“自组织临界控制”的版本间的差异

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在应用物理学中, 自组织临界控制Controlling self-organized criticality的概念是指对自组织系统能量耗散过程的控制。这种控制的目的是减少自组织系统发生能量耗散爆发(通常称为雪崩)的概率和规模。一个自组织临界系统中转变到较低的能量状态时的能量耗散对社会来说可能是代价高昂的,因为它取决于各种规模的雪崩,这些雪崩的规模通常遵循 幂律分布Power law distribution,而大规模雪崩具有伤害性和破坏性。
 
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<ref name=cajand10a>{{cite journal
In [[applied physics]], the concept of '''controlling self-organized criticality''' refers to the control of processes by which a [[self-organized]] system dissipates [[energy]]. The objective of the control is to reduce the probability of occurrence of and size of [[energy dissipation]] bursts, often called ''avalanches'', of self-organized systemsDissipation of energy in a [[self-organized criticality|self-organized critical]] system into a lower energy state can be costly for society, since it depends on avalanches of all sizes usually following a kind of [[power law]] distribution and large avalanches can be damaging and disruptive.
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| author = D. O. Cajueiro and R. F. S. Andrade
 +
| year = 2010
 +
| title = Controlling self-organized criticality in sandpile models
 +
| journal = Physical Review E
 +
| volume = 81
 +
| pages = 015102#R
 +
| doi=10.1103/physreve.81.015102
 +
|arxiv = 1305.6648 |bibcode = 2010PhRvE..81a5102C }}</ref><ref name=cajand10b>{{cite journal
 +
| author = D. O. Cajueiro and R. F. S. Andrade
 +
| year = 2010
 +
  | title = Controlling self-organized criticality in complex networks
 +
| journal = European Physical Journal B
 +
| volume = 77
 +
| pages = 291–296
 +
| doi=10.1140/epjb/e2010-00229-8
 +
|arxiv = 1305.6656 |bibcode = 2010EPJB...77..291C }}</ref><ref name=cajand10c>{{cite journal
 +
| author = D. O. Cajueiro and R. F. S. Andrade
 +
| year = 2010
 +
| title = Dynamical programming approach for controlling the directed Abelian Dhar-Ramaswamy model
 +
| journal = Physical Review E
 +
| volume = 82
 +
| pages = 031108
 +
| doi=10.1103/physreve.82.031108
 +
|arxiv = 1305.6668 |bibcode = 2010PhRvE..82c1108C }}</ref>
  
In applied physics, the concept of controlling self-organized criticality refers to the control of processes by which a self-organized system dissipates energy. The objective of the control is to reduce the probability of occurrence of and size of energy dissipation bursts, often called avalanches, of self-organized systems.  Dissipation of energy in a self-organized critical system into a lower energy state can be costly for society, since it depends on avalanches of all sizes usually following a kind of power law distribution and large avalanches can be damaging and disruptive.
 
  
在应用物理学中,<font color="#ff8000"> 自组织临界控制Controlling self-organized criticality</font>的概念是指对自组织系统能量耗散过程的控制。这种控制的目的是减少自组织系统发生能量耗散爆发(通常称为雪崩)的概率和规模。一个自组织临界系统中转变到较低的能量状态时的能量耗散对社会来说可能是代价高昂的,因为它取决于各种规模的雪崩,这些雪崩的规模通常遵循<font color="#ff8000"> 幂律分布Power law distribution</font>,而大规模雪崩具有伤害性和破坏性。
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==计划==
 
 
 
 
 
 
 
 
 
== Schemes ==
 
 
 
== Schemes ==
 
 
 
计划
 
 
 
 
 
 
 
 
 
 
 
Several strategies have been proposed to deal with the issue of controlling self-organized criticality:
 
 
 
Several strategies have been proposed to deal with the issue of controlling self-organized criticality:
 
  
 
已经有一些策略被提出用于处理自组织临界状态的控制问题:
 
已经有一些策略被提出用于处理自组织临界状态的控制问题:
  
 
 
 
 
#''The design of controlled avalanches.'' [[Daniel O. Cajueiro]] and [[Roberto F. S. Andrade]] show that if well-formulated small and medium avalanches are exogenously triggered in the system, the energy of the system is released in a way that large avalanches are rarer.
 
 
The design of controlled avalanches. Daniel O. Cajueiro and Roberto F. S. Andrade show that if well-formulated small and medium avalanches are exogenously triggered in the system, the energy of the system is released in a way that large avalanches are rarer.
 
 
可控雪崩的设计。Daniel O.Cajueiro和Roberto F.S.Andrade表明,如果系统中成熟的中小规模雪崩是靠外部触发的,那么系统的能量会以不同于大规模雪崩的方式释放出来。 <ref name="cajand10a" /><ref name="cajand10b" /><ref name="cajand10c" />
 
可控雪崩的设计。Daniel O.Cajueiro和Roberto F.S.Andrade表明,如果系统中成熟的中小规模雪崩是靠外部触发的,那么系统的能量会以不同于大规模雪崩的方式释放出来。 <ref name="cajand10a" /><ref name="cajand10b" /><ref name="cajand10c" />
  
#'' The modification of the degree of interdependence of the network where the avalanche spreads.'' [[Charles D. Brummitt]], [[Raissa M. D'Souza]] and [[E. A. Leicht]] show that the dynamics of self-organized critical systems on [[complex network]]s depend on connectivity of the complex network. They find that while some connectivity is beneficial (since it suppresses the largest cascades in the system), too much connectivity gives space for the development of very large cascades and increases the size of capacity of the system.{{cite journal
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#''雪崩蔓延地区网络相互依赖程度的修正:Charles D. Brummitt,Raissa M. D'Souza和 E. A. Leicht 证明了复杂网络上自组织临界系统的动力学依赖于复杂网络的连通性。他们发现,虽然有些连通性是有益的(因为它压制了系统中最大的级联) ,但过多的连通性为大型级联的发展提供了空间,并增加了系统的容量。<ref name=cajand10a>{{cite journal
 
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  | author = D. O. Cajueiro and R. F. S. Andrade
The modification of the degree of interdependence of the network where the avalanche spreads. Charles D. Brummitt, Raissa M. D'Souza and E. A. Leicht show that the dynamics of self-organized critical systems on complex networks depend on connectivity of the complex network. They find that while some connectivity is beneficial (since it suppresses the largest cascades in the system), too much connectivity gives space for the development of very large cascades and increases the size of capacity of the system.<ref name=brum12>{{cite journal
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  | year = 2010
 
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  | title = Controlling self-organized criticality in sandpile models
雪崩蔓延地区网络相互依赖程度的修正:Charles D. Brummitt,Raissa M. D'Souza和 E. A. Leicht 证明了复杂网络上自组织临界系统的动力学依赖于复杂网络的连通性。他们发现,虽然有些连通性是有益的(因为它压制了系统中最大的级联) ,但过多的连通性为大型级联的发展提供了空间,并增加了系统的容量。<ref name=brum12>
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  | journal = Physical Review E
 
+
  | volume = 81
  | author = C. D. Brummitt, R. M. D'Souza and E. A. Leicht
+
  | pages = 015102#R
 
+
  | doi=10.1103/physreve.81.015102
| author = C. D. Brummitt, R. M. D'Souza and E. A. Leicht
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|arxiv = 1305.6648 |bibcode = 2010PhRvE..81a5102C }}</ref><ref name=cajand10b>{{cite journal
 
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  | author = D. O. Cajueiro and R. F. S. Andrade
作者 c. d. Brummitt,r. m. d’ souza and e. a. Leicht
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  | year = 2010
 
+
  | title = Controlling self-organized criticality in complex networks
  | year = 2012
+
  | journal = European Physical Journal B
 
+
| volume = 77
  | year = 2012
+
  | pages = 291–296
 
+
  | doi=10.1140/epjb/e2010-00229-8
2012年
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|arxiv = 1305.6656 |bibcode = 2010EPJB...77..291C }}</ref><ref name=cajand10c>{{cite journal
 
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| author = D. O. Cajueiro and R. F. S. Andrade
  | title = Suppressing cascades of load in interdependent networks
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| year = 2010
 
+
| title = Dynamical programming approach for controlling the directed Abelian Dhar-Ramaswamy model
  | title = Suppressing cascades of load in interdependent networks
+
  | journal = Physical Review E
 
+
  | volume = 82
在相互依赖的网络中抑制负载级联
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  | pages = 031108
 
+
  | doi=10.1103/physreve.82.031108
  | journal = PNAS
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|arxiv = 1305.6668 |bibcode = 2010PhRvE..82c1108C }}</ref>
 
 
  | journal = PNAS
 
 
 
美国科学院院刊
 
 
 
  | volume = 109
 
 
 
  | volume = 109
 
 
 
第109卷
 
 
 
  | pages = E680–E689
 
 
 
  | pages = E680–E689
 
 
 
| 页 E680-E689
 
 
 
  | doi=10.1073/pnas.1110586109
 
 
 
  | doi=10.1073/pnas.1110586109
 
 
 
10.1073 / pnas. 1110586109
 
 
 
|arxiv = 1106.4499 |bibcode = 2012PNAS..109E.680B
 
 
 
|arxiv = 1106.4499 |bibcode = 2012PNAS..109E.680B
 
 
 
|arxiv = 1106.4499 |bibcode = 2012PNAS..109E.680B
 
 
 
  | pmid=22355144
 
 
 
  | pmid=22355144
 
 
 
22355144
 
 
 
  | pmc=3311366}}</ref>
 
 
 
  | pmc=3311366}}</ref>
 
 
 
3311366} / ref
 
 
 
#'' The modification of the deposition process of the self-organized system.''  [[Pierre-Andre Noel]], Charles D. Brummitt and Raissa M. D'Souza show that it is possible to control the self-organized system by modifying the natural deposition process of the self-organized system adjusting the place where the avalanche starts.<ref name=noel13>{{cite journal
 
 
 
The modification of the deposition process of the self-organized system. Pierre-Andre Noel, Charles D. Brummitt and Raissa M. D'Souza show that it is possible to control the self-organized system by modifying the natural deposition process of the self-organized system adjusting the place where the avalanche starts.<ref name=noel13>{{cite journal
 
 
 
自组织系统沉积工艺的改进。 Pierre-Andre Noel、Charles D. Brummitt和Raissa M. D'Souza 指出,通过改变自组织系统的自然沉积过程来调整雪崩开始的位置,有可能实现对自组织系统的控制。
 
  
 +
#''自组织系统沉积工艺的改进。 Pierre-Andre Noel、Charles D. Brummitt和Raissa M. D'Souza 指出,通过改变自组织系统的自然沉积过程来调整雪崩开始的位置,有可能实现对自组织系统的控制。
 +
<ref name=noel13>{{cite journal
 
  | author = P. A. Noel, C. D. Brummitt and R. M. D'Souza
 
  | author = P. A. Noel, C. D. Brummitt and R. M. D'Souza
 
| author = P. A. Noel, C. D. Brummitt and R. M. D'Souza
 
 
作者 p. a. Noel,c. d. Brummitt and r. m. d’ souza
 
 
 
  | year = 2013
 
  | year = 2013
 
| year = 2013
 
 
2013年
 
 
| title = Controlling self-organized criticality on networks using models that self-organize
 
 
 
  | title = Controlling self-organized criticality on networks using models that self-organize
 
  | title = Controlling self-organized criticality on networks using models that self-organize
 
| 标题:使用自组织模型的网络上的自组织临界控制
 
 
| journal = Physical Review Letters
 
 
 
  | journal = Physical Review Letters
 
  | journal = Physical Review Letters
 
物理评论快报
 
 
 
  | volume = 111
 
  | volume = 111
 
| volume = 111
 
 
第111卷
 
 
 
  | pages = 078701
 
  | pages = 078701
 
| pages = 078701
 
 
078701页
 
 
| doi=10.1103/physrevlett.111.078701
 
 
 
  | doi=10.1103/physrevlett.111.078701
 
  | doi=10.1103/physrevlett.111.078701
 
10.1103 / physrvlett. 111.078701
 
 
|arxiv = 1305.1877 |bibcode = 2013PhRvL.111g8701N
 
 
 
|arxiv = 1305.1877 |bibcode = 2013PhRvL.111g8701N
 
|arxiv = 1305.1877 |bibcode = 2013PhRvL.111g8701N
 
|arxiv = 1305.1877 |bibcode = 2013PhRvL.111g8701N
 
 
 
  | pmid=23992086}}</ref>
 
  | pmid=23992086}}</ref>
  
| pmid=23992086}}</ref>
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#''动态修改级联故障的局部阈值;在一个电力传输网络模型中, H .霍夫曼 和 D. W. 佩顿 证明,要么随机升级线路(类似于预防性维护) ,要么将破损线路更新到到随机破损阈值,都可以抑制自组织临界性。
 
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<ref name=hh2014>{{cite journal
23992086} / ref
 
 
 
#'' Dynamically modifying the local thresholds of cascading failures.'' In a model of an electric transmission network, Heiko Hoffmann and David W. Payton demonstrated that either randomly upgrading lines (sort of like preventive maintenance) or upgrading broken lines to a random breakage threshold suppresses self-organized criticality.{{cite journal
 
 
 
Dynamically modifying the local thresholds of cascading failures. In a model of an electric transmission network, Heiko Hoffmann and  David W. Payton  demonstrated that either randomly upgrading lines (sort of like preventive maintenance) or upgrading broken lines to a random breakage threshold suppresses self-organized criticality.<ref name=hh2014>{{cite journal
 
 
 
动态修改级联故障的局部阈值;在一个电力传输网络模型中, H .霍夫曼 和 D. W. 佩顿 证明,要么随机升级线路(类似于预防性维护) ,要么将破损线路更新到到随机破损阈值,都可以抑制自组织临界性。 <ref name=hh2014>
 
  --~~~人名第一次出现用中文 后续用英文
 
2014{ cite journal
 
 
 
| author = H. Hoffmann and D. W. Payton
 
 
 
 
  | author = H. Hoffmann and D. W. Payton
 
  | author = H. Hoffmann and D. W. Payton
 
作者 H. Hoffmann 和 D. W. Payton
 
 
 
  | year = 2014
 
  | year = 2014
 
| year = 2014
 
 
2014年
 
 
| title = Suppressing cascades in a self-organized-critical model with non-contiguous spread of failures
 
 
 
  | title = Suppressing cascades in a self-organized-critical model with non-contiguous spread of failures
 
  | title = Suppressing cascades in a self-organized-critical model with non-contiguous spread of failures
 
在具有非连续分布故障的自组织临界模型中抑制级联
 
 
| journal = Chaos, Solitons and Fractals
 
 
 
  | journal = Chaos, Solitons and Fractals
 
  | journal = Chaos, Solitons and Fractals
 
混沌,孤子和分形
 
 
 
  | volume = 67
 
  | volume = 67
 
| volume = 67
 
 
第67卷
 
 
| pages = 87–93
 
 
 
  | pages = 87–93
 
  | pages = 87–93
 
第87-93页
 
 
 
  | doi=10.1016/j.chaos.2014.06.011
 
  | doi=10.1016/j.chaos.2014.06.011
 
| doi=10.1016/j.chaos.2014.06.011
 
 
10.1016 / j.chaos. 2014.06.011
 
 
| bibcode = 2014CSF....67...87H
 
 
 
| bibcode = 2014CSF....67...87H
 
| bibcode = 2014CSF....67...87H
 +
}}</ref>
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显然,这些策略破坏了大型临界集群的自我组织。在这里,一个临界集群是一组即将发生故障的传输线,一旦触发条件,这些传输线就会完全崩溃。
  
| bibcode 2014CSF... 67... 87H
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==应用==
  
Apparently, these strategies undermine the self-organization of large critical clusters. Here, a critical cluster is a collection of transmission lines that are near the failure threshold and that collapse entirely if triggered.
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在自然界或社会中发生的一些事件中,这些控制观念可能有助于避免它们:<ref name="cajand10a" /><ref name="cajand10b" /><ref name="cajand10c" /><ref name="brum12" /><ref name="noel13" /><ref name="hh2014" />
  
}}</ref> Apparently, these strategies undermine the self-organization of large critical clusters. Here, a critical cluster is a collection of transmission lines that are near the failure threshold and that collapse entirely if triggered.
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#由水坝、水库或相互连接的山谷系统造成的洪水。
  
  }}</ref>显然,这些策略破坏了大型临界集群的自我组织。在这里,一个临界集群是一组即将发生故障的传输线,一旦触发条件,这些传输线就会完全崩溃。
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#在雪山上发生的雪崩。
 +
   
 +
#易受闪电或火柴照明影响的地区发生的森林火灾。
  
 +
#电网中发生的减载级联(断电的一种形式)。OPA模型可以用来研究不同的临界控制技术。
  
 +
#互联网交换结构中的级联故障。
  
 +
#缺血级联反应,在血液供应不足的时刻释放毒素的一系列生化反应。
  
 +
#金融系统的系统性风险。
  
== Applications ==
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#核能系统中的游离。
 
 
== Applications ==
 
 
 
应用
 
 
 
 
 
 
 
 
 
 
 
There are several events that arise in nature or society where these ideas of control may help to avoid them:
 
 
 
There are several events that arise in nature or society where these ideas of control may help to avoid them:
 
 
 
在自然界或社会中发生的一些事件中,这些控制观念可能有助于避免它们:<ref name="cajand10a" /><ref name="cajand10b" /><ref name="cajand10c" /><ref name="brum12" /><ref name="noel13" /><ref name="hh2014" />
 
 
 
 
 
 
 
 
 
 
 
# [[Flood]] caused by systems of dams and reservoirs or interconnected valleys.
 
 
 
Flood caused by systems of dams and reservoirs or interconnected valleys.
 
 
 
由水坝、水库或相互连接的山谷系统造成的洪水。
 
 
 
# Snow avalanches that take place in snow hills.
 
 
 
Snow avalanches that take place in snow hills.
 
 
 
在雪山上发生的雪崩。
 
 
 
# Forest fires in areas susceptible to a lightning bolt or a match lighting.
 
 
 
Forest fires in areas susceptible to a lightning bolt or a match lighting.
 
 
 
易受闪电或火柴照明影响的地区发生的森林火灾。
 
 
 
# [[Cascading failure|Cascades of load shedding]] that take place in power grids (a type of [[power outage]]). The [[power outage#OPA model|OPA model]] is used to study different techniques for criticality control.
 
 
 
Cascades of load shedding that take place in power grids (a type of power outage). The OPA model is used to study different techniques for criticality control.
 
 
 
电网中发生的减载级联(断电的一种形式)。OPA模型可以用来研究不同的临界控制技术。
 
 
 
# [[Cascading failure]] in the internet switching fabric.
 
 
 
Cascading failure in the internet switching fabric.
 
 
 
互联网交换结构中的级联故障。
 
 
 
# [[Ischemic cascade]]s, a series of biochemical reactions releasing toxins during moments of inadequate blood supply.
 
 
 
Ischemic cascades, a series of biochemical reactions releasing toxins during moments of inadequate blood supply.
 
 
 
缺血级联反应,在血液供应不足的时刻释放毒素的一系列生化反应。
 
 
 
# [[Systemic risk]] in financial systems.
 
 
 
Systemic risk in financial systems.
 
 
 
金融系统的系统性风险。
 
 
 
# [[Criticality accident|Excursions in nuclear energy systems]].
 
 
 
Excursions in nuclear energy systems.
 
 
 
核能系统中的游离。
 
 
 
 
 
 
 
 
 
 
 
The failure cascades in electrical transmission and financial sectors occur because economic forces cause these systems to operate near a critical point, where avalanches of indeterminate size become possible.
 
 
 
The failure cascades in electrical transmission and financial sectors occur because economic forces cause these systems to operate near a critical point, where avalanches of indeterminate size become possible.
 
  
 
发生电力传输和金融部门的故障级联是因为经济力量使这些系统在临界点附近运行,在那里可能发生规模不确定的雪崩。
 
发生电力传输和金融部门的故障级联是因为经济力量使这些系统在临界点附近运行,在那里可能发生规模不确定的雪崩。
  
 +
==参见==
  
 +
*[[阿贝尔沙堆模型 Abelian sandpile model]]
  
 +
*[[复杂网络 Complex network]]s
  
 +
*[[自组织临界 Self-organized criticality]]
  
== See also ==
+
==参考资料==
 
 
== See also ==
 
 
 
参见
 
 
 
 
 
 
 
 
 
 
 
*[[Abelian sandpile model]]
 
 
 
阿贝尔沙堆模型
 
 
 
*[[Complex network]]s
 
复杂网络
 
 
 
 
 
*[[Self-organized criticality]]
 
 
 
自组织临界
 
 
 
 
 
 
 
 
 
 
 
== References ==
 
 
 
== References ==
 
 
 
参考资料
 
 
 
 
 
 
 
 
 
  
 
{{reflist}}
 
{{reflist}}
  
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----
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本中文词条由[[用户:Henry|Henry豆]]翻译,[[用户:打豆豆|打豆豆]]编辑,[[用户:Fernando|Fernando]审校,欢迎在讨论页面留言。
  
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'''本词条内容源自wikipedia及公开资料,遵守 CC3.0协议。'''
  
 
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[[Category:应用和跨学科物理学]]
 
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[[Category:控制理论]]
 
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[[Category:混沌理论]]
 
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[[Category:自我组织]]
[[Category:Applied and interdisciplinary physics]]
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[[Category:关键现象]]
 
 
Category:Applied and interdisciplinary physics
 
 
 
类别: 应用和跨学科物理学
 
 
 
[[Category:Control theory]]
 
 
 
Category:Control theory
 
 
 
范畴: 控制理论
 
 
 
[[Category:Chaos theory]]
 
 
 
Category:Chaos theory
 
 
 
范畴: 混沌理论
 
 
 
[[Category:Self-organization]]
 
 
 
Category:Self-organization
 
 
 
类别: 自我组织
 
 
 
[[Category:Critical phenomena]]
 
 
 
Category:Critical phenomena
 
 
 
范畴: 关键现象
 
 
 
<noinclude>
 
 
 
<small>This page was moved from [[wikipedia:en:Self-organized criticality control]]. Its edit history can be viewed at [[自组织临界控制/edithistory]]</small></noinclude>
 
 
 
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2020年12月12日 (六) 21:59的版本

在应用物理学中, 自组织临界控制Controlling self-organized criticality的概念是指对自组织系统能量耗散过程的控制。这种控制的目的是减少自组织系统发生能量耗散爆发(通常称为雪崩)的概率和规模。一个自组织临界系统中转变到较低的能量状态时的能量耗散对社会来说可能是代价高昂的,因为它取决于各种规模的雪崩,这些雪崩的规模通常遵循 幂律分布Power law distribution,而大规模雪崩具有伤害性和破坏性。 [1][2][3]


计划

已经有一些策略被提出用于处理自组织临界状态的控制问题:

可控雪崩的设计。Daniel O.Cajueiro和Roberto F.S.Andrade表明,如果系统中成熟的中小规模雪崩是靠外部触发的,那么系统的能量会以不同于大规模雪崩的方式释放出来。 [1][2][3]

  1. 雪崩蔓延地区网络相互依赖程度的修正:Charles D. Brummitt,Raissa M. D'Souza和 E. A. Leicht 证明了复杂网络上自组织临界系统的动力学依赖于复杂网络的连通性。他们发现,虽然有些连通性是有益的(因为它压制了系统中最大的级联) ,但过多的连通性为大型级联的发展提供了空间,并增加了系统的容量。[1][2][3]
  1. 自组织系统沉积工艺的改进。 Pierre-Andre Noel、Charles D. Brummitt和Raissa M. D'Souza 指出,通过改变自组织系统的自然沉积过程来调整雪崩开始的位置,有可能实现对自组织系统的控制。

[4]

  1. 动态修改级联故障的局部阈值;在一个电力传输网络模型中, H .霍夫曼 和 D. W. 佩顿 证明,要么随机升级线路(类似于预防性维护) ,要么将破损线路更新到到随机破损阈值,都可以抑制自组织临界性。

[5] 显然,这些策略破坏了大型临界集群的自我组织。在这里,一个临界集群是一组即将发生故障的传输线,一旦触发条件,这些传输线就会完全崩溃。

应用

在自然界或社会中发生的一些事件中,这些控制观念可能有助于避免它们:[1][2][3][6][4][5]

  1. 由水坝、水库或相互连接的山谷系统造成的洪水。
  1. 在雪山上发生的雪崩。
  1. 易受闪电或火柴照明影响的地区发生的森林火灾。
  1. 电网中发生的减载级联(断电的一种形式)。OPA模型可以用来研究不同的临界控制技术。
  1. 互联网交换结构中的级联故障。
  1. 缺血级联反应,在血液供应不足的时刻释放毒素的一系列生化反应。
  1. 金融系统的系统性风险。
  1. 核能系统中的游离。

发生电力传输和金融部门的故障级联是因为经济力量使这些系统在临界点附近运行,在那里可能发生规模不确定的雪崩。

参见

参考资料

  1. 1.0 1.1 1.2 1.3 D. O. Cajueiro and R. F. S. Andrade (2010). "Controlling self-organized criticality in sandpile models". Physical Review E. 81: 015102#R. arXiv:1305.6648. Bibcode:2010PhRvE..81a5102C. doi:10.1103/physreve.81.015102.
  2. 2.0 2.1 2.2 2.3 D. O. Cajueiro and R. F. S. Andrade (2010). "Controlling self-organized criticality in complex networks". European Physical Journal B. 77: 291–296. arXiv:1305.6656. Bibcode:2010EPJB...77..291C. doi:10.1140/epjb/e2010-00229-8.
  3. 3.0 3.1 3.2 3.3 D. O. Cajueiro and R. F. S. Andrade (2010). "Dynamical programming approach for controlling the directed Abelian Dhar-Ramaswamy model". Physical Review E. 82: 031108. arXiv:1305.6668. Bibcode:2010PhRvE..82c1108C. doi:10.1103/physreve.82.031108.
  4. 4.0 4.1 P. A. Noel, C. D. Brummitt and R. M. D'Souza (2013). "Controlling self-organized criticality on networks using models that self-organize". Physical Review Letters. 111: 078701. arXiv:1305.1877. Bibcode:2013PhRvL.111g8701N. doi:10.1103/physrevlett.111.078701. PMID 23992086.
  5. 5.0 5.1 H. Hoffmann and D. W. Payton (2014). "Suppressing cascades in a self-organized-critical model with non-contiguous spread of failures". Chaos, Solitons and Fractals. 67: 87–93. Bibcode:2014CSF....67...87H. doi:10.1016/j.chaos.2014.06.011.
  6. 引用错误:无效<ref>标签;未给name属性为brum12的引用提供文字

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