更改

尺度不变分布的例子有帕累托分布分布和 Zipfian 分布。

尺度不变分布的例子有帕累托分布分布和 Zipfian 分布。

===Scale invariant Tweedie distributions===

===Scale invariant Tweedie distributions 标度不变的Tweedie分布===

'''[[Tweedie distributions]]''' are a special case of '''[[exponential dispersion model]]s''', a class of statistical models used to describe error distributions for the [[generalized linear model]] and characterized by [[Closure (mathematics)|closure]] under additive and reproductive convolution as well as under scale transformation.<ref name="Jørgensen1997">{{cite book |last=Jørgensen |first=B. |year=1997 |title=The Theory of Dispersion Models |publisher=Chapman & Hall |location=London |isbn=978-0412997112 }}</ref>  These include a number of common distributions: the [[normal distribution]], [[Poisson distribution]] and [[gamma distribution]], as well as more unusual distributions like the compound Poisson-gamma distribution, positive [[stable distribution]]s, and extreme stable distributions.

'''[[Tweedie distributions]]''' are a special case of '''[[exponential dispersion model]]s''', a class of statistical models used to describe error distributions for the [[generalized linear model]] and characterized by [[Closure (mathematics)|closure]] under additive and reproductive convolution as well as under scale transformation.<ref name="Jørgensen1997">{{cite book |last=Jørgensen |first=B. |year=1997 |title=The Theory of Dispersion Models |publisher=Chapman & Hall |location=London |isbn=978-0412997112 }}</ref>  These include a number of common distributions: the [[normal distribution]], [[Poisson distribution]] and [[gamma distribution]], as well as more unusual distributions like the compound Poisson-gamma distribution, positive [[stable distribution]]s, and extreme stable distributions.

Consequent to their inherent scale invariance Tweedie [[random variable]]s ''Y'' demonstrate a [[variance]] var(''Y'') to [[mean]] E(''Y'')  power law:

Consequent to their inherent scale invariance Tweedie [[random variable]]s ''Y'' demonstrate a [[variance]] var(''Y'') to [[mean]] E(''Y'')  power law:

正如中心极限定理需要某些类型的随机变量作为聚焦点来收敛正态分布和表示白噪声一样，Tweedie 收敛定理需要某些非高斯随机变量来表示1/f 噪声和波动尺度。

正如中心极限定理需要某些类型的随机变量作为聚焦点来收敛正态分布和表示白噪声一样，Tweedie 收敛定理需要某些非高斯随机变量来表示1/f 噪声和波动尺度。

===Cosmology===<!-- 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21物理宇宙学，宇宙微波背景辐射空间分布的功率谱接近于尺度不变函数。尽管在数学上这意味着光谱是一个幂定律，但在宇宙学中“尺度不变量”这一术语表明，原始涨落的振幅，作为波数的函数，是近似常数，即。一个平坦的光谱。这种模式与宇宙膨胀的提议是一致的。

在21物理宇宙学，宇宙微波背景辐射空间分布的功率谱接近于尺度不变函数。尽管在数学上这意味着光谱是一个幂定律，但在宇宙学中“尺度不变量”这一术语表明，原始涨落的振幅，作为波数的函数，是近似常数，即。一个平坦的光谱。这种模式与宇宙膨胀的提议是一致的。

==Scale invariance in classical field theory==

==Scale invariance in classical field theory 经典场论中的标度不变性==

[[Classical field theory]] is generically described by a field, or set of fields,  ''φ'', that depend on coordinates, ''x''. Valid field configurations are then determined by solving [[differential equations]] for ''φ'', and these equations are known as [[field equation]]s.

[[Classical field theory]] is generically described by a field, or set of fields,  ''φ'', that depend on coordinates, ''x''. Valid field configurations are then determined by solving [[differential equations]] for ''φ'', and these equations are known as [[field equation]]s.

: lambda ^ { Delta } varphi (lambda x).

: lambda ^ { Delta } varphi (lambda x).

===Scale invariance of field configurations===

===Scale invariance of field configurations 场结构中的标度不变性===

For a particular field configuration,  ''φ''(''x''),  to be scale-invariant, we require that

For a particular field configuration,  ''φ''(''x''),  to be scale-invariant, we require that

:<math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math>

:<math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math>

我们注意到这个条件相当有限制性。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，对称性被称为自发破缺。

我们注意到这个条件相当有限制性。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，对称性被称为自发破缺。

===Classical electromagnetism===

===Classical electromagnetism 经典电磁学 ===

 

 

 

An example of a scale-invariant classical field theory is [[electromagnetic field|electromagnetism]] with no charges or currents. The fields are the electric and magnetic fields, '''E'''('''x''',''t'') and '''B'''('''x''',''t''), while their field equations are [[Maxwell's equations]].

An example of a scale-invariant classical field theory is [[electromagnetic field|electromagnetism]] with no charges or currents. The fields are the electric and magnetic fields, '''E'''('''x''',''t'') and '''B'''('''x''',''t''), while their field equations are [[Maxwell's equations]].

给出了麦克斯韦方程组 e (x，t)和 b (x，t)的解，证明了 e (λx，λt)和 b (λx，λt)也是解。

给出了麦克斯韦方程组 e (x，t)和 b (x，t)的解，证明了 e (λx，λt)和 b (λx，λt)也是解。

===Massless scalar field theory===

===Massless scalar field theory 无质量标量场理论===

Another example of a scale-invariant classical field theory is the massless [[scalar field theory|scalar field]] (note that the name [[scalar (physics)|scalar]] is unrelated to scale invariance). The scalar field, {{math|''φ''('''''x''''', ''t'')}} is a function of a set of spatial variables, '''''x''''', and a time variable, {{mvar|t}}.

Another example of a scale-invariant classical field theory is the massless [[scalar field theory|scalar field]] (note that the name [[scalar (physics)|scalar]] is unrelated to scale invariance). The scalar field, {{math|''φ''('''''x''''', ''t'')}} is a function of a set of spatial variables, '''''x''''', and a time variable, {{mvar|t}}.

名称 massless 是指在字段方程中没有一个 propto m ^ 2 varphi 项。这样的术语通常被称为“质量”术语，它打破了上述变换下的不变性。在相对论场理论中，质量尺度在物理上等同于一个固定长度的尺度: l = frac { hbar }{ mc } ，因此大质量纯量场理论不具有尺度不变性也就不足为奇了。

名称 massless 是指在字段方程中没有一个 propto m ^ 2 varphi 项。这样的术语通常被称为“质量”术语，它打破了上述变换下的不变性。在相对论场理论中，质量尺度在物理上等同于一个固定长度的尺度: l = frac { hbar }{ mc } ，因此大质量纯量场理论不具有尺度不变性也就不足为奇了。

====φ⁴ theory====

====φ⁴ theory φ4理论====

The field equations in the examples above are all [[linear]] in the fields, which has meant that the [[scaling dimension]], {{mvar|Δ}}, has not been so important. However, one usually requires that the scalar field [[action (physics)|action]] is dimensionless, and this fixes the [[scaling dimension]] of {{mvar|φ}}. In particular,

The field equations in the examples above are all [[linear]] in the fields, which has meant that the [[scaling dimension]], {{mvar|Δ}}, has not been so important. However, one usually requires that the scalar field [[action (physics)|action]] is dimensionless, and this fixes the [[scaling dimension]] of {{mvar|φ}}. In particular,

:<math>\Delta=\frac{D-2}{2},</math>

:<math>\Delta=\frac{D-2}{2},</math>

关键在于参数必须是无量纲的，否则就会在理论中引入一个固定长度的标度: 对于4理论，这只是 = 4的情况。注意，在这些转换下，函数的参数是不变的。

关键在于参数必须是无量纲的，否则就会在理论中引入一个固定长度的标度: 对于4理论，这只是 = 4的情况。注意，在这些转换下，函数的参数是不变的。

==Scale invariance in quantum field theory==

==Scale invariance in quantum field theory 量子场论中的标度不变性==

The scale-dependence of a [[quantum field theory]] (QFT) is characterised by the way its [[coupling constant|coupling parameters]] depend on the energy-scale of a given physical process. This energy dependence is described by the [[renormalization group]], and is encoded in the [[beta-function]]s of the theory.

The scale-dependence of a [[quantum field theory]] (QFT) is characterised by the way its [[coupling constant|coupling parameters]] depend on the energy-scale of a given physical process. This energy dependence is described by the [[renormalization group]], and is encoded in the [[beta-function]]s of the theory.

对于具有尺度不变性的 QFT，其耦合参数必须与能量尺度无关，这可以用理论中 β 函数的消失来表示。这种理论也被称为相应重整化群流的不动点。Zinn-Justin (2010) Scholarpedia 文章“批判现象: 场域理论方法”。

对于具有尺度不变性的 QFT，其耦合参数必须与能量尺度无关，这可以用理论中 β 函数的消失来表示。这种理论也被称为相应重整化群流的不动点。Zinn-Justin (2010) Scholarpedia 文章“批判现象: 场域理论方法”。

===Quantum electrodynamics===

===Quantum electrodynamics 量子电动力学===

A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since [[photon]]s are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.

A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since [[photon]]s are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.

然而，在自然界中，电磁场是与带电粒子耦合的，比如电子。描述光子和带电粒子相互作用的量子力学理论是量子电动力学的，而且这个理论不是标度不变的。我们可以从 QED beta 函数中看到这一点。这告诉我们电荷(在理论上是耦合参数)随着能量的增加而增加。因此，尽管没有带电粒子的量子化电磁场是尺度不变的，QED 却不是尺度不变的。

然而，在自然界中，电磁场是与带电粒子耦合的，比如电子。描述光子和带电粒子相互作用的量子力学理论是量子电动力学的，而且这个理论不是标度不变的。我们可以从 QED beta 函数中看到这一点。这告诉我们电荷(在理论上是耦合参数)随着能量的增加而增加。因此，尽管没有带电粒子的量子化电磁场是尺度不变的，QED 却不是尺度不变的。

===Massless scalar field theory===

===Massless scalar field theory 无质量标量场理论===

Free, massless [[scalar field (quantum field theory)|quantized scalar field theory]] has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the [[Gaussian fixed point]].

Free, massless [[scalar field (quantum field theory)|quantized scalar field theory]] has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the [[Gaussian fixed point]].

尽管量子化的无质量 φ4不具有尺度不变性，但除了高斯不动点理论外，还存在尺度不变量子化标量场理论。一个例子是下面的 Wilson-Fisher 定点。

尽管量子化的无质量 φ4不具有尺度不变性，但除了高斯不动点理论外，还存在尺度不变量子化标量场理论。一个例子是下面的 Wilson-Fisher 定点。

===Conformal field theory===

===Conformal field theory 共形场论===

Scale-invariant QFTs are almost always invariant under the full [[conformal symmetry]], and the study of such QFTs is [[conformal field theory]] (CFT). [[operator (physics)|Operators]] in a CFT have a well-defined [[scaling dimension]], analogous to the [[scaling dimension]], ''∆'', of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as [[anomalous scaling dimension]]s.

Scale-invariant QFTs are almost always invariant under the full [[conformal symmetry]], and the study of such QFTs is [[conformal field theory]] (CFT). [[operator (physics)|Operators]] in a CFT have a well-defined [[scaling dimension]], analogous to the [[scaling dimension]], ''∆'', of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as [[anomalous scaling dimension]]s.

在完全共形对称条件下，尺度不变量子图几乎都是不变的，对此类量子图的研究是共形场论的。CFT 中的算子有一个定义良好的尺度维度，类似于上面讨论的经典场的尺度维度 something。然而，CFT 中算子的尺度维数与经典理论中场的尺度维数不同。在 CFT 中出现的附加贡献被称为异常尺度维数。

在完全共形对称条件下，尺度不变量子图几乎都是不变的，对此类量子图的研究是共形场论的。CFT 中的算子有一个定义良好的尺度维度，类似于上面讨论的经典场的尺度维度 something。然而，CFT 中算子的尺度维数与经典理论中场的尺度维数不同。在 CFT 中出现的附加贡献被称为异常尺度维数。

===Scale and conformal anomalies===

===Scale and conformal anomalies 标度与共性异常===

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>{{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，即使相应的经典场论是尺度不变的(或共形不变的) ，量子场论的耦合参数也可以是尺度相关的。如果是这种情况，经典尺度(或共形)不变性被称为反常的。一个经典的尺度不变场理论，其中尺度不变性被量子效应打破，提供了早期宇宙近乎指数膨胀的解释，称为宇宙暴涨，只要该理论可以通过摄动理论研究。

上面的 φ4理论例子表明，即使相应的经典场论是尺度不变的(或共形不变的) ，量子场论的耦合参数也可以是尺度相关的。如果是这种情况，经典尺度(或共形)不变性被称为反常的。一个经典的尺度不变场理论，其中尺度不变性被量子效应打破，提供了早期宇宙近乎指数膨胀的解释，称为宇宙暴涨，只要该理论可以通过摄动理论研究。

==Phase transitions==

==Phase transitions 相变==

In [[statistical mechanics]], as a system undergoes a [[phase transition]], its fluctuations are described by a scale-invariant [[statistical field theory]]. For a system in equilibrium (i.e. time-independent) in {{mvar|D}} spatial dimensions, the corresponding statistical field theory is formally similar to a {{mvar|D}}-dimensional CFT. The scaling dimensions in such problems are usually referred to as [[critical exponent]]s, and one can in principle compute these exponents in the appropriate CFT.

In [[statistical mechanics]], as a system undergoes a [[phase transition]], its fluctuations are described by a scale-invariant [[statistical field theory]]. For a system in equilibrium (i.e. time-independent) in {{mvar|D}} spatial dimensions, the corresponding statistical field theory is formally similar to a {{mvar|D}}-dimensional CFT. The scaling dimensions in such problems are usually referred to as [[critical exponent]]s, and one can in principle compute these exponents in the appropriate CFT.

在21统计力学，当一个系统经历相变时，它的波动用一个尺度不变的统计场理论来描述。对于一个处于平衡状态的系统(即。时间无关)的空间维度，相应的统计场理论形式上类似于一维 CFT。这类问题中的标度维数通常被称为临界指数，原则上可以在适当的 CFT 中计算这些指数。

在21统计力学，当一个系统经历相变时，它的波动用一个尺度不变的统计场理论来描述。对于一个处于平衡状态的系统(即。时间无关)的空间维度，相应的统计场理论形式上类似于一维 CFT。这类问题中的标度维数通常被称为临界指数，原则上可以在适当的 CFT 中计算这些指数。

===The Ising model===

===The Ising model Ising模型===

An example that links together many of the ideas in this article is the phase transition of the [[Ising model]], a simple model of [[ferromagnet]]ic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a {{mvar|D}}-dimensional periodic lattice. Associated with each lattice site is a [[magnetic moment]], or [[spin (physics)|spin]], and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)

An example that links together many of the ideas in this article is the phase transition of the [[Ising model]], a simple model of [[ferromagnet]]ic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a {{mvar|D}}-dimensional periodic lattice. Associated with each lattice site is a [[magnetic moment]], or [[spin (physics)|spin]], and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)

在这个临界温度下，人们想要计算的物理量的一个例子就是距离分开的自旋之间的相互关系。这有一个通用的行为: : g (r) propto frac {1}{ r ^ { D-2 + eta }} ，对于某个特定的 eta 值，这是一个临界指数的例子。

在这个临界温度下，人们想要计算的物理量的一个例子就是距离分开的自旋之间的相互关系。这有一个通用的行为: : g (r) propto frac {1}{ r ^ { D-2 + eta }} ，对于某个特定的 eta 值，这是一个临界指数的例子。

====CFT description====

====CFT description CFT描述====

The fluctuations at temperature {{math|''T_c''}} are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the '''Wilson-Fisher fixed point''', a particular scale-invariant [[scalar field (quantum field theory)|scalar field theory]].

The fluctuations at temperature {{math|''T_c''}} are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the '''Wilson-Fisher fixed point''', a particular scale-invariant [[scalar field (quantum field theory)|scalar field theory]].

某些二维 CFTs 的异常维数可能与随机游动的典型分形维数有关，其中随机游动是通过施拉姆-洛沃纳演化(SLE)定义的。正如我们上面所看到的，CFTs 描述了相变的物理过程，因此我们可以把某些相变的临界指数与这些分形维数联系起来。例子包括二维临界 Ising 模型和更一般的二维临界 Potts 模型。关联其他2 d cft 与系统性红斑狼疮是一个积极的研究领域。

某些二维 CFTs 的异常维数可能与随机游动的典型分形维数有关，其中随机游动是通过施拉姆-洛沃纳演化(SLE)定义的。正如我们上面所看到的，CFTs 描述了相变的物理过程，因此我们可以把某些相变的临界指数与这些分形维数联系起来。例子包括二维临界 Ising 模型和更一般的二维临界 Potts 模型。关联其他2 d cft 与系统性红斑狼疮是一个积极的研究领域。

==Universality==

==Universality 普适性==

A phenomenon known as [[universality (dynamical systems)|universality]] is seen in a large variety of physical systems. It expresses the idea that different microscopic physics can give rise to the same scaling behaviour at a phase transition. A canonical example of universality involves the following two systems:

A phenomenon known as [[universality (dynamical systems)|universality]] is seen in a large variety of physical systems. It expresses the idea that different microscopic physics can give rise to the same scaling behaviour at a phase transition. A canonical example of universality involves the following two systems:

* The [[Ising model]] phase transition, described above.

* The [[Ising model]] phase transition, described above.

关键的观察结果是，对于所有这些不同的系统来说，它们的行为类似于相变，并且可以用统计力学的语言和尺度不变的统计场理论来描述它们。

关键的观察结果是，对于所有这些不同的系统来说，它们的行为类似于相变，并且可以用统计力学的语言和尺度不变的统计场理论来描述它们。

==Other examples of scale invariance==

==Other examples of scale invariance 尺度不变性的其他例子==

 

===Newtonian fluid mechanics with no applied forces 无应力牛顿流体力学===

 

= = 尺度不变性的其他例子 = =  

 

 

===Newtonian fluid mechanics with no applied forces===

 

 

Under certain circumstances, [[fluid mechanics]] is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, <math>\mathbf{u}(\mathbf{x},t)</math>, the fluid density, <math>\rho(\mathbf{x},t)</math>, and the fluid pressure, <math>P(\mathbf{x},t)</math>. These fields must satisfy both the [[Navier–Stokes equation]] and the [[continuity equation#Fluid dynamics|continuity equation]]. For a [[Newtonian fluid]] these take the respective forms

Under certain circumstances, [[fluid mechanics]] is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, <math>\mathbf{u}(\mathbf{x},t)</math>, the fluid density, <math>\rho(\mathbf{x},t)</math>, and the fluid pressure, <math>P(\mathbf{x},t)</math>. These fields must satisfy both the [[Navier–Stokes equation]] and the [[continuity equation#Fluid dynamics|continuity equation]]. For a [[Newtonian fluid]] these take the respective forms

:<math>\rho\frac{\partial \mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla \mathbf{u} = -\nabla P+\mu \left(\nabla^2 \mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot\mathbf{u}\right)\right)</math>

:<math>\rho\frac{\partial \mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla \mathbf{u} = -\nabla P+\mu \left(\nabla^2 \mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot\mathbf{u}\right)\right)</math>

为了推导这些方程的尺度不变性，我们指定了一个状态方程，将流体压力与流体密度联系起来。状态方程取决于流体的类型和它所处的条件。例如，我们考虑满足以下条件的等温理想气体: p = c _ s ^ 2 rho，其中 c _ s 是流体中声速。给定这个状态方程，Navier-Stokes 和连续性方程在变换下是不变的: x right tarrow lambda x，: t right tarrow lambda ^ 2 t，: rho right tarrow lambda ^ {-1} rho，: mathbf { u } right tarrow mathbf { u }。给定解 mathbf { u }(mathbf { x } ，t)和 rho (mathbf { x } ，t) ，我们自动有 lambda mathbf { u }(lambda mathbf { x } ，lambda ^ 2 t)和 lambda rho (lambda mathbf { x } ，lambda ^ 2 t)也是解。

为了推导这些方程的尺度不变性，我们指定了一个状态方程，将流体压力与流体密度联系起来。状态方程取决于流体的类型和它所处的条件。例如，我们考虑满足以下条件的等温理想气体: p = c _ s ^ 2 rho，其中 c _ s 是流体中声速。给定这个状态方程，Navier-Stokes 和连续性方程在变换下是不变的: x right tarrow lambda x，: t right tarrow lambda ^ 2 t，: rho right tarrow lambda ^ {-1} rho，: mathbf { u } right tarrow mathbf { u }。给定解 mathbf { u }(mathbf { x } ，t)和 rho (mathbf { x } ，t) ，我们自动有 lambda mathbf { u }(lambda mathbf { x } ，lambda ^ 2 t)和 lambda rho (lambda mathbf { x } ，lambda ^ 2 t)也是解。

===Computer vision===

===Computer vision 计算机视觉===

{{Main article|Scale space}}

{{Main article|Scale space}}

In [[computer vision]] and [[biological vision]], scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed.<ref name=Lin13PONE>[https://dx.doi.org/10.1371/journal.pone.0066990 Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.]</ref>   

In [[computer vision]] and [[biological vision]], scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed.<ref name=Lin13PONE>[https://dx.doi.org/10.1371/journal.pone.0066990 Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.]</ref>   

}}</ref><ref name=Lin14CompVis>T. Lindeberg (2014) [http://www.csc.kth.se/~tony/abstracts/Lin14-ScSel-CompVisRefGuide.html "Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701-713.]</ref>

}}</ref><ref name=Lin14CompVis>T. Lindeberg (2014) [http://www.csc.kth.se/~tony/abstracts/Lin14-ScSel-CompVisRefGuide.html "Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701-713.]</ref>

Examples of applications include [[blob detection]], [[corner detection]], [[ridge detection]], and object recognition via the [[scale-invariant feature transform]].

Examples of applications include [[blob detection]], [[corner detection]], [[ridge detection]], and object recognition via the [[scale-invariant feature transform]].

In computer vision and biological vision, scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed.Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.   

In computer vision and biological vision, scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed.Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.   

在计算机视觉和生物视觉中，由于透视图像映射和物体在世界上有不同的物理大小，缩放变换产生。在这些领域中，图像尺度不变性是指当图像域的局部尺度改变时保持不变的图像数据的局部图像描述符或视觉表示。视觉操作在感受野水平的不变性，PLoS ONE 8(7) : e66990。检测标准化导数响应的局部极大值为从图像数据中获取尺度不变性提供了一个通用框架。林德伯格(2014)“比例选择”，《计算机视觉: 参考指南》 ，(k. Ikeuchi，编辑) ，Springer，701-713页。应用的例子包括斑点检测、角检测、脊线检测和通过尺度不变特征转换识别系统进行的目标识别。

在计算机视觉和生物视觉中，由于透视图像映射和物体在世界上有不同的物理大小，缩放变换产生。在这些领域中，图像尺度不变性是指当图像域的局部尺度改变时保持不变的图像数据的局部图像描述符或视觉表示。视觉操作在感受野水平的不变性，PLoS ONE 8(7) : e66990。检测标准化导数响应的局部极大值为从图像数据中获取尺度不变性提供了一个通用框架。林德伯格(2014)“比例选择”，《计算机视觉: 参考指南》 ，(k. Ikeuchi，编辑) ，Springer，701-713页。应用的例子包括斑点检测、角检测、脊线检测和通过尺度不变特征转换识别系统进行的目标识别。

==See also==

==See also 另见==

*[[Inverse square potential]]  

*[[Inverse square potential]]  

*[[Power law]]

*[[Power law]]

*Power law

*Power law

*Scale-free network

*Scale-free network

* 逆平方势  

* 逆平方势  

*  

*  

==References==

==References 参考文献==

{{Reflist}}

{{Reflist}}

==Further reading==

==Further reading 拓展阅读==

*{{cite book |last=Zinn-Justin |first=Jean |title=Quantum Field Theory and Critical Phenomena |publisher=Oxford University Press |year=2002 }} Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.

*{{cite book |last=Zinn-Justin |first=Jean |title=Quantum Field Theory and Critical Phenomena |publisher=Oxford University Press |year=2002 }} Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.

*{{cite book |first1=P. |last1=DiFrancesco |first2=P. |last2=Mathieu |first3=D. |last3=Senechal |title=Conformal Field Theory |publisher=Springer-Verlag |year=1997 }}

*{{cite book |first1=P. |last1=DiFrancesco |first2=P. |last2=Mathieu |first3=D. |last3=Senechal |title=Conformal Field Theory |publisher=Springer-Verlag |year=1997 }}