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第204行： − 在数学中，淬致无序比退火无序更难分析，因为热平均和噪声通常起着非常不同的作用。事实上，这个问题是如此的困难，以至于很少有已知的技术可以处理任何一个问题，大多数解决方案都依赖于近似值。最常用的是+ 第220行：
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第239行： − 该领域第二常用的技术是'''<font color="#ff8000">生成函数分析 Generating Functional Analysis</font>'''。这种方法是基于'''<font color="#ff8000">路线积分 Path Integrals</font>'''的，虽然这通常比复制过程更难应用，但原则上是完全精确的，+
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此词条暂由小竹凉翻译，未经人工整理和审校，带来阅读不便，请见谅。
此词条暂由小竹凉翻译，未经人工整理和审校，带来阅读不便，请见谅。
由CecileLi初步审校{{Unreferenced|date=February 2011}}
由CecileLi初步审校{{Unreferenced|date=February 2011}}于2020.11.19再次审校，若有遗漏敬请谅解
* the demagnetization of iron by heating above the [[Curie temperature]]: ferromagnetic-paramagnetic transition, loss of magnetic order.
* the demagnetization of iron by heating above the [[Curie temperature]]: ferromagnetic-paramagnetic transition, loss of magnetic order.
铁在'''<font color="#ff8000">受到居里点 Curie Temperature</font>'''以上温度加热时就会逐渐退磁: 铁磁-顺磁转变时会导致磁有序的损失
铁在'''<font color="#ff8000">受到居里点 Curie Temperature</font>'''以上温度加热时就会逐渐退磁: 铁磁-顺磁转变时会导致磁有序的耗损。
The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long range order or short range order.
The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long range order or short range order.
这种顺序既可以是完全晶体'''<font color="#ff8000">空间群 Space Group</font>'''的对称或关联的。根据相关系数随距离衰减的程度，我们可以说'''<font color="#ff8000">长程有序 Long Range Order</font>'''或'''<font color="#ff8000">短程有序 Short Range Order</font>'''。
这种顺序既可以是完全晶体'''<font color="#ff8000">空间群 Space Group</font>'''的对称的也可以是关联的。根据相关系数随距离衰减的程度，我们可以说'''<font color="#ff8000">长程有序 Long Range Order</font>'''或'''<font color="#ff8000">短程有序 Short Range Order</font>'''。
It is a thermodynamic entropy concept often displayed by a second-order phase transition. Generally speaking, high thermal energy is associated with disorder and low thermal energy with ordering, although there have been violations of this. Ordering peaks become apparent in diffraction experiments at low energy.
It is a thermodynamic entropy concept often displayed by a second-order phase transition. Generally speaking, high thermal energy is associated with disorder and low thermal energy with ordering, although there have been violations of this. Ordering peaks become apparent in diffraction experiments at low energy.
这是一个'''<font color="#ff8000">熵 Thermodynamics Entropy</font>'''的概念，通常表现为一个二阶'''<font color="#ff8000">相变 Phase Transition</font>'''。一般来说，高热能与无序有关，低热能与有序有关，但有违背这一规律的现象存在。在低能衍射实验中，有序峰变得明显。
这是一个'''<font color="#ff8000">熵 Thermodynamics Entropy</font>'''的概念，通常表现为一个二阶'''<font color="#ff8000">相变 Phase Transition</font>'''。一般来说，高热能与无序有关，低热能与有序有关，但有违背这一规律的现象存在。在低能衍射实验中，有序峰十分明显。
<math>G(x,x') = \langle s(x),s(x') \rangle. \, </math>
<math>G(x,x') = \langle s(x),s(x') \rangle. \, </math>
: <math>G(x,x') = \langle s(x),s(x') \rangle. \, </math>
设: <math>G(x,x') = \langle s(x),s(x') \rangle. \, </math>
This function is equal to unity when <math>x=x'</math> and decreases as the distance <math>|x-x'|</math> increases. Typically, it decays exponentially to zero at large distances, and the system is considered to be disordered. But if the correlation function decays to a constant value at large <math>|x-x'|</math> then the system is said to possess long-range order. If it decays to zero as a power of the distance then it is called quasi-long-range order (for details see Chapter 11 in the textbook cited below. See also Berezinskii–Kosterlitz–Thouless transition). Note that what constitutes a large value of <math>|x-x'|</math> is understood in the sense of asymptotics.
This function is equal to unity when <math>x=x'</math> and decreases as the distance <math>|x-x'|</math> increases. Typically, it decays exponentially to zero at large distances, and the system is considered to be disordered. But if the correlation function decays to a constant value at large <math>|x-x'|</math> then the system is said to possess long-range order. If it decays to zero as a power of the distance then it is called quasi-long-range order (for details see Chapter 11 in the textbook cited below. See also Berezinskii–Kosterlitz–Thouless transition). Note that what constitutes a large value of <math>|x-x'|</math> is understood in the sense of asymptotics.
当<math>x=x'</math>时，这个函数等于单位数量，当距离<math>|x-x'|</math>增加时，函数值减少。通常情况下，它在很大距离上'''<font color="#ff8000">呈指数衰减 Decays Exponentially</font>'''为零，系统被认为是无序的。但是如果相关函数(量子场论)衰变为一个常数值，那么这个系统就被认为具有远程有序。如果它衰变成为零以作为距离的幂，那么它被称为准远程有序(详见下面引用的教科书第11章)。参见'''<font color="#ff8000">Berezinskii–Kosterlitz–Thouless过渡 Berezinskii–Kosterlitz–Thouless Transition</font>''')。请注意，构成较大的<math>|x-x'|</math>的值可以理解为渐近性。
当<math>x=x'</math>时，这个函数等于单位数量，当距离<math>|x-x'|</math>增加时，函数值减少。通常情况下，当它在很大程度上'''<font color="#ff8000">呈指数衰减 Decays Exponentially</font>'''为零时，系统就被认为是无序的。但如果相关函数(量子场论)衰变为一个常数值，那么这个系统就被认为具有远程有序。如果它衰变成为零以作为距离的幂，那么它被称为准远程有序(详见下面引用的教科书第11章)。参见'''<font color="#ff8000">Berezinskii–Kosterlitz–Thouless过渡 Berezinskii–Kosterlitz–Thouless Transition</font>''')。请注意，构成较大的<math>|x-x'|</math>的值可以理解为渐近性。
==Quenched disorder==
==Quenched disorder==
In statistical physics, a system is said to present quenched disorder when some parameters defining its behavior are random variables which do not evolve with time, i.e. they are quenched or frozen. Spin glasses are a typical example. It is opposite to annealed disorder, where the random variables are allowed to evolve themselves.
In statistical physics, a system is said to present quenched disorder when some parameters defining its behavior are random variables which do not evolve with time, i.e. they are quenched or frozen. Spin glasses are a typical example. It is opposite to annealed disorder, where the random variables are allowed to evolve themselves.
在'''<font color="#ff8000">统计物理学 Statistical Physics</font>'''中，当定义系统行为的某些参数是不随时间演化的'''<font color="#ff8000">随机变量 Random Variable</font>'''时，系统称为淬致无序。它们被'''<font color="#ff8000">淬火 Quenched</font>'''或者''冷冻''。'''<font color="#ff8000">自旋玻璃 Spin Glass</font>'''就是一个典型的例子。与'''<font color="#ff8000">退火无序 Annealed Disorder</font>'''相反，随机变量允许自身进化。
在'''<font color="#ff8000">统计物理学 Statistical Physics</font>'''中，当定义系统行为的某些参数是不随时间演化的'''<font color="#ff8000">随机变量 Random Variable</font>'''时，系统称为淬致无序。它们被'''<font color="#ff8000">淬火 Quenched</font>'''或者''冷冻''。'''<font color="#ff8000">自旋玻璃 Spin Glass</font>'''就是一个典型的例子。与'''<font color="#ff8000">退火无序 Annealed Disorder</font>'''相反，它允许随机变量的自我进化。
In mathematical terms, quenched disorder is harder to analyze than its annealed counterpart, since the thermal and the noise averaging play very different roles. In fact, the problem is so hard that few techniques to approach each are known, most of them relying on approximations. The most used are
In mathematical terms, quenched disorder is harder to analyze than its annealed counterpart, since the thermal and the noise averaging play very different roles. In fact, the problem is so hard that few techniques to approach each are known, most of them relying on approximations. The most used are
在数学中，由于热平均和噪声通常起着非常不同的作用，淬致无序比退火无序更难分析。事实上，这个问题太过困难以至于很少有已知的技术可以处理，而现有的大多数解决方案都依赖于近似值。最常用的是
the Cavity method; although these give results in accord with experiments in a large range of problems, they are not generally proven to be a rigorous mathematical procedure.
the Cavity method; although these give results in accord with experiments in a large range of problems, they are not generally proven to be a rigorous mathematical procedure.
'''<font color="#ff8000">谐振腔法 Cavity Method</font>'''：虽然这些方法给出的结果与许多问题的实验结果相一致，但它们通常不是一个可证明的严格数学过程。
'''<font color="#ff8000">谐振腔法 Cavity Method</font>'''
the Cavity method 谐振腔法
虽然这些方法给出的结果与许多问题的实验结果相一致，但它们通常不是一个可证明的严格数学过程。
More recently it has been shown by rigorous methods, however, that at least in the archetypal spin-glass model (the so-called [[Spin_glass#The_model_of_Sherrington_and_Kirkpatrick|Sherrington–Kirkpatrick model]]) the replica based solution is indeed exact.
More recently it has been shown by rigorous methods, however, that at least in the archetypal spin-glass model (the so-called [[Spin_glass#The_model_of_Sherrington_and_Kirkpatrick|Sherrington–Kirkpatrick model]]) the replica based solution is indeed exact.
The second most used technique in this field is generating functional analysis. This method is based on path integrals, and is in principle fully exact, although generally more difficult to apply than the replica procedure.
The second most used technique in this field is generating functional analysis. This method is based on path integrals, and is in principle fully exact, although generally more difficult to apply than the replica procedure.
该领域次常用的技术是'''<font color="#ff8000">生成函数分析 Generating Functional Analysis</font>'''。这种方法是基于'''<font color="#ff8000">路线积分 Path Integrals</font>'''的，虽然这通常比复制过程更难应用，但原则上是完全精确的，