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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。
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>
The concept was put forward by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] ("BTW") in a paper<ref name=Bak1987>
{{cite journal
{{cite journal
| author = [[Per Bak|Bak, P.]], [[Chao Tang|Tang, C.]] and [[Kurt Wiesenfeld|Wiesenfeld, K.]]
| author = [[Per Bak|Bak, P.]], [[Chao Tang|Tang, C.]] and [[Kurt Wiesenfeld|Wiesenfeld, K.]]
| year = 1987
| year = 1987
| title = Self-organized criticality: an explanation of 1/''f'' noise
| title = Self-organized criticality: an explanation of 1/''f'' noise
| journal = [[Physical Review Letters]]
| journal = [[Physical Review Letters]]
| volume = 59
| volume = 59
| issue = 4
| issue = 4
| pages = 381–384
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| doi = 10.1103/PhysRevLett.59.381
| pmid = 10035754
| pmid = 10035754
| bibcode=1987PhRvL..59..381B
| bibcode=1987PhRvL..59..381B
}}
}}
Papercore summary: [https://archive.is/20130704122906/http://papercore.org/Bak1987 http://papercore.org/Bak1987].</ref>
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>
published in 1987 in ''[[Physical Review Letters]]'', and is considered to be one of the mechanisms by which [[complexity]]<ref name=Bak1995>
{{cite journal
{{cite journal
| author = [[Per Bak|Bak, P.]], and [[Maya Paczuski|Paczuski, M.]]
| author = [[Per Bak|Bak, P.]], and [[Maya Paczuski|Paczuski, M.]]
| year = 1995
| year = 1995
| title = Complexity, contingency, and criticality
| title = Complexity, contingency, and criticality
| journal =Proc Natl Acad Sci U S A
| journal =Proc Natl Acad Sci U S A
| volume = 92
| volume = 92
| pages = 6689–6696
| pages = 6689–6696
| pmid = 11607561
| pmid = 11607561
| doi = 10.1073/pnas.92.15.6689
| doi = 10.1073/pnas.92.15.6689
| issue = 15
| issue = 15
| pmc = 41396
| pmc = 41396
|bibcode = 1995PNAS...92.6689B }}</ref> arises in nature. Its concepts have been applied across fields as diverse as [[geophysics]],<ref name=SmalleyTurcotteSolla85>
|bibcode = 1995PNAS...92.6689B }}</ref> arises in nature. Its concepts have been applied across fields as diverse as [[geophysics]],<ref name=SmalleyTurcotteSolla85>
{{cite journal
{{cite journal
|author1=Smalley, R. F., Jr. |author2=Turcotte, D. L. |author3=Solla, S. A. | year = 1985
|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
| title = A renormalization group approach to the stick-slip behavior of faults
| journal = Journal of Geophysical Research
| journal = Journal of Geophysical Research
| bibcode = 1985JGR....90.1894S
| bibcode = 1985JGR....90.1894S
| doi = 10.1029/JB090iB02p01894
| doi = 10.1029/JB090iB02p01894
| volume = 90
| volume = 90
| issue = B2
| issue = B2
| pages = 1894
| 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>
|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
{{cite journal
|author1=K. Linkenkaer-Hansen |author2=V. V. Nikouline |author3=J. M. Palva |author4=R. J. Ilmoniemi. |name-list-style=amp | year = 2001
|author1=K. Linkenkaer-Hansen |author2=V. V. Nikouline |author3=J. M. Palva |author4=R. J. Ilmoniemi. |last-author-amp=yes | year = 2001
| title = Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations
| title = Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations
| journal = J. Neurosci.
| journal = J. Neurosci.
| volume = 21
| volume = 21
| pages = 1370–1377
| pages = 1370–1377
| pmid = 11160408
| pmid = 11160408
| issue = 4
| issue = 4
|doi=10.1523/JNEUROSCI.21-04-01370.2001 |pmc=6762238 }}</ref><ref name=Beggs2003>
|doi=10.1523/JNEUROSCI.21-04-01370.2001 |pmc=6762238 }}</ref><ref name=Beggs2003>
{{cite journal
{{cite journal
|author1=J. M. Beggs |author2=D. Plenz
|author1=J. M. Beggs |author2=D. Plenz
|name-list-style=amp | year = 2006
|lastauthoramp=yes | year = 2006
| title = Neuronal Avalanches in Neocortical Circuits
| title = Neuronal Avalanches in Neocortical Circuits
| journal = J. Neurosci.
| journal = J. Neurosci.
| volume = 23
| volume = 23
|issue=35
|issue=35
|pages=11167–77
|pages=11167–77
|doi=10.1523/JNEUROSCI.23-35-11167.2003
|doi=10.1523/JNEUROSCI.23-35-11167.2003
|pmid=14657176
|pmid=14657176
|pmc=6741045
|pmc=6741045
}}</ref><ref name=Chialvo2004>
}}</ref><ref name=Chialvo2004>
{{cite journal
{{cite journal
| author =Chialvo, D. R.
| author =Chialvo, D. R.
| year = 2004
| year = 2004
| title = Critical brain networks
| title = Critical brain networks
| journal = Physica A
| journal = Physica A
| volume = 340
| volume = 340
| issue =4
| issue =4
| pages = 756–765
| pages = 756–765
| doi = 10.1016/j.physa.2004.05.064
| doi = 10.1016/j.physa.2004.05.064
|arxiv = cond-mat/0402538 |bibcode = 2004PhyA..340..756R | author-link = Dante R. Chialvo
|arxiv = cond-mat/0402538 |bibcode = 2004PhyA..340..756R | author-link = Dante R. Chialvo
}}</ref> and others.
}}</ref> 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>
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。'''<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==
== 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.
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.
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>
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
{{cite journal
| author = [[Chao Tang|Tang, C.]] and [[Per Bak|Bak, P.]]
| author = [[Chao Tang|Tang, C.]] and [[Per Bak|Bak, P.]]
| year = 1988
| year = 1988
| title = Critical exponents and scaling relations for self-organized critical phenomena
| title = Critical exponents and scaling relations for self-organized critical phenomena
| journal = [[Physical Review Letters]]
| journal = [[Physical Review Letters]]
| volume = 60
| volume = 60
| issue = 23
| issue = 23
| pages = 2347–2350
| pages = 2347–2350
| doi = 10.1103/PhysRevLett.60.2347
| doi = 10.1103/PhysRevLett.60.2347
| bibcode= 1988PhRvL..60.2347T
| bibcode= 1988PhRvL..60.2347T
| pmid=10038328
| pmid=10038328
}}
}}
</ref><ref name=Tang1988b>
</ref><ref name=Tang1988b>
{{cite journal
{{cite journal
| author = [[Chao Tang|Tang, C.]] and [[Per Bak|Bak, P.]]
| author = [[Chao Tang|Tang, C.]] and [[Per Bak|Bak, P.]]
| year = 1988
| year = 1988
| title = Mean field theory of self-organized critical phenomena
| title = Mean field theory of self-organized critical phenomena
| journal = [[Journal of Statistical Physics]]
| journal = [[Journal of Statistical Physics]]
| volume = 51
| volume = 51
| issue = 5–6
| issue = 5–6
| pages = 797–802
| pages = 797–802
| doi = 10.1007/BF01014884
| doi = 10.1007/BF01014884
| bibcode= 1988JSP....51..797T
| bibcode= 1988JSP....51..797T
| url = https://zenodo.org/record/1232502
| url = https://zenodo.org/record/1232502
| type = Submitted manuscript
| 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.
</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) ，以及研究出现 '''<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确定任意算法是否显示 '''<font color="#ff8000"> SOC</font>''' 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 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>
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
{{cite journal
|author1=Turcotte, D. L. |author2=Smalley, R. F., Jr. |author3=Solla, S. A. | year = 1985
|author1=Turcotte, D. L. |author2=Smalley, R. F., Jr. |author3=Solla, S. A. | year = 1985
| title = Collapse of loaded fractal trees
| title = Collapse of loaded fractal trees
| journal = Nature
| journal = Nature
| doi= 10.1038/313671a0
| doi= 10.1038/313671a0
| volume = 313
| volume = 313
| issue = 6004
| issue = 6004
| pages = 671–672|bibcode = 1985Natur.313..671T
| 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>
}}</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
{{cite journal
| pmid = 22815496
| pmid = 22815496
|date=Jul 2012
|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
|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
| volume = 32
| issue = 29
| issue = 29
| pages = 9817–23
| pages = 9817–23
| doi = 10.1523/JNEUROSCI.5990-11.2012
| doi = 10.1523/JNEUROSCI.5990-11.2012
| journal = Journal of Neuroscience
| journal = Journal of Neuroscience
| pmc=3553543
| 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.
}}</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. 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.<ref name=Hoffmann2018>
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>
{{cite journal
{{cite journal
| author = [[H. Hoffmann|Hoffmann, H.]] and [[D. W. Payton|Payton, D. W.]]
| author = [[H. Hoffmann|Hoffmann, H.]] and [[D. W. Payton|Payton, D. W.]]
| year = 2018
| year = 2018
| title = Optimization by Self-Organized Criticality
| title = Optimization by Self-Organized Criticality
| journal = [[Scientific Reports]]
| journal = [[Scientific Reports]]
| volume = 8
| volume = 8
| issue = 1
| issue = 1
| pages = 2358
| pages = 2358
| doi=10.1038/s41598-018-20275-7
| doi=10.1038/s41598-018-20275-7
| pmid = 29402956
| pmid = 29402956
| pmc = 5799203
| pmc = 5799203
| bibcode = 2018NatSR...8.2358H
| bibcode = 2018NatSR...8.2358H
}}</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 [[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。
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。
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.[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.<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>
{{cite journal
{{cite journal
| author = [[M. A. Moret|Moret, M. A.]] and [[G. Zebende|Zebende, G.]]
| author = [[M. A. Moret|Moret, M. A.]] and [[G. Zebende|Zebende, G.]]
| year = 2007
| year = 2007
| title = Amino acid hydrophobicity and accessible surface area
| title = Amino acid hydrophobicity and accessible surface area
| journal = [[Phys. Rev. E]]
| journal = [[Phys. Rev. E]]
| volume = 75
| volume = 75
| issue = 1
| issue = 1
| pages = 011920
| pages = 011920
| doi=10.1103/PhysRevE.75.011920
| doi=10.1103/PhysRevE.75.011920
| pmid = 17358197
| pmid = 17358197
| bibcode = 2007PhRvE..75a1920M
| bibcode = 2007PhRvE..75a1920M
}}</ref>
}}</ref>
These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.<ref name=Phillips2014>
These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.<ref name=Phillips2014>
{{cite journal
{{cite journal
| author = Phillips, J. C.
| author = Phillips, J. C.
| year = 2014
| year = 2014
| title = Fractals and self-organized criticality in proteins
| title = Fractals and self-organized criticality in proteins
| journal = Physica A
| journal = Physica A
| volume = 415
| volume = 415
| pages = 440–448
| pages = 440–448
| doi=10.1016/j.physa.2014.08.034
| doi=10.1016/j.physa.2014.08.034
| bibcode = 2014PhyA..415..440P
| bibcode = 2014PhyA..415..440P
| author-link = J. C. Phillips
| author-link = J. C. Phillips
}}</ref>
}}</ref>
'''<font color="#ff8000"> 无标度网络Scale-free networks</font>'''最近引起的兴奋为 '''<font color="#ff8000"> SOC</font>'''相关研究提出了一些有趣的新问题: 许多不同的 '''<font color="#ff8000"> SOC</font>'''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.<ref name=Bak1987/> 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.<ref name=Bak1987/> However,
it has been argued that this model would actually generate 1/f<sup>2</sup> noise rather than 1/f noise.<ref name=Jensen1989>
it has been argued that this model would actually generate 1/f<sup>2</sup> noise rather than 1/f noise.<ref name=Jensen1989>
{{cite journal
{{cite journal
| author = [[H. J. Jensen|Jensen, H. J.]], [[K. Christensen|Christensen, K.]] and [[H. C. Fogedby|Fogedby, H. C.]]
| author = [[H. J. Jensen|Jensen, H. J.]], [[K. Christensen|Christensen, K.]] and [[H. C. Fogedby|Fogedby, H. C.]]
| year = 1989
| year = 1989
| title = 1/f noise, distribution of lifetimes, and a pile of sand
| title = 1/f noise, distribution of lifetimes, and a pile of sand
| journal = [[Phys. Rev. B]]
| journal = [[Phys. Rev. B]]
| volume = 40
| volume = 40
| issue = 10
| issue = 10
| pages = 7425–7427
| pages = 7425–7427
| doi=10.1103/physrevb.40.7425
| doi=10.1103/physrevb.40.7425
| pmid = 9991162
| pmid = 9991162
|bibcode = 1989PhRvB..40.7425J }}
|bibcode = 1989PhRvB..40.7425J }}
</ref>
</ref>
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
generally produce 1/f<sup>a</sup> spectra, with a<2.<ref name=Laurson2005>
generally produce 1/f<sup>a</sup> spectra, with a<2. <ref name=Laurson2005>
{{cite journal |author1=Laurson, Lasse |author2=Alava, Mikko J. |author3=Zapperi, Stefano |title=Letter: Power spectra of self-organized critical sand piles |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=0511 |id=L001 |date=15 September 2005 }}</ref>
{{cite journal |author1=Laurson, Lasse |author2=Alava, Mikko J. |author3=Zapperi, Stefano |title=Letter: Power spectra of self-organized critical sand piles |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=0511 |id=L001 |date=15 September 2005 }}</ref>
Other simulation models were proposed later that could produce true 1/f noise,<ref name=Maslov1999>
Other simulation models were proposed later that could produce true 1/f noise,<ref name=Maslov1999>
{{cite journal
{{cite journal
| author = [[S. Maslov|Maslov, S.]], [[C. Tang|Tang, C.]] and [[Y. –C. Zhang|Zhang, Y. - C.]]
| author = [[S. Maslov|Maslov, S.]], [[C. Tang|Tang, C.]] and [[Y. –C. Zhang|Zhang, Y. - C.]]
| year = 1999
| year = 1999
| title = 1/f noise in Bak-Tang-Wiesenfeld models on narrow stripes
| title = 1/f noise in Bak-Tang-Wiesenfeld models on narrow stripes
| journal = [[Phys. Rev. Lett.]]
| journal = [[Phys. Rev. Lett.]]
| volume = 83
| volume = 83
| issue = 12
| issue = 12
| pages = 2449–2452
| pages = 2449–2452
| doi=10.1103/physrevlett.83.2449
| doi=10.1103/physrevlett.83.2449
|arxiv = cond-mat/9902074 |bibcode = 1999PhRvL..83.2449M }}
|arxiv = cond-mat/9902074 |bibcode = 1999PhRvL..83.2449M }}
</ref> and experimental sandpile models were observed to yield 1/f noise.<ref name=Frette1996>
</ref> and experimental sandpile models were observed to yield 1/f noise.<ref name=Frette1996>
{{cite journal
{{cite journal
| author = [[V.Frette|Frette, V.]], [[K. Christiansen|Christinasen, K.]], [[A. Malthe-Sørenssen|Malthe-Sørenssen, A.]], [[J. Feder|Feder, J]], [[T. Jøssang|Jøssang, T]] and [[P. Meakin|Meaken, P]]
| author = [[V.Frette|Frette, V.]], [[K. Christiansen|Christinasen, K.]], [[A. Malthe-Sørenssen|Malthe-Sørenssen, A.]], [[J. Feder|Feder, J]], [[T. Jøssang|Jøssang, T]] and [[P. Meakin|Meaken, P]]
| year = 1996
| year = 1996
| title = Avalanche dynamics in a pile of rice
| title = Avalanche dynamics in a pile of rice
| journal = [[Nature (journal)|Nature]]
| journal = [[Nature (journal)|Nature]]
| volume = 379
| volume = 379
| issue = 6560
| issue = 6560
| pages = 49–52
| pages = 49–52
| doi =10.1038/379049a0
| doi =10.1038/379049a0
| bibcode= 1996Natur.379...49F}}
| bibcode= 1996Natur.379...49F}}
</ref> In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon [[information theory]],<ref name=Dewar2003>
</ref> In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon [[information theory]]<ref name=Dewar2003>
{{cite journal
{{cite journal
| author = Dewar, R.
| author = Dewar, R.
| year = 2003
| year = 2003
| 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
| journal =J. Phys. A: Math. Gen.
| journal =J. Phys. A: Math. Gen.
| volume = 36
| volume = 36
| pages =631–641
| pages =631–641
| doi = 10.1088/0305-4470/36/3/303
| doi = 10.1088/0305-4470/36/3/303
| issue = 3
| issue = 3
|bibcode = 2003JPhA...36..631D|arxiv = cond-mat/0005382 | author-link = R Dewar
|bibcode = 2003JPhA...36..631D|arxiv = cond-mat/0005382 | author-link = R Dewar
}}</ref>
[[mean field theory]],<ref name=Vespignani1998>
}}</ref>,
[[mean field theory]]<ref name=Vespignani1998>
{{cite journal
{{cite journal
| author = [[Alessandro Vespignani|Vespignani, A.]], and [[Stefano Zapperi|Zapperi, S.]]
| author = [[Alessandro Vespignani|Vespignani, A.]], and [[Stefano Zapperi|Zapperi, S.]]
| year = 1998
| year = 1998
| title = How self-organized criticality works: a unified mean-field picture
| title = How self-organized criticality works: a unified mean-field picture
| journal =Phys. Rev. E
| journal =Phys. Rev. E
| volume = 57
| volume = 57
| pages =6345–6362
| pages =6345–6362
| doi = 10.1103/physreve.57.6345
| doi = 10.1103/physreve.57.6345
| issue = 6
| issue = 6
| bibcode = 1998PhRvE..57.6345V
| bibcode = 1998PhRvE..57.6345V
| arxiv = cond-mat/9709192
| arxiv = cond-mat/9709192
| hdl = 2047/d20002173
| hdl = 2047/d20002173
}}</ref>
the [[convergence of random variables]],<ref name=Kendal2015>
}}</ref>,
the [[convergence of random variables]]<ref name=Kendal2015>
{{cite journal
{{cite journal
| author = Kendal, WS
| author = Kendal, WS
| year = 2015
| year = 2015
| title = Self-organized criticality attributed to a central limit-like convergence effect
| title = Self-organized criticality attributed to a central limit-like convergence effect
| journal =Physica A
| journal =Physica A
| volume = 421
| volume = 421
| pages =141–150
| pages =141–150
| doi = 10.1016/j.physa.2014.11.035
| doi = 10.1016/j.physa.2014.11.035
|bibcode =2015PhyA..421..141K | author-link = Wayne Kendal
|bibcode =2015PhyA..421..141K | author-link = Wayne Kendal
}}</ref>
}}</ref>,
and cluster formation.<ref name=Hoffmann2018b>
and cluster formation.<ref name=Hoffmann2018b>
{{cite journal
{{cite journal
| author = Hoffmann, H.
| author = Hoffmann, H.
| year = 2018
| year = 2018
| title = Impact of Network Topology on Self-Organized Criticality
| title = Impact of Network Topology on Self-Organized Criticality
| journal = Phys. Rev. E
| journal = Phys. Rev. E
| volume = 97
| volume = 97
| pages =022313
| pages =022313
| pmid = 29548239
| pmid = 29548239
| doi = 10.1103/PhysRevE.97.022313
| doi = 10.1103/PhysRevE.97.022313
| issue = 2
| issue = 2
| bibcode =2018PhRvE..97b2313H
| bibcode =2018PhRvE..97b2313H
| author-link = Heiko Hoffmann
| author-link = Heiko Hoffmann
| doi-access = free
| doi-access = free
}}</ref> A continuous model of self-organised criticality is proposed by using [[tropical geometry]].<ref>{{Cite journal|last=Kalinin|first=N.|last2=Guzmán-Sáenz|first2=A.|last3=Prieto|first3=Y.|last4=Shkolnikov|first4=M.|last5=Kalinina|first5=V.|last6=Lupercio|first6=E.|date=2018-08-15|title=Self-organized criticality and pattern emergence through the lens of tropical geometry|journal=Proceedings of the National Academy of Sciences|volume=115|issue=35|language=en|pages=E8135–E8142|doi=10.1073/pnas.1805847115|issn=0027-8424|pmid=30111541|pmc=6126730|arxiv=1806.09153}}</ref>
}}</ref> A continuous model of self-organised criticality is proposed by using [[tropical geometry]].<ref>{{Cite journal|last=Kalinin|first=N.|last2=Guzmán-Sáenz|first2=A.|last3=Prieto|first3=Y.|last4=Shkolnikov|first4=M.|last5=Kalinina|first5=V.|last6=Lupercio|first6=E.|date=2018-08-15|title=Self-organized criticality and pattern emergence through the lens of tropical geometry|journal=Proceedings of the National Academy of Sciences|volume=115|issue=35|language=en|pages=E8135–E8142|doi=10.1073/pnas.1805847115|issn=0027-8424|pmid=30111541|pmc=6126730|arxiv=1806.09153}}</ref>
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'''<font color="#ff8000"> 自组织临界Self-organised criticality</font>'''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 ==
== Examples of self-organized critical dynamics自组织临界动力学的例子 ==