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第470行：
第470行： − * The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks). − − The set of different microscopic theories described by the same scale-invariant theory is known as a universality class. Other examples of systems which belong to a universality class are: − * Avalanches in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales. − * The frequency of network outages on the Internet, as a function of size and duration. − * The frequency of citations of journal articles, considered in the network of all citations amongst all papers, as a function of the number of citations in a given paper. − * The formation and propagation of cracks and tears in materials ranging from steel to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale. − * The electrical breakdown of dielectrics, which resemble cracks and tears. − * The percolation of fluids through disordered media, such as petroleum through fractured rock beds, or water through filter paper, such as in chromatography. Power-law scaling connects the rate of flow to the distribution of fractures. − * The diffusion of molecules in solution, and the phenomenon of diffusion-limited aggregation. 第493行：
第483行： − − The key observation is that, for all of these different systems, the behaviour resembles a phase transition, and that the language of statistical mechanics and scale-invariant statistical field theory may be applied to describe them. − + 第505行：
第493行： − Under certain circumstances, fluid mechanics is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, \mathbf{u}(\mathbf{x},t), the fluid density, \rho(\mathbf{x},t), and the fluid pressure, P(\mathbf{x},t). These fields must satisfy both the Navier–Stokes equation and the continuity equation. For a Newtonian fluid these take the respective forms+ − :\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) − :\frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0 − where \mu is the dynamic viscosity. − 在一定条件下，流体力学是一种尺度不变的经典场论。流场包括流体流动速度 mathbf { u }(mathbf { x } ，t)、流体密度 rho (mathbf { x } ，t)和流体压力 p (mathbf { x } ，t)。这些场必须同时满足 Navier-Stokes 方程和连续性方程。对于牛顿流体，它们有各自的形式: rho frac { partial mathbf { u }{ partial t } + rho mathabla nabla mathbf { u } =-nabla p + mu left (nabla ^ 2 mathbf { u } + frac {1} nabla left (nabla cbf { u }{ u }右)) : frac { partial rho }{ partial t } + nabla cabla cleft (rho mathbf { u }右) = 0其中 mu 是动力粘度。+ + + + + 第532行：
第521行： − 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。+ + + + + + + + + + +
→Universality 普适性
* The [[percolation]] of fluids through disordered media, such as [[petroleum]] through fractured rock beds, or water through filter paper, such as in [[chromatography]]. Power-law scaling connects the rate of flow to the distribution of fractures.
* The [[percolation]] of fluids through disordered media, such as [[petroleum]] through fractured rock beds, or water through filter paper, such as in [[chromatography]]. Power-law scaling connects the rate of flow to the distribution of fractures.
* The [[diffusion]] of [[molecule]]s in [[Solution (chemistry)|solution]], and the phenomenon of [[diffusion-limited aggregation]].
* The [[diffusion]] of [[molecule]]s in [[Solution (chemistry)|solution]], and the phenomenon of [[diffusion-limited aggregation]].
* The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).
* The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).
The key observation is that, for all of these different systems, the behaviour resembles a [[phase transition]], and that the language of statistical mechanics and scale-invariant [[statistical field theory]] may be applied to describe them.
The key observation is that, for all of these different systems, the behaviour resembles a [[phase transition]], and that the language of statistical mechanics and scale-invariant [[statistical field theory]] may be applied to describe them.
最关键的是，对于所有这些不同的系统来说，它们的行为都类似于相变，并且可以用统计力学的方式和标度不变的统计场论来描述。
最关键的是，对于所有这些不同的系统来说，它们的行为都类似于相变，并且可以用统计力学的方式和标度不变的统计场论来描述。
==Other examples of scale invariance 尺度不变性的其他例子==
==Other examples of scale invariance 标度不变性的其他实例==
===Newtonian fluid mechanics with no applied forces 无应力牛顿流体力学===
===Newtonian fluid mechanics with no applied forces 无应力牛顿流体力学===
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
where <math>\mu</math> is the [[dynamic viscosity#Viscosity .28dynamic viscosity.29: .CE.BC|dynamic viscosity]].
where <math>\mu</math> is the [[dynamic viscosity#Viscosity .28dynamic viscosity.29: .CE.BC|dynamic viscosity]].
在一定条件下，流体力学是一种标度不变的经典场论。流场包括流体流动速度<math>\mathbf{u}(\mathbf{x},t)</math>、流体密度<math>\rho(\mathbf{x},t)</math>和流体压力<math>P(\mathbf{x},t)</math>。这些场必须同时满足'''Navier–Stokes equation 纳维-斯托克斯方程'''和'''Continuity Equation 连续性方程'''。对于'''Newtonian Fluid 牛顿流体'''，它们有各自的形式：
<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>\frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0</math>
其中<math>\mu</math>是'''Dynamic Viscosity 动态黏度'''。
In order to deduce the scale invariance of these equations we specify an [[equation of state]], relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider the [[isothermal]] [[ideal gas]], which satisfies
In order to deduce the scale invariance of these equations we specify an [[equation of state]], relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider the [[isothermal]] [[ideal gas]], which satisfies
\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t) and \lambda\rho(\lambda\mathbf{x},\lambda^2 t) are also solutions.
\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t) and \lambda\rho(\lambda\mathbf{x},\lambda^2 t) are also solutions.
为了推导这些方程的尺度不变性，我们指定一个状态方程，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及其所处的条件。例如，我们考虑等温理想气体，它满足：
<math>P=c_s^2\rho,</math>，
其中<math>c_s</math>是流体中声速。给定这个状态方程，纳维-斯托克斯方程和连续性方程在进行如下变换时是不变的：
:<math>x\rightarrow\lambda x,</math>
:<math>t\rightarrow\lambda^2 t,</math>
:<math>\rho\rightarrow\lambda^{-1} \rho,</math>
:<math>\mathbf{u}\rightarrow\mathbf{u}.</math>.
:已知解<math>\mathbf{u}(\mathbf{x},t)</math>和<math>\rho(\mathbf{x},t)</math>，我们自然可以得到<math>\lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2 t)</math>和<math>\lambda\rho(\lambda\mathbf{x},\lambda^2 t)</math>也是解。
===Computer vision 计算机视觉===
===Computer vision 计算机视觉===