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非晶体材料的热容在接近玻璃相变温度时具有这样的行为，其中通用临界指数α= 0.59。类似的行为适用于相关长度，但使用指数需要改为ν而不是α。

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The exponent is positive. This is different with . Its actual value depends on the type of phase transition we are considering.

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~~指数是正的。这是不同的。它的实际价值取决于我们正在考虑的相变类型。~~

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这里指数ν为正数。这与α不同。它的实际值取决于我们正在考虑的相变类型。

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It is widely believed that the critical exponents are the same above and below the critical temperature. It has now been shown that this is not necessarily true: When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then some exponents (such as <math>\gamma </math>, the exponent of the susceptibility) are not identical.

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It is widely believed that the critical exponents are the same above and below the critical temperature. It has now been shown that this is not necessarily true: When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then some exponents (such as <math>\gamma </math>, the exponent of the susceptibility) are not identical.<ref>{{cite journal|last1=Leonard|first1=F.|last2=Delamotte|first2=B.|year = 2015|title=Critical exponents can be different on the two sides of a transition| url = | journal = Phys. Rev. Lett. | volume = 115 | issue = 20| page = 200601 | arxiv = 1508.07852|bibcode = 2015PhRvL.115t0601L| doi = 10.1103/PhysRevLett.115.200601 |pmid=26613426}}</ref>

It is widely believed that the critical exponents are the same above and below the critical temperature. It has now been shown that this is not necessarily true: When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then some exponents (such as <math>\gamma </math>, the exponent of the susceptibility) are not identical.

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人们普遍认为，临界指数在临界温度上下是相同的。现在已经证明，这并不一定是正确的: 当一个连续对称被显式地分解为离散对称由无关的(在重整化群意义上)各向异性，然后一些指数(如数学伽马 / 数学，易感性指数)是不相同的。

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之前普遍认为，临界指数在临界温度上下浮动的时候都是相同的。但是现已证明其不一定正确：因不相关的各向异性（在重整化群理论意义上）将连续对称属性清晰地分解为离散对称属性时，则某些指数（例如γ，磁化率指数）不相同。

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For −1 < α < 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the [[lambda transition]] from a normal state to the [[superfluid]] state, for which experiments have found {{mvar|α}} = -0.013±0.003.At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample.This experimental value of α agrees with theoretical predictions based on variational perturbation theory.

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For −1 < α < 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the lambda transition from a normal state to the superfluid state, for which experiments have found = -0.013±0.003.At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample. This experimental value of α agrees with theoretical predictions based on variational perturbation theory.

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For −1 < α < 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the [[lambda transition]] from a normal state to the [[superfluid]] state, for which experiments have found {{mvar|α}} = -~~0.013±0.003.~~

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当-1 <α<0时，热容在相变温度下具有“扭结”性质。这是液氦在从正常状态到超流体状态的λ相变行为，为此实验发现α= -0.013±0.003。采取了至少一次在轨道卫星的零重力条件下进行，以最小化样品中的压力差。α的这个实验值与基于变分微扰理论的预测相符。

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~~For −1 <~~ α < 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the lambda transition from a normal state to the superfluid state, for which experiments have found = -0.013±0.~~003.~~

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~~对于 -10，热容在转变温度处有一个“扭结”。这是 Lambda相变中液氦从正常状态到超流体状态的行为，实验发现-0.0130.003。~~

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At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample.<ref>{{cite journal | doi=10.1103/PhysRevB.68.174518 | title=Specific heat of liquid helium in zero gravity very near the lambda point | year=2003 | last1=Lipa | first1=J. | last2=Nissen | first2=J. | last3=Stricker | first3=D. | last4=Swanson | first4=D. | last5=Chui | first5=T. | journal=Physical Review B | volume=68 | issue=17| pages=174518 |arxiv = cond-mat/0310163 |bibcode = 2003PhRvB..68q4518L }}</ref> This experimental value of α agrees with theoretical predictions based on [[variational perturbation theory]].<ref>{{cite journal | doi=10.1103/PhysRevD.60.085001 | title=Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions | year=1999 | last1=Kleinert | first1=Hagen | journal=Physical Review D | volume=60 | issue=8| pages=085001 |arxiv = hep-th/9812197 |bibcode = 1999PhRvD..60h5001K }}</ref>

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At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample. This experimental value of α agrees with theoretical predictions based on variational perturbation theory.

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~~在轨道卫星的零重力条件下至少进行了一次实验，以尽量减小样品中的压力差。这个实验值与基于变分摄动理论的理论预测一致。~~

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For 0 < < 1, the heat capacity diverges at the transition temperature (though, since < 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensional Ising model for uniaxial magnets, detailed theoretical studies have yielded the exponent ∼ +0.110.

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对于01，热容在转变温度处发生分化(尽管自1以来，焓保持有限)。这种行为的一个例子是三维铁磁相变。在单轴磁体的三维伊辛模型中，详细的理论研究得到了指数∼0.110。

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当0 <α<1时，热容在相变温度处发散（然而由于α<1，焓保持有限）。这种行为的一个例子是3D铁磁相变。在单轴磁体的三维伊辛模型中，进行了详细的理论研究并得出了指数α≈+0.110。

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Some model systems do not obey a power-law behavior. For example, mean field theory predicts a finite discontinuity of the heat capacity at the transition temperature, and the two-dimensional Ising model has a logarithmic divergence. However, these systems are limiting cases and an exception to the rule. Real phase transitions exhibit power-law behavior.

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有些模型系统不遵守幂律行为。例如，平均场理论预测在转变温度下热容有限的不连续性，而二维伊辛模型具有对数发散性。然而，这些系统是有限的情况和规则的例外。实际相变呈现幂律行为。

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当然也存在一些模型系统不遵循幂律行为。例如，平均场理论预测了相变温度下热容量的有限不连续性，而二维伊辛模型则具有对数发散。但是，这些系统存在有限，是规则的例外。实际的相变仍然表现出幂律行为。

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Several other critical exponents, , and , are defined, examining the power law behavior of a measurable physical quantity near the phase transition. Exponents are related by scaling relations, such as

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~~定义了其他几个临界指数，，和，检验了在相变附近可测量的物理量的幂律行为。指数通过缩放关系相关，例如~~

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另外还存在其他几个关键指数β，γ，δ，ν和η，主要是定义检查相变附近可测物理量的幂律行为。它们通过比例关系相互联系，例如：

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~~:<math>\beta=\gamma/(\delta-1) , \qquad \nu=\gamma/(2-\eta)</math>.~~

<math>\beta=\gamma/(\delta-1) , \qquad \nu=\gamma/(2-\eta)</math>.

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~~数学 beta gamma / ( delta-1) qquad nu gamma / (2- eta) / math。~~

It can be shown that there are only two independent exponents, e.g. {{mvar|ν}} and {{mvar|η}}.

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It can be shown that there are only two independent exponents, e.g. and .

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上式可以证明只有两个独立的指数，例如 ν和η。

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~~可以证明只有两个独立的指数，例如。及。~~

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It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known as universality. For example, the critical exponents at the liquid–gas critical point have been found to be independent of the chemical composition of the fluid.

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在不同系统中出现的相变通常具有相同的临界指数，这是一个值得注意的事实。这种现象被称为普遍性。例如，已经发现液气临界点的临界指数与流体的化学成份无关。

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值得注意的是，在不同系统中发生的相变通常具有相同的关键指数集。这种现象称为通用性。例如，已发现液气临界点的临界指数与流体的化学组成无关。

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More impressively, but understandably from above, they are an exact match for the critical exponents of the ferromagnetic phase transition in uniaxial magnets. Such systems are said to be in the same universality class. Universality is a prediction of the renormalization group theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and are insensitive to the underlying microscopic properties of the system. Again, the divergence of the correlation length is the essential point.

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更令人印象深刻的是，从上面可以理解，它们与单轴磁体中铁磁相变的临界指数完全匹配。这样的系统据说属于同一个普遍性类别。普适性是重整化群相变理论的一种预测，该理论认为相变附近系统的热力学性质只依赖于少量的特征，如维数和对称性，并且对系统的潜在微观性质不敏感。同样，相关长度的散度是本质点。

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上方这个现象很特别但是可以理解，它们与单轴磁体中铁磁相变的临界指数完全匹配。这类系统因此被认为属于同一通用性类别。这里的通用性是相变的重整化群理论的一种预测，指出系统在相变附近的热力学性质仅取决于少数特征，例如维数和对称性，并且对系统的基本微观特性不敏感。同样，相关长度的差异性是非常重要的。

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===Critical slowing down and other phenomena===

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=== Critical slowing down and other phenomena 临界慢化和其他现象 ===

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~~临界减速和其他现象~~

There are also other critical phenomena; e.g., besides ''static functions'' there is also ''critical dynamics''. As a consequence, at a phase transition one may observe critical slowing down or ''speeding up''. The large ''static universality classes'' of a continuous phase transition split into smaller ''dynamic universality'' classes. In addition to the critical exponents, there are also universal relations for certain static or dynamic functions of the magnetic fields and temperature differences from the critical value.

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There are also other critical phenomena; e.g., besides static functions there is also critical dynamics. As a consequence, at a phase transition one may observe critical slowing down or speeding up. The large static universality classes of a continuous phase transition split into smaller dynamic universality classes. In addition to the critical exponents, there are also universal relations for certain static or dynamic functions of the magnetic fields and temperature differences from the critical value.

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还有其他关键现象，例如，除了静态函数之外，还有关键动力学。因此，在相变过程中，我们可以观察到关键的减速或加速。连续相变的大静态普适性类分裂为较小的动态普适性类。除了临界指数外，磁场的某些静态或动态函数与临界值之间的温度差也存在普遍关系。

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在相变过程中，还存在其他临界现象。例如除了静态函数外，还存在关键的临界动态。因此在相变发生时，人们可能会观察到严重的减速或加速现象。连续相变的静态通用类可以分成更小的动态通用类。除了临界指数外，磁场的某些静态或动态函数以及与临界值的温差也存在通用关系。

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=== Percolation theory 渗流理论===

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Another phenomenon which shows phase transitions and critical exponents is [[percolation theory|percolation]]. The simplest example is perhaps percolation in a two dimensional square lattice. Sites are randomly occupied with probability p. For small values of p the occupied sites form only small clusters. At a certain threshold pc a giant cluster is formed and we have a second-order phase transition. The behavior of P∞ near pc is, P∞~(p-pc)β, where β is a critical exponent.

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~~===Percolation theory===~~

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~~逾渗理论~~

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Another phenomenon which shows phase transitions and critical exponents is [[percolation theory|percolation]]. The simplest example is perhaps percolation in a two dimensional square lattice. Sites are randomly occupied with probability p. For small values of p the occupied sites form only small clusters. At a certain threshold pc a giant cluster is formed and we have a second-order phase transition.~~<ref>{{cite book |title= Fractals and Disordered Systems|author= Armin Bunde and [[Shlomo Havlin]] |year= 1996|publisher= Springer|url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php}}</ref>~~ The behavior of P∞ near pc is, P∞~(p-pc)β, where β is a critical exponent.

Another phenomenon which shows phase transitions and critical exponents is percolation. The simplest example is perhaps percolation in a two dimensional square lattice. Sites are randomly occupied with probability p. For small values of p the occupied sites form only small clusters. At a certain threshold pc a giant cluster is formed and we have a second-order phase transition. The behavior of P∞ near pc is, P∞~(p-pc)β, where β is a critical exponent.

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另一个显示相变和临界指数的现象是逾渗。最简单的例子也许是二维正方形格子中的渗流。位点以概率 p 随机占据。对于 p 的小值，占据的位点只形成小的集群。在一定的阈值条件下，形成了一个巨大的团簇，并且发生了二阶相变。P 子∞ / 子在 p 子 c / 子附近的行为是，p 子∞ / 子 ~ (p-p 子 c / 子) sup / sup，这里是临界指数。

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显示相变和临界指数的另一种现象是渗流。最简单的例子是发生在二维方格中的渗流。其中每一个格子以概率p标记。对于较小的p值，标记的格子仅形成较小的团簇。但是当p达到某个阈值pc时会形成一个巨大的团簇，此时发生二阶相变。pc附近的P∞行为是P∞〜（p − pc）β，其中β是一个临界指数。

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~~! -- 相变的平均场理论 --~~

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~~! -- 波动 --~~

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~~! -- 相变的重整化群理论 --~~

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~~! -- om sairam--~~

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~~===Phase transitions in biological systems===~~

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生物系统中的相变

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=== Phase transitions in biological systems 生物系统中的相变 ===

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Phase transitions play many important roles in biological systems. Examples include the [[lipid bilayer]] formation, the [[Coil–globule transition|coil-globule transition]] in the process of [[protein folding]] and [[DNA melting]], liquid crystal-like transitions in the process of [[DNA condensation]], and cooperative ligand binding to DNA and proteins with the character of phase transition.<ref>{{cite journal | doi=10.1080/07391102.2000.10506578 | pmid=10798534 | title=Long-range interactions between ligands bound to a DNA molecule give rise to adsorption with the character of phase transition of the first kind | year=2000| author = D.Y. Lando and V.B. Teif| journal=J. Biomol. Struct. Dynam. | volume=17 | issue=5 | pages=903–911 }}</ref>

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Phase transitions play many important roles in biological systems. Examples include the [[lipid bilayer]] formation, the [[Coil–globule transition|coil-globule transition]] in the process of [[protein folding]] and [[DNA melting]], liquid crystal-like transitions in the process of [[DNA condensation]], and cooperative ligand binding to DNA and proteins with the character of phase transition.

Phase transitions play many important roles in biological systems. Examples include the lipid bilayer formation, the coil-globule transition in the process of protein folding and DNA melting, liquid crystal-like transitions in the process of DNA condensation, and cooperative ligand binding to DNA and proteins with the character of phase transition.

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相变在生物体系中起着重要的作用。其中包括脂质双分子层的形成，蛋白质折叠和 DNA 熔融过程中的线圈-球状转变，DNA 凝聚过程中的液晶转变，以及具有相变特征的配体与 DNA 和蛋白质的合作结合。

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相变在生物系统中也起到许多重要的作用。比如包括脂质双层的形成，蛋白质折叠和DNA解链过程中的坍塌转变，DNA缩合过程中的液晶转变，以及具有相变特征的配体与DNA和蛋白质的结合配体。

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In ''biological membranes'', gel to liquid crystalline phase transitions play a critical role in physiological functioning of biomembranes. In gel phase, due to low fluidity of membrane lipid fatty-acyl chains, membrane proteins have restricted movement and thus are restrained in exercise of their physiological role. Plants depend critically on photosynthesis by chloroplast thylakoid membranes which are exposed cold environmental temperatures. Thylakoid membranes retain innate fluidity even at relatively low temperatures because of high degree of fatty-acyl disorder allowed by their high content of linolenic acid, 18-carbon chain with 3-double bonds. Gel-to-liquid crystalline phase transition temperature of biological membranes can be determined by many techniques including calorimetry, fluorescence, [[spin label]] [[electron paramagnetic resonance]] and [[NMR]] by recording measurements of the concerned parameter by at series of sample temperatures. A simple method for its determination from 13-C NMR line intensities has also been proposed.

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In ''biological membranes'', gel to liquid crystalline phase transitions play a critical role in physiological functioning of biomembranes. In gel phase, due to low fluidity of membrane lipid fatty-acyl chains, membrane proteins have restricted movement and thus are restrained in exercise of their physiological role. Plants depend critically on photosynthesis by chloroplast thylakoid membranes which are exposed cold environmental temperatures. Thylakoid membranes retain innate fluidity even at relatively low temperatures because of high degree of fatty-acyl disorder allowed by their high content of linolenic acid, 18-carbon chain with 3-double bonds.<ref>{{cite journal | last1 = YashRoy | first1 = R.C. | year = 1987 | title = 13-C NMR studies of lipid fatty acyl chains of chloroplast membranes | url = https://www.researchgate.net/publication/230822408 | journal = Indian Journal of Biochemistry and Biophysics | volume = 24 | issue = 6| pages = 177–178 }}</ref> Gel-to-liquid crystalline phase transition temperature of biological membranes can be determined by many techniques including calorimetry, fluorescence, [[spin label]] [[electron paramagnetic resonance]] and [[NMR]] by recording measurements of the concerned parameter by at series of sample temperatures. A simple method for its determination from 13-C NMR line intensities has also been proposed.<ref>{{cite journal | last1 = YashRoy | first1 = R C | year = 1990 | title = Determination of membrane lipid phase transition temperature from 13-C NMR intensities | url = https://www.researchgate.net/publication/20790042 | journal = Journal of Biochemical and Biophysical Methods | volume = 20 | issue = 4| pages = 353–356 | pmid = 2365951 |doi=10.1016/0165-022X(90)90097-V }}</ref>

In biological membranes, gel to liquid crystalline phase transitions play a critical role in physiological functioning of biomembranes. In gel phase, due to low fluidity of membrane lipid fatty-acyl chains, membrane proteins have restricted movement and thus are restrained in exercise of their physiological role. Plants depend critically on photosynthesis by chloroplast thylakoid membranes which are exposed cold environmental temperatures. Thylakoid membranes retain innate fluidity even at relatively low temperatures because of high degree of fatty-acyl disorder allowed by their high content of linolenic acid, 18-carbon chain with 3-double bonds. Gel-to-liquid crystalline phase transition temperature of biological membranes can be determined by many techniques including calorimetry, fluorescence, spin label electron paramagnetic resonance and NMR by recording measurements of the concerned parameter by at series of sample temperatures. A simple method for its determination from 13-C NMR line intensities has also been proposed.

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在生物膜中，凝胶到液晶的相变对生物膜的生理功能起着至关重要的作用。在凝胶期，由于膜脂脂肪酰基链的低流动性，膜蛋白的运动受到限制，从而限制了其生理作用的发挥。植物主要依靠暴露在低温环境下的叶绿体类囊体膜的光合作用。类囊体膜即使在相对较低的温度下也能保持固有的流动性，这是由于其高含量的亚麻酸、18碳链和3- 双键所致的高度脂肪酰基无序所致。生物膜的凝胶-液晶相变温度可以通过量热法、荧光法、自旋标记电子自旋共振法和核磁共振(NMR)等多种方法测定。本文还提出了一种从13-C ~~NMR 谱线强度测定其含量的简便方法。~~

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在生物膜中，凝胶到液晶的相变在生物膜的生理机能中起关键作用。在凝胶相中，由于膜脂质脂肪酰基链的流动性低，膜蛋白的运动受到限制，因此在行使其生理作用方面受到限制。植物非常依赖于暴露于寒冷环境温度下叶绿体类囊体膜的光合作用。类囊体膜即使在相对较低的温度下也能保持固有的流动性，这是由于其高含量的亚麻酸，带有3个双键的18碳链允许高度的脂肪酰基紊乱。基于众多技术，包括量热法，荧光法，自旋标记电子顺磁共振和NMR，通过记录一系列样品温度下有关参数的测量值，来确定生物膜的凝胶到液晶的相变温度。同时还提出了一种由13-C NMR谱线强度测定的简单方法。

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It has been proposed that some biological systems might lie near critical points. Examples include [[neural network]]s in the salamander retina, bird flocks gene expression networks in Drosophila, and protein folding. However, it is not clear whether or not alternative reasons could explain some of the phenomena supporting arguments for criticality. It has also been suggested that biological organisms share two key properties of phase transitions: the change of macroscopic behavior and the coherence of a system at a critical point.

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It has been proposed that some biological systems might lie near critical points. Examples include neural networks in the salamander retina, bird flocks gene expression networks in Drosophila, and protein folding. However, it is not clear whether or not alternative reasons could explain some of the phenomena supporting arguments for criticality. It has also been suggested that biological organisms share two key properties of phase transitions: the change of macroscopic behavior and the coherence of a system at a critical point.

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It has been proposed that some biological systems might lie near critical points. Examples include [[neural network]]s in the salamander retina,<ref>{{cite arXiv|eprint=1407.5946|last1=Tkacik|first1=Gasper|title=Thermodynamics for a network of neurons: Signatures of criticality|last2=Mora|first2=Thierry|last3=Marre|first3=Olivier|last4=Amodei|first4=Dario|last5= Berry II|first5=Michael J.|last6=Bialek|first6=William|class=q-bio.NC|year=2014}}</ref> bird flocks<ref>{{cite journal|last1=Bialek|first1=W|last2=Cavagna|first2=A|last3=Giardina|first3=I|title = Social interactions dominate speed control in poising natural flocks near criticality|journal=PNAS|volume=111|issue=20|pages=7212–7217|year = 2014|bibcode=2014PNAS..111.7212B|doi=10.1073/pnas.1324045111|pmid=24785504|pmc=4034227|arxiv=1307.5563}}</ref>

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曾经有观点认为生物系统可能位于临界点附近。类似包括蝾螈视网膜中的神经网络，果蝇中的鸟群基因表达网络和蛋白质折叠。但是，尚不清楚替代原因是否可以解释某些现象来支持关键性论证。另一个观点认为，生物有机体具有两个重要的相变特性：宏观行为的变化和系统在临界点的一致性。

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~~It has been proposed that some biological systems might lie near critical points. Examples include neural networks in the salamander retina, bird flocks~~

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~~有人提出，一些生物系统可能处于临界点附近。例如蝾螈视网膜中的神经网络，鸟群~~

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gene expression networks in Drosophila,<ref>{{cite journal | last1=Krotov |first1=D|last2=Dubuis|first2=J O|last3=Gregor|first3=T|last4=Bialek|first4=W|title = Morphogenesis at criticality|journal = PNAS|year = 2014|doi=10.1073/pnas.1324186111|pmid=24516161|pmc=3956198|volume=111|issue=10|pages=3683–3688|arxiv=1309.2614|bibcode=2014PNAS..111.3683K}}</ref> and protein folding.<ref>{{cite journal|last1=Mora|first1=Thierry|last2=Bialek|first2=William|title = Are biological systems poised at criticality?|journal = Journal of ~~Statistical Physics|volume=144|issue=2|pages=268–302|year = 2011 |arxiv=1012.2242 |doi= 10.1007/s10955-011-0229-4|bibcode=2011JSP...144..268M}}</ref> However, it~~ is ~~not clear whether or not alternative reasons could explain some~~ of ~~the phenomena supporting arguments for criticality.<ref>{{cite journal |last1=Schwab|first1=David J|last2=Nemenman|first2=Ilya|last3=Mehta|first3=Pankaj|title = Zipf's law and criticality~~ in ~~multivariate data without fine~~-~~tuning|journal = Physical Review Letters|volume=113|issue=6|pages=068102|year = 2014 |arxiv=1310.0448 |bibcode= 2014PhRvL.113f8102S|doi= 10~~.~~1103/PhysRevLett.113.068102|pmid=25148352|pmc=5142845}}</ref>~~ It has ~~also~~ been ~~suggested~~ that ~~biological organisms share two key properties of phase transitions:~~ the ~~change of macroscopic behavior and~~ the ~~coherence~~ of ~~a system at a critical point.<ref~~>~~{{Cite journal|last=Longo|first=G~~.~~|last2=Montévil|first2=M.|date=2011~~-08-01|title=From physics to biology by extending criticality and symmetry breakings|url=https://www.academia.edu/23155991|journal=Progress in Biophysics and Molecular Biology|series=Systems Biology and Cancer|volume=106|issue=2|pages=340–347|doi=10.1016/j.pbiomolbio.~~2011.03.005|pmid=21419157|via=|arxiv=1103.1833}}</ref>~~

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The characteristic feature of second order phase transitions is the appearance of fractals in some scale-free properties. It has long been known that protein globules are shaped by interactions with water. There are 20 amino acids that form side groups on protein peptide chains range from hydrophilic to hydrophobic, causing the former to lie near the globular surface, while the latter lie closer to the globular center. Twenty fractals were discovered in solvent associated surface areas of > 5000 protein segments [39]. The existence of these fractals proves that proteins function near critical points of second-order phase transitions.

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~~gene expression networks~~ in ~~Drosophila, and protein folding. However, it is not clear whether or not alternative reasons could explain~~ some ~~of the phenomena supporting arguments for criticality~~. It has ~~also~~ been ~~suggested~~ that ~~biological organisms share two key properties of phase transitions:~~ the ~~change~~ of ~~macroscopic behavior and the coherence~~ of ~~a system at a~~ critical ~~point~~.

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The characteristic feature of second order phase transitions is the appearance of fractals in some scale-free properties. It has long been known that protein globules are shaped by interactions with water. There are 20 amino acids that form side groups on protein peptide chains range from hydrophilic to hydrophobic, causing the former to lie near the globular surface, while the latter lie closer to the globular center. Twenty fractals were discovered in solvent associated surface areas of > 5000 protein segments [39]. The existence of these fractals proves that proteins function near critical points of second-order phase transitions.

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果蝇的基因表达网络和蛋白质折叠。然而，尚不清楚替代原因是否可以解释支持临界性论点的一些现象。还有人提出，生物有机体具有相变的两个关键属性: 宏观行为的变化和系统在临界点的一致性。

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二阶相变的特征是在某些无标度特性中出现了分形。众所周知，蛋白质球是通过与水相互作用而形成的。蛋白质肽链上形成侧基的氨基酸有20种，范围从亲水性到疏水性，使前者位于球状表面附近，而后者更靠近球状中心。在与溶剂相关的表面积大于5000个蛋白质片段的区域中发现了二十个分形。这些分形的存在证明了蛋白质在二阶相变的临界点附近起作用。

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In groups of organisms in stress (when approaching critical transitions), correlations tend to increase, while at the same time, fluctuations also increase. This effect is supported by many experiments and observations of groups of people, mice, trees, and grassy plants.

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In groups of organisms in stress (when approaching critical transitions), correlations tend to increase, while at the same time, fluctuations also increase. This effect is supported by many experiments and observations of groups of people, mice, trees, and grassy plants.<ref>{{cite journal|last1=Gorban|first1=A.N.|last2=Smirnova|first2=E.V.|last3=Tyukina|first3=T.A.|title=Correlations, risk and crisis: From physiology to finance|journal=Physica A: Statistical Mechanics and Its Applications|date=August 2010|volume=389|issue=16|pages=3193–3217|url=https://www.researchgate.net/publication/222687003|doi=10.1016/j.physa.2010.03.035|arxiv=0905.0129|bibcode=2010PhyA..389.3193G}}</ref>

In groups of organisms in stress (when approaching critical transitions), correlations tend to increase, while at the same time, fluctuations also increase. This effect is supported by many experiments and observations of groups of people, mice, trees, and grassy plants.

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在受到压力的生物群体中(当接近临界转变时) ，相关性倾向于增加，同时波动也增加。这种效应得到了许多实验和观察的支持，这些实验和观察对象包括人群、老鼠、树木和草本植物。

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在处于压力下的生物群中（接近关键转变时），相关性趋于增加，而与此同时，波动也会增加。许多人和小鼠，树木和草类植物的实验和观察都支持这种效果。

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== Experimental 实验性 ==

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==Experimental==

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~~实验性的~~

A variety of methods are applied for studying the various effects. Selected examples are:

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A variety of methods are applied for studying the various effects. Selected examples are:

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研究各种效果的方法多种多样。部分示例如下：

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~~各种方法被用来研究各种效应。选定的例子如下:~~

* [[Thermogravimetry]] (very common)

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* [[X-ray diffraction]]

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* [[Neutron diffraction]]

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* [[Raman Spectroscopy]]

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* [[SQUID]] (measurement of magnetic transitions)

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* [[Hall effect]] (measurement of magnetic transitions)

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* [[Mössbauer spectroscopy]] (simultaneous measurement of magnetic and non-magnetic transitions. Limited up to about 800-1000 °C)

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* [[Perturbed angular correlation]] (simultaneous measurement of magnetic and non-magnetic transitions. No temperature limits. Over 2000 °C already performed, theoretical possible up to the highest crystal material, such as [[tantalum hafnium carbide]] 4215 °C.)

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•热重量分析法（非常常见）

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•X射线衍射法

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•中子衍射

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•拉曼光谱法

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•SQUID（磁跃迁测量）

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•霍尔效应（磁跃迁测量）

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•穆斯堡尔光谱法（同时测量磁性和非磁性跃迁。限制在大约800–1000°C的温度下）

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•扰动角关联（同时测量磁性和非磁性跃迁。没有温度限制。已经执行了超过2000°C的操作，理论上可能达到最高晶体材料，例如钽碳化carbide 4215°C。）

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== See also 其他参考资料 ==

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==See also==

参见

第605行：第516行： −

* [[Allotropy]]

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* [[Allotropy 同素异形体]]

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* [[Autocatalytic reactions and order creation 自动催化反应和序生成]]

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* [[Crystal growth 晶体生长]]

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** [[Abnormal grain growth 谷物异常生长]]

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* [[Autocatalytic reactions and order creation]]

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* [[Differential scanning calorimetry 差示扫描量热法]]

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* [[Diffusionless transformations 无扩散相变]]

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+

* [[Ehrenfest equations 埃伦费斯特方程]]

−

+

* [[Jamming (physics) 阻塞（物理）]]

−

* [[Crystal growth]]

+

* [[Kelvin probe force microscope 开尔文探针力显微镜]]

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+

* [[Landau theory of second order phase transition]] 朗道二阶相变理论

−

+

* [[Laser-heated pedestal growth 激光加热基座法]]

−

+

* [[List of states of matter 物质状态列表]]

−

** [[Abnormal grain growth]]

+

* [[Micro-pulling-down 微下拉法]]

−

+

* [[Percolation theory 渗流理论]]

−

+

** [[Continuum percolation theory 连续介质渗流理论]]

−

+

* [[Superfluid film 超流体膜]]

−

* [[Differential scanning calorimetry]]

+

* [[Superradiant phase transition 超辐射相变]]

−

+

* [[Topological quantum field theory 拓扑量子场论]]

−

* [[Diffusionless transformations]]

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* [[Ehrenfest equations]]

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* [[Jamming (physics)]]

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* [[Kelvin probe force microscope]]

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* [[Landau theory]] of second order phase ~~transitions~~

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* [[Laser-heated pedestal growth]]

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* [[List of states of matter]]

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* [[Micro-pulling-down]]

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* [[Percolation theory]]

−

** [[Continuum percolation theory]]

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* [[Superfluid film]]

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* [[Superradiant phase transition]]

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* [[Topological quantum field theory]]

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第693行：第546行： −

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== Further reading 扩展阅读 ==

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==Further reading==

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~~进一步阅读~~

* [[Philip Warren Anderson|Anderson, P.W.]], ''Basic Notions of Condensed Matter Physics'', [[Perseus Publishing]] (1997).

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* [[Amir Faghri|Faghri, A.]], and [[Yuwen Zhang|Zhang, Y.]], [https://www.springer.com/gp/book/9783030221362 Fundamentals of Multiphase Heat Transfer and Flow], [[Springer Nature]] Switzerland AG, 2020.

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* {{cite journal | last1 = Fisher | first1 = M.E. | authorlink = Michael E. Fisher | year = 1974 | title = The renormalization group in the theory of critical behavior | url = | journal = Rev. Mod. Phys. | volume = 46 | issue = 4| pages = 597–616 | doi=10.1103/revmodphys.46.597|bibcode = 1974RvMP...46..597F }}

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* Goldenfeld, N., ''Lectures on Phase Transitions and the Renormalization Group'', Perseus Publishing (1992).

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*{{citation |year=2008 |author=Ivancevic, Vladimir G |author2=Ivancevic, Tijana T |title=Chaos, Phase Transitions, Topology Change and Path Integrals |url=https://books.google.com/books?id=wpsPgHgtxEYC&printsec=frontcover&dq=complex+nonlinearity |place=Berlin |publisher=Springer |isbn=978-3-540-79356-4 |accessdate=14 March 2013 }}

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* M.R.Khoshbin-e-Khoshnazar, ''Ice Phase Transition as a sample of finite system phase transition'', (Physics Education(India)Volume 32. No. 2, Apr - Jun 2016)[http://www.physedu.in/uploads/publication/23/371/4.-Ice-Phase-transition-as-a-sample-of-finite-system-phase--transition.pdf]

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* [[Hagen Kleinert|Kleinert, H.]], ''Gauge Fields in Condensed Matter'', Vol. I, "[[:de:Supraflüssigkeit|Superfluid]] and [[vortex|Vortex lines]]; Disorder Fields, [[Phase Transition]]s,", pp. 1–742, [https://archive.is/20060514143926/http://www.worldscibooks.com/physics/0356.htm World Scientific (Singapore, 1989)]; Paperback {{ISBN|9971-5-0210-0}} (readable online [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html physik.fu-berlin.de])

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* [[Hagen Kleinert|Kleinert, H.]] and Verena Schulte-Frohlinde, ''Critical Properties of φ4-Theories'', [https://web.archive.org/web/20080226151023/http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback {{ISBN|981-02-4659-5}}'' (readable online [http://www.physik.fu-berlin.de/~kleinert/b8 here]).''

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* {{cite journal | last1 = Kogut | first1 = J. | authorlink2 = Kenneth G. Wilson | last2 = Wilson | first2 = K | year = 1974 | title = The Renormalization Group and the epsilon-Expansion | url = | journal = Phys. Rep. | volume = 12 | issue = 2| pages = 75–199 |bibcode = 1974PhR....12...75W |doi = 10.1016/0370-1573(74)90023-4 }}

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* Krieger, Martin H., ''Constitutions of matter : mathematically modelling the most everyday of physical phenomena'', [[University of Chicago Press]], 1996. Contains a detailed pedagogical discussion of [[Lars Onsager|Onsager]]'s solution of the 2-D Ising Model.

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* [[Lev Davidovich Landau|Landau, L.D.]] and [[Evgeny Mikhailovich Lifshitz|Lifshitz, E.M.]], ''Statistical Physics Part 1'', vol. 5 of ''[[Course of Theoretical Physics]]'', Pergamon Press, 3rd Ed. (1994).

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* Mussardo G., "Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics", Oxford University Press, 2010.

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*[[Manfred R. Schroeder|Schroeder, Manfred R.]], ''Fractals, chaos, power laws : minutes from an infinite paradise'', New York: [[W. H. Freeman]], 1991. Very well-written book in "semi-popular" style—not a textbook—aimed at an audience with some training in mathematics and the physical sciences. Explains what scaling in phase transitions is all about, among other things.

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* H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena'' (Oxford University Press, Oxford and New York 1971).

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* Yeomans J. M., ''Statistical Mechanics of Phase Transitions'', Oxford University Press, 1992.

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• Anderson, P.W., 《凝聚态物理的基本概念》, Perseus Publishing (1997).

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• Faghri, A., and Zhang, Y., 《多相传热与流动基础》, Springer Nature Switzerland AG, 2020.

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• Fisher, M.E. (1974). “临界行为理论中的再规范化”. Rev. Mod. Phys. 46 (4): 597–616. Bibcode:1974RvMP...46..597F. doi:10.1103/revmodphys.46.597.

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• Goldenfeld, N., 《关于相变和重整化群的讲座》, Perseus Publishing (1992).

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• Ivancevic, Vladimir G; Ivancevic, Tijana T (2008), 《混沌，相变，拓扑变化和路径积分》, Berlin: Springer, ISBN 978-3-540-79356-4, retrieved 14 March 2013

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• M.R.Khoshbin-e-Khoshnazar, 《冰的相变作为有限系统相变的一个样本》, (Physics Education(India)Volume 32. No. 2, Apr - Jun 2016)[1]

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• Kleinert, H., 《凝聚物质的规范场》, Vol. I, “超流体和涡旋线；无序场，相变”, pp. 1–742, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (readable online physik.fu-berlin.de)

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• Kleinert, H. and Verena Schulte-Frohlinde, 《φ4-理论的临界性质》, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4659-5 (readable online here).

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• Kogut, J.; Wilson, K (1974). “重整化群和epsilon扩展”. Phys. Rep. 12 (2): 75–199. Bibcode:1974PhR....12...75W. doi:10.1016/0370-1573(74)90023-4.

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• Krieger, Martin H., 《物质的构成：对物理现象进行最日常的数学建模》, University of Chicago Press, 1996. 其中包含对Onsager的二维伊辛模型解决方案的详细教学讨论。

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• Landau, L.D. and Lifshitz, E.M., 《统计物理学》 Part 1, vol. 5 of Course of Theoretical Physics, Pergamon Press, 3rd Ed. (1994).

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• Mussardo G., “统计场论。统计物理学的精确求解模型导论”，Oxford University Press, 2010.

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• Schroeder, Manfred R., 《分形，混沌，幂定律：距离无限天堂的分钟路程》, New York: W. H. Freeman, 1991. 写得很好的“半大众”风格的书，而不是教科书，旨在介绍数学和物理科学方面的知识。解释相变的标定是什么。

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• H. E. Stanley, 《相变和临界现象导论》(Oxford University Press, Oxford and New York 1971).

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• Yeomans J. M., 《相变的统计力学》, Oxford University Press, 1992.

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== External links 相关链接 ==

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==External links==

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~~外部链接~~

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*[http://www.ibiblio.org/e-notes/Perc/contents.htm Interactive Phase Transitions on lattices] with Java applets 使用Java小程序在晶格上进行交互式相变

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*[http://www.sklogwiki.org/SklogWiki/index.php/Universality_classes Universality classes 通用类] from Sklogwiki

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*[http://www.ibiblio.org/e-notes/Perc/contents.htm Interactive Phase Transitions on lattices] with Java applets

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*[http://www.sklogwiki.org/SklogWiki/index.php/Universality_classes Universality classes] from Sklogwiki

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[[Category:Phase transitions| ]]

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[[Category:Concepts in physics]]

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