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随机序列由Tweedie分布控制，并通过展开箱的方法进行评估，在方差-均值幂律和幂律自相关之间表现出双条件关系。'''Wiener–Khinchin Theorem 维纳-辛钦定理'''进一步表明，在这些条件下，对于任何具有方差-均值幂律的序列，也会出现1/f噪声<ref name="Kendal2011" />。

随机序列由Tweedie分布控制，并通过展开箱的方法进行评估，在方差-均值幂律和幂律自相关之间表现出双条件关系。'''Wiener–Khinchin Theorem 维纳-辛钦定理'''进一步表明，在这些条件下，对于任何具有方差-均值幂律的序列，也会出现1/f噪声<ref name="Kendal2011" />。

The [[Tweedie distributions|'''Tweedie convergence theorem''']] provides a hypothetical explanation for the wide manifestation of fluctuation scaling and ''1/f'' noise.<ref name="Jørgensen1994">{{cite journal |last1=Jørgensen |first1=B. |last2=Martinez |first2=J. R. |last3=Tsao |first3=M. |year=1994 |title=Asymptotic behaviour of the variance function |journal=[[Scandinavian Journal of Statistics|Scand J Statist]] |volume=21 |issue=3 |pages=223–243 |jstor=4616314 }}</ref> It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a [[natural exponential family|variance function]] that comes within the [[Attractor|domain of attraction]] of a Tweedie model.  Almost all distribution functions with finite [[cumulant|cumulant generating functions]] qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form.  Hence many probability distributions have variance functions that express this [[Asymptotic expansion|asymptotic behavior]], and  the Tweedie distributions become foci of convergence for a wide range of data types.<ref name="Kendal2011" />

The [[Tweedie distributions|'''Tweedie convergence theorem''']] provides a hypothetical explanation for the wide manifestation of fluctuation scaling and ''1/f'' noise.<ref name="Jørgensen1994">Jørgensen, B.; Martinez, J. R.; Tsao, M. (1994). "Asymptotic behaviour of the variance function". ''Scand J Statist''. '''21''' (3): 223–243. JSTOR 4616314.</ref> It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a [[natural exponential family|variance function]] that comes within the [[Attractor|domain of attraction]] of a Tweedie model.  Almost all distribution functions with finite [[cumulant|cumulant generating functions]] qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form.  Hence many probability distributions have variance functions that express this [[Asymptotic expansion|asymptotic behavior]], and  the Tweedie distributions become foci of convergence for a wide range of data types.<ref name="Kendal2011" />

'''Tweedie Convergence Theorem Tweedie 收敛定理'''为涨落标度和1/f噪声的广泛出现提供了一个假设性解释。本质上，它要求任何一个可以渐近地显示方差-均值幂律的指数弥散模型，需要在Tweedie模型的吸引域内表达一个方差函数。几乎所有具有有限累积母函数的分布函数都符合指数弥散模型，而大多数指数弥散模型都表现出这种形式的方差函数。因此，许多概率分布都有表达这种渐近行为的方差函数，而Tweedie分布成为了不同数据类型收敛的焦点。

'''Tweedie Convergence Theorem Tweedie 收敛定理'''为涨落标度和1/f噪声的广泛出现提供了一个假设性解释<ref name="Jørgensen1994" />。本质上，它要求任何一个可以渐近地显示方差-均值幂律的指数弥散模型，需要在Tweedie模型的吸引域内表达一个方差函数。几乎所有具有有限累积母函数的分布函数都符合指数弥散模型，而大多数指数弥散模型都表现出这种形式的方差函数。因此，许多概率分布都有表达这种渐近行为的方差函数，而Tweedie分布成为了不同数据类型收敛的焦点<ref name="Kendal2011" />。

Much as the [[central limit theorem]] requires certain kinds of random variables to have as a focus of convergence the [[normal distribution|Gaussian distribution]] and express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.<ref name="Kendal2011" />

Much as the [[central limit theorem]] requires certain kinds of random variables to have as a focus of convergence the [[normal distribution|Gaussian distribution]] and express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.<ref name="Kendal2011" />

正如中心极限定理要求某些类型的随机变量以高斯分布为收敛焦点并表示白噪声一样，Tweedie收敛定理要求某些非高斯随机变量来表达1/f噪声和涨落标度。

正如中心极限定理要求某些类型的随机变量以高斯分布为收敛焦点并表示白噪声一样，Tweedie收敛定理要求某些非高斯随机变量来表达1/f噪声和涨落标度<ref name="Kendal2011" />。

===Cosmology 宇宙学===<!-- This section is linked from [[Cosmic inflation]] -->

===Cosmology 宇宙学===<!-- This section is linked from [[Cosmic inflation]] -->

In [[physical cosmology]], the power spectrum of the spatial distribution of the [[cosmic microwave background]] is near to being a scale-invariant function. Although in mathematics this means that the spectrum is a power-law, in cosmology the term "scale-invariant" indicates that the amplitude, {{math|''P''(''k'')}}, of [[primordial fluctuations]] as a function of [[wave number]], {{mvar|k}}, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of [[cosmic inflation]].

In [[physical cosmology]], the power spectrum of the spatial distribution of the [[cosmic microwave background]] is near to being a scale-invariant function. Although in mathematics this means that the spectrum is a power-law, in cosmology the term "scale-invariant" indicates that the amplitude, {{math|''P''(''k'')}}, of [[primordial fluctuations]] as a function of [[wave number]], {{mvar|k}}, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of [[cosmic inflation]].

在'''Physical Cosmology 宇宙物理学'''，'''Cosmic Microwave Background 宇宙微波背景'''的空间分布功率频谱近似于标度不变函数。尽管在数学上这意味着该频谱服从幂律，但在宇宙学中“标度不变”一词表明，'''Primordial Fluctuations 原始涨落'''的振幅{{math|''P''(''k'')}}，作为波数{{mvar|k}}的函数，是近似常数，也就是一个平谱。这种模式与'''Cosmic Inflation 宇宙膨胀论'''的主张是一致的。

在'''Physical Cosmology 宇宙物理学'''，'''Cosmic Microwave Background 宇宙微波背景'''的空间分布功率频谱近似于标度不变函数。尽管在数学上这意味着该频谱服从幂律，但在宇宙学中“标度不变”一词表明，'''Primordial Fluctuations 原始涨落'''的振幅{{math|''P''(''k'')}}，作为波数{{mvar|k}}的函数，是近似常数，也就是一个平谱。这种模式与'''Cosmic Inflation 宇宙膨胀论'''的主张是一致的。

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will '''not''' be scale-invariant, and in such cases the symmetry is said to be [[spontaneously broken]].

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will '''not''' be scale-invariant, and in such cases the symmetry is said to be [[spontaneously broken]].

我们注意到这个条件限制性很强。一般来说，即使是标度不变场方程的解也不是标度不变的，在这种情况下，对称性出现'''Spontaneously Broken 自发破缺'''。

我们注意到这个条件限制性很强。一般来说，即使是标度不变场方程的解也不是标度不变的，在这种情况下，对称性出现'''Spontaneously Broken 自发破缺'''。

量子场论(QFT)的标度依赖性的特征是其耦合参数依赖于给定物理过程的能量标度。这种能量依赖由重正化群描述，并编码在理论的'''Beta-function β函数'''中。

量子场论(QFT)的标度依赖性的特征是其耦合参数依赖于给定物理过程的能量标度。这种能量依赖由重正化群描述，并编码在理论的'''Beta-function β函数'''中。

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as [[Renormalization group|fixed points]] of the corresponding renormalization group flow.<ref>[[Jean Zinn-Justin|J. Zinn-Justin]] (2010) Scholarpedia article [http://www.scholarpedia.org/article/Critical_Phenomena:_field_theoretical_approach "Critical Phenomena: field theoretical approach"].</ref>

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as [[Renormalization group|fixed points]] of the corresponding renormalization group flow.<ref name=":0">J. Zinn-Justin (2010) Scholarpedia article "Critical Phenomena: field theoretical approach".</ref>

===Quantum electrodynamics 量子电动力学===

===Quantum electrodynamics 量子电动力学===

===Scale and conformal anomalies 标度与共形异常===

===Scale and conformal anomalies 标度与共形异常===

The φ⁴ theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be [[conformal anomaly|anomalous]]. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called [[Inflation (cosmology)|cosmic inflation]], as long as the theory can be studied through [[perturbation theory]].<ref>{{cite journal|last=Salvio, Strumia|title=Agravity|journal=JHEP  |volume=2014 |issue=6|pages=080|date=2014-03-17|url=http://inspirehep.net/record/1286134|arxiv = 1403.4226|bibcode = 2014JHEP...06..080S|doi=10.1007/JHEP06(2014)080}}</ref>

The φ⁴ theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be [[conformal anomaly|anomalous]]. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called [[Inflation (cosmology)|cosmic inflation]], as long as the theory can be studied through [[perturbation theory]].<ref name=":1">{{cite journal|last=Salvio, Strumia|title=Agravity|journal=JHEP  |volume=2014 |issue=6|pages=080|date=2014-03-17|url=http://inspirehep.net/record/1286134|arxiv = 1403.4226|bibcode = 2014JHEP...06..080S|doi=10.1007/JHEP06(2014)080}}</ref>

上面的φ⁴理论例子表明，量子场论的耦合参数可以是标度依赖的，即使相应的经典场论是标度不变(或共形不变)。如果是这种情况，则称经典标度(或共形)不变性为异常。经典的标度不变场论，当量子效应打破其中的标度不变性，可以为接近指数级膨胀的早期宇宙提供了一种解释，即为'''Cosmic Inflation 宇宙膨胀'''，只要该理论可以通过'''Perturbation Theory 微扰理论'''研究。

上面的φ⁴理论例子表明，量子场论的耦合参数可以是标度依赖的，即使相应的经典场论是标度不变(或共形不变)。如果是这种情况，则称经典标度(或共形)不变性为异常。经典的标度不变场论，当量子效应打破其中的标度不变性，可以为接近指数级膨胀的早期宇宙提供了一种解释，即为'''Cosmic Inflation 宇宙膨胀'''，只要该理论可以通过'''Perturbation Theory 微扰理论'''研究<ref name=":1" />。

==Phase transitions 相变==

==Phase transitions 相变==

*[[Scale-free network]]

*[[Scale-free network]]

*Inverse square potential

* 逆平方势

* 幂律

*Scale-free network

* 无尺度网络定律

 

* 无尺度网络定律  

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