile of rice | title = Avalanche dynamics in a pile of rice | 题目: 大米堆中的雪崩动力学 | journal = Nature | journal = Nature 自然》杂志 | volume = 379 | volume = 379 第379卷 | issue = 6560 | issue = 6560 第6560期 | pages = 49–52 | pages = 49–52 第49-52页 | doi =10.1038/379049a0 | doi =10.1038/379049a0 10.1038 / 379049a0 | bibcode= 1996Natur.379...49F}} | bibcode= 1996Natur.379...49F}} 1996 / natur. 379... 49F }
  9. 9.0 9.1 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 dewar2003 {{cite journal {{cite journal {引用期刊 | author = Dewar, R. | author = Dewar, R. 作者杜瓦,r。 | year = 2003 | year = 2003 2003年 | title = Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states | title = Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states 非平衡态中涨落定理、最大产生熵和自组织临界性的信息论解释 | journal =J. Phys. A: Math. Gen. | journal =J. Phys. A: Math. Gen. | j 杂志。女名女子名。答: 数学。将军。 | volume = 36 | volume = 36 第36卷 | pages =631–641 | pages =631–641 631-- 641 | pmid = | pmid = 我不会让你失望的 | doi = 10.1088/0305-4470/36/3/303 | doi = 10.1088/0305-4470/36/3/303 10.1088 / 0305-4470 / 36 / 3 / 303 | issue = 3 | issue = 3 第三期 | pmc = | pmc = 我会的,我会的,我会的 |bibcode = 2003JPhA...36..631D|arxiv = cond-mat/0005382 | author-link = R Dewar |bibcode = 2003JPhA...36..631D|arxiv = cond-mat/0005382 | author-link = R Dewar | bibcode 2003JPhA... 36. . 631 d | arxiv cond-mat / 0005382 | author-link r Dewar }}
  10. 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.
  11. 11.0 11.1 引用错误:无效<ref>标签;未给name属性为TurcotteSmalleySolla85的引用提供文字




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] Check |citeseerx= value (help). doi:10.1016/0375-9601(95)00372-A. Text " 生命系统中的自组织临界性

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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"] Check |url= value (help). 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+%2F+pnas.+92.15.6689 10.1073/pnas.92.15.6689 10.1073 / pnas. 92.15.6689] Check |doi= value (help). PMC [//www.ncbi.nlm.nih.gov/pmc/articles/PMC41396%0A%0A41396 41396 41396] Check |pmc= value (help). PMID [//pubmed.ncbi.nlm.nih.gov/11607561

11607561 11607561 11607561] Check |pmid= value (help). Text " 标题复杂性、偶然性和临界性

" ignored (help); line feed character in |pmid= at position 9 (help); line feed character in |author= at position 25 (help); line feed character in |journal= at position 59 (help); line feed character in |year= at position 5 (help); line feed character in |url= at position 48 (help); line feed character in |pmc= at position 6 (help); line feed character in |volume= at position 3 (help); line feed character in |doi= at position 24 (help); line feed character in |pages= at position 16 (help); line feed character in |issue= at position 3 (help); Check date values in: |year= (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+%2F+physrvlett.+71.4083 10.1103/PhysRevLett.71.4083 10.1103 / physrvlett. 71.4083] Check |doi= value (help). PMID [//pubmed.ncbi.nlm.nih.gov/10055149

10055149 10055149 10055149] Check |pmid= value (help). Text " 简单演化模型中的间断平衡和临界性

" ignored (help); line feed character in |pmid= at position 9 (help); line feed character in |author= at position 24 (help); line feed character in |journal= at position 24 (help); line feed character in |year= at position 5 (help); line feed character in |volume= at position 3 (help); line feed character in |issue= at position 3 (help); line feed character in |pages= at position 16 (help); line feed character in |doi= at position 28 (help); Check date values in: |year= (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]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+%2F+physrvlett.+59.381 10.1103/PhysRevLett.59.381 10.1103 / physrvlett. 59.381] Check |doi= value (help). PMID [//pubmed.ncbi.nlm.nih.gov/10035754

10035754 10035754 10035754] Check |pmid= value (help). Text " 题目自组织临界性: 数学噪音的解释

" ignored (help); line feed character in |pmid= at position 9 (help); line feed character in |author= at position 37 (help); line feed character in |journal= at position 24 (help); line feed character in |year= at position 5 (help); line feed character in |volume= at position 3 (help); line feed character in |issue= at position 2 (help); line feed character in |pages= at position 14 (help); line feed character in |doi= at position 27 (help); Check date values in: |year= (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+%2F+PhysRevA.+38.364 10.1103/PhysRevA.38.364 10.1103 / PhysRevA. 38.364] Check |doi= value (help). PMID [//pubmed.ncbi.nlm.nih.gov/9900174

9900174 9900174 9900174] Check |pmid= value (help). line feed character in |pmid= at position 8 (help); line feed character in |author= at position 37 (help); line feed character in |journal= at position 18 (help); line feed character in |title= at position 27 (help); line feed character in |year= at position 5 (help); line feed character in |volume= at position 3 (help); line feed character in |issue= at position 2 (help); line feed character in |pages= at position 14 (help); line feed character in |doi= at position 24 (help); Check date values in: |year= (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

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第10期): 10412

10412页. Bibcode:1986JGR....9110412K. doi:10.1029/JB091iB10p10412. Text " bibcode 1986JGR... 9110412K

    " ignored (help); Text " doi 10.1029 / JB091iB10p10412
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}}

}}



  • 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. line feed character in |year= at position 5 (help); line feed character in |volume= at position 3 (help); line feed character in |title= at position 43 (help); line feed character in |author= at position 19 (help); line feed character in |pages= at position 6 (help); line feed character in |journal= at position 30 (help); Check date values in: |year= (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+%2F+0034-4885+%2F+62+%2F+10+%2F+201 10.1088/0034-4885/62/10/201 10.1088 / 0034-4885 / 62 / 10 / 201] Check |doi= value (help). Text " bibcode 1999RPPh... 62.1377 t " ignored (help); Text " 作者链接 Donald l. Turcotte

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}}
}}

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. Text " arxiv physics / 0611069 " ignored (help); Text " bibcode 2007 phla. . 360. . 717 n " ignored (help); Text " author-link md.Nurujjaman / a.N. Sekar Iyengar

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}}
}}

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


This page was moved from wikipedia:en:Self-organized criticality. Its edit history can be viewed at 自组织临界性/edithistory