相变

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Jie（讨论 | 贡献）2020年9月7日 (一) 01:46的版本
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此图显示了不同相变的命名法

The term phase transition (or phase change) is most commonly used to describe transitions between solid, liquid, and gaseous states of matter, as well as plasma in rare cases. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change, often discontinuously, as a result of the change of external conditions, such as temperature, pressure, or others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition. Phase transitions commonly occur in nature and are used today in many technologies.

The term phase transition (or phase change) is most commonly used to describe transitions between solid, liquid, and gaseous states of matter, as well as plasma in rare cases. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change, often discontinuously, as a result of the change of external conditions, such as temperature, pressure, or others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition. Phase transitions commonly occur in nature and are used today in many technologies.

相变Phase transition (or phase change)一词常用于描述物质固态，液态和气态之间的转变，在极少数情况下还涉及到等离子体。热力学系统的相和物质的状态一样，均具有统一的物理特性。由于外部条件（例如温度，压力或其他）的变化，在给定介质的相变过程中，它的某些属性通常发生不连续地变化。例如，液体在加热到沸点的时候可能会变成气体，导致体积大小突然改变。当这种转变发生时，外部条件的度量称为相变。相变通常发生在自然界，不过如今越来越多地被用于科技行业。


Types of phase transition 相变的种类

该图为典型的相图。其中虚线部分表示了水的反常行为

Examples of phase transitions include: Examples of phase transitions include: 关于相变的例子包括：


  • The transitions between the solid, liquid, and gaseous phases of a single component, due to the effects of temperature and/or pressure:

由于温度和/或压力的影响，单个成分物质的固态，液态和气态之间会产生相互变化：

参见蒸汽压和相图
一小块固体氩在快速溶解，同时显示出从固体到液体以及从液体到气体的转变。
二氧化碳（红色）和水（蓝色）的相图比较，解释了它们在1个大气压下的不同相变


Condensed matter physics
文件:QuantumPhaseTransition.svg
Phases模板:· Phase transition模板:· QCP
States of matter
Solid模板:· Liquid模板:· Gas模板:· Plasma模板:· Bose–Einstein condensate模板:· Bose gas模板:· Fermionic condensate模板:· Fermi gas模板:· Fermi liquid模板:· Supersolid模板:· Superfluidity模板:· Luttinger liquid 模板:· Time crystal


  • A eutectic transformation, in which a two-component single-phase liquid is cooled and transforms into two solid phases. The same process, but beginning with a solid instead of a liquid is called a eutectoid transformation.

共晶转变指的是一类互溶液体（由两种不同成分组成的单相液体）经过冷却后，转变成为两个不同的固相。同样的过程，由一类固态开始转变成为两个不同的固相，则称为共析转变。


  • A metastable to equilibrium phase transformation. A metastable polymorph which forms rapidly due to lower surface energy will transform to an equilibrium phase given sufficient thermal input to overcome an energetic barrier.

由于较低的表面能而迅速形成的亚稳多晶体会逐渐趋向一种平衡相，前提是需要足够的热输入以克服能量位垒。


  • A peritectic transformation, in which a two-component single-phase solid is heated and transforms into a solid phase and a liquid phase.

包晶转变，指的是一类单相固体（包含两种不同成分）经过加热后转变为一种固相和一种液相的过程。


  • A spinodal decomposition, in which a single phase is cooled and separates into two different compositions of that same phase.

亚稳相分解，指的是一个单相经过冷却后分离为同相的两种不同成分的物质。


在固体和液体之间过渡的中间相，例如“液晶”相之一。


磁性材料在居里点（居里温度）时，铁磁和顺磁相之间的转变。


在各种有序，相称或不相称的磁性结构（如锑化铈中）之间的转变。


马氏体转变是碳钢的众多相变之一，是典型的位移相变。


晶体结构的变化，例如铁在不同温度，不同处理方式下铁素体和奥氏体之间的转变。


有序到无序的过渡，例如α-钛铝化物。


  • The dependence of the adsorption geometry on coverage and temperature, such as for hydrogen on iron (110).

吸附几何结构对覆盖率和温度的依赖性，例如氢对铁（110）的依赖性。


  • The emergence of superconductivity in certain metals and ceramics when cooled below a critical temperature.

当冷却到临界温度以下时，某些金属和陶瓷会出现超导现象。


  • The transition between different molecular structures (polymorphs, allotropes or [[polyamorphism|polyamorphs模板:Not a typo]]), especially of solids, such as between an amorphous structure and a crystal structure, between two different crystal structures, or between two amorphous structures.

不同分子结构（同质多形体，同素异形体或非晶多形体）之间的过渡，特别是固体之间的过渡，例如非晶结构和晶体结构之间，两种不同晶体结构之间或两种非晶结构之间的过渡。


玻色子流体的量子凝聚（玻色–爱因斯坦凝聚）。液态氦中的超流体转变就是一个例子。


  • The breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled.

早期的宇宙随着温度的降温，物理学定律的对称性破裂。


  • Isotope fractionation occurs during a phase transition, the ratio of light to heavy isotopes in the involved molecules changes. When water vapor condenses (an equilibrium fractionation), the heavier water isotopes (18O and 2H) become enriched in the liquid phase while the lighter isotopes (16O and 1H) tend toward the vapor phase.

同位素分馏发生在相变过程中，所涉及分子中的轻同位素与重同位素的比率会发生变化。当水蒸气冷凝（平衡分馏）时，较重的水同位素（18O和2H）在液相中富集，而较轻的同位素（16O和1H）则趋向于气相。


Phase transitions occur when the thermodynamic free energy of a system is non-analytic for some choice of thermodynamic variables (cf. phases). This condition generally stems from the interactions of a large number of particles in a system, and does not appear in systems that are too small. It is important to note that phase transitions can occur and are defined for non-thermodynamic systems, where temperature is not a parameter. Examples include: quantum phase transitions, dynamic phase transitions, and topological (structural) phase transitions. In these types of systems other parameters take the place of temperature. For instance, connection probability replaces temperature for percolating networks.

Phase transitions occur when the thermodynamic free energy of a system is non-analytic for some choice of thermodynamic variables (cf. phases). This condition generally stems from the interactions of a large number of particles in a system, and does not appear in systems that are too small. It is important to note that phase transitions can occur and are defined for non-thermodynamic systems, where temperature is not a parameter. Examples include: quantum phase transitions, dynamic phase transitions, and topological (structural) phase transitions. In these types of systems other parameters take the place of temperature. For instance, connection probability replaces temperature for percolating networks.

当一个系统的热力学自由能对于某些热力学变量（参见相）选择不解析时，就会发生相变。这种情况通常是由于系统中存在大量粒子相互作用，如果系统太小，则不太会出现。值得注意的是，相变的发生和定义同样可以针对于非热力学系统，并且不将温度作为参数。例如：量子相变，动态相变和拓扑（结构）相变。在这些类型的系统中，其他参数代替了温度。例如，连接概率代替渗滤网络Percolating networks的温度。



At the phase transition point (for instance, boiling point) the two phases of a substance, liquid and vapor, have identical free energies and therefore are equally likely to exist. Below the boiling point, the liquid is the more stable state of the two, whereas above the gaseous form is preferred.

At the phase transition point (for instance, boiling point) the two phases of a substance, liquid and vapor, have identical free energies and therefore are equally likely to exist. Below the boiling point, the liquid is the more stable state of the two, whereas above the gaseous form is preferred.

在相变点（例如，沸点）下，一种物质的两个相（液体和蒸气）具有相同的自由能，因此它们可以同时存在。而当温度低于沸点时，液体在两者中状态更稳定，因此相比较气态更趋近于液态存在。



It is sometimes possible to change the state of a system diabatically (as opposed to adiabatically) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state is metastable, i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs in superheating, supercooling, and supersaturation, for example.

It is sometimes possible to change the state of a system diabatically (as opposed to adiabatically) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state is metastable, i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs in superheating, supercooling, and supersaturation, for example.

有时可以通过传热方式（注意不是绝热方式）改变系统状态，使系统状态可以通过相变点而不会经历相变。因此该系统会处于亚稳态，是指比较于相变发生过后的状态没有那么稳定，但也不是说不稳定。

Classifications 分类

Ehrenfest classification 埃伦费斯特分类法

Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy as a function of other thermodynamic variables.[1] Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition. First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable.[2] The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free energy with respect to pressure. Second-order phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy.[2] These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with respect to the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higher-order phase transitions.

Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy as a function of other thermodynamic variables. Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition. First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free energy with respect to pressure. Second-order phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy. These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with respect to the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higher-order phase transitions.

保罗·埃伦费斯特Paul Ehrenfest根据热力学自由能与其他热力学变量的函数关系对相变进行了分类。根据他的方法，可以将相变按照转变时的不连续自由能的最低导数标记。一阶相变相对于某些热力学变量，表现出自由能的一阶导数不连续。各种固/液/气的转变都归为一阶转变，因为它们都涉及到密度的不连续变化，这是自由能相对于压力的一阶导数（一阶导数的逆函数）。而二阶相变在一阶导数中是连续的（有序参数，即自由能相对于外部场的一阶导数，在整个转变过程中是连续的），但在自由能的二阶导数中表现出不连续性。比如包括铁等材料中的铁磁相变，其中磁化强度是自由能相对于施加磁场强度的一阶导数，随着温度降低到居里温度以下，磁化强度将从零开始连续增加。而磁化率，是自由能相对于磁场的二阶导数，它的变化则是不连续的。以此类推，按照埃伦费斯特的分类方法，原则上可以存在第三，第四和更高阶的相变。


The Ehrenfest classification implicitly allows for continuous phase transformations, where the bonding character of a material changes, but there is no discontinuity in any free energy derivative. An example of this occurs at the supercritical liquid–gas boundaries.

The Ehrenfest classification implicitly allows for continuous phase transformations, where the bonding character of a material changes, but there is no discontinuity in any free energy derivative. An example of this occurs at the supercritical liquid–gas boundaries.

埃伦费斯特分类法隐含了连续的相变，其中材料的成键特征发生了变化，但任何自由能导数都没有间断。比如说超临界液气的边界。


Modern classifications 现代分类法

In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:[1]

In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:

在现代分类方案中，将相变分为两大类，类似于埃伦费斯特的命名方式：


First-order phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. During this process, the temperature of the system will stay constant as heat is added: the system is in a "mixed-phase regime" in which some parts of the system have completed the transition and others have not. Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into vapor, but forms a turbulent mixture of liquid water and vapor bubbles). Imry and Wortis showed that quenched disorder can broaden a first-order transition. That is, the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling.

First-order phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. During this process, the temperature of the system will stay constant as heat is added: the system is in a "mixed-phase regime" in which some parts of the system have completed the transition and others have not. Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into vapor, but forms a turbulent mixture of liquid water and vapor bubbles). Imry and Wortis showed that quenched disorder can broaden a first-order transition. That is, the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling.

一阶相变是那些涉及潜伏热的相变。在这种相变过程中，系统会吸收或释放每体积固定（通常是大量）的能量。在此过程中，系统的温度将随着热量的增加而保持恒定：系统处于“混合相状态”，其中系统的某些部分已完成转变，而其他部分尚未完成。常见的例子是冰的融化或水的沸腾（水不会立即变成蒸气，而是形成液态水和蒸气气泡的湍流混合物）。物理学家Imry和Wortis研究表明，淬火无序可以视为一阶转变。即在有限的温度范围内完成了相的转变，但是诸如过冷和过热的现象仍然存在，并且在热循环中观察到了滞后。


Second-order phase transitions are also called "continuous phase transitions". They are characterized by a divergent susceptibility, an infinite correlation length, and a power law decay of correlations near criticality. Examples of second-order phase transitions are the ferromagnetic transition, superconducting transition (for a Type-I superconductor the phase transition is second-order at zero external field and for a Type-II superconductor the phase transition is second-order for both normal-state—mixed-state and mixed-state—superconducting-state transitions) and the superfluid transition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperature which enables accurate detection using differential scanning calorimetry measurements. Lev Landau gave a phenomenological theory of second-order phase transitions.

Second-order phase transitions are also called "continuous phase transitions". They are characterized by a divergent susceptibility, an infinite correlation length, and a power law decay of correlations near criticality. Examples of second-order phase transitions are the ferromagnetic transition, superconducting transition (for a Type-I superconductor the phase transition is second-order at zero external field and for a Type-II superconductor the phase transition is second-order for both normal-state—mixed-state and mixed-state—superconducting-state transitions) and the superfluid transition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperature which enables accurate detection using differential scanning calorimetry measurements. Lev Landau gave a phenomenological theory of second-order phase transitions.

二阶相变，或称为“连续相变”，它们的特征是敏感度发散，相关长度无限以及接近临界的相关性幂律衰减。二阶相变的有关例子是铁磁相变，超导相变（对于I型超导体，在零外场下的相变是二阶的;对于II型超导体，常态到混合态，以及混合态到超导状态的转变都是二阶的）和超流体转换。另外，关于非晶体材料。其热膨胀和热容属性在玻璃相变温度下变化相当突然，这与粘度属性恰恰相反，从而可以使用差示扫描量热法来精确检测。列夫·兰道Lev Landau后来研究出二阶相变的现象学理论。


Apart from isolated, simple phase transitions, there exist transition lines as well as multicritical points, when varying external parameters like the magnetic field or composition.

Apart from isolated, simple phase transitions, there exist transition lines as well as multicritical points, when varying external parameters like the magnetic field or composition.

当改变诸如磁场或成分之类的外部参数时，除了独立简单的相变之外，还存在跃迁谱线以及多个临界点。


Several transitions are known as infinite-order phase transitions.They are continuous but break no symmetries. The most famous example is the Kosterlitz–Thouless transition in the two-dimensional XY model. Many quantum phase transitions, e.g., in two-dimensional electron gases, belong to this class.

Several transitions are known as infinite-order phase transitions.They are continuous but break no symmetries. The most famous example is the Kosterlitz–Thouless transition in the two-dimensional XY model. Many quantum phase transitions, e.g., in two-dimensional electron gases, belong to this class.

另外还存在其他相变类型例如无序相变。无序相变是连续的但并不破坏对称性。最著名的例子是二维XY模型中的Kosterlitz-Thouless相变。除此之外二维电子气中的量子相变也都属于此类。


The liquid–glass transition is observed in many polymers and other liquids that can be supercooled far below the melting point of the crystalline phase. This is atypical in several respects. It is not a transition between thermodynamic ground states: it is widely believed that the true ground state is always crystalline. Glass is a quenched disorder state, and its entropy, density, and so on, depend on the thermal history. Therefore, the glass transition is primarily a dynamic phenomenon: on cooling a liquid, internal degrees of freedom successively fall out of equilibrium. Some theoretical methods predict an underlying phase transition in the hypothetical limit of infinitely long relaxation times. No direct experimental evidence supports the existence of these transitions.

The liquid–glass transition is observed in many polymers and other liquids that can be supercooled far below the melting point of the crystalline phase. This is atypical in several respects. It is not a transition between thermodynamic ground states: it is widely believed that the true ground state is always crystalline. Glass is a quenched disorder state, and its entropy, density, and so on, depend on the thermal history. Therefore, the glass transition is primarily a dynamic phenomenon: on cooling a liquid, internal degrees of freedom successively fall out of equilibrium. Some theoretical methods predict an underlying phase transition in the hypothetical limit of infinitely long relaxation times. No direct experimental evidence supports the existence of these transitions.

在过度冷却至远低于结晶相熔点的聚合物和其他液体中，观察到了液体-玻璃转变。该现象仓多个方面考虑均属于非典型的相变过程。它不是热力学基态之间的转变：人们普遍认为，真正的基态始终是晶体。玻璃是淬火无序状态，其熵，密度等取决于热历史。因此，玻璃相变主要是一种动态现象：冷却液体时，内部自由度会逐渐失去平衡。一些理论方法预测其潜在相变发生在无限长时间的假象极限内。但是目前并不存在直接的实验证据来支持这些相变的存在。


The gelation transition of colloidal particles has been shown to be a second-order phase transition under nonequilibrium conditions. 在非平衡条件下，胶体粒子的凝胶化转变被认为是二级相变。


Characteristic properties 特征属性

Phase coexistence 相共存

A disorder-broadened first-order transition occurs over a finite range of temperatures where the fraction of the low-temperature equilibrium phase grows from zero to one (100%) as the temperature is lowered. This continuous variation of the coexisting fractions with temperature raised interesting possibilities. On cooling, some liquids vitrify into a glass rather than transform to the equilibrium crystal phase. This happens if the cooling rate is faster than a critical cooling rate, and is attributed to the molecular motions becoming so slow that the molecules cannot rearrange into the crystal positions. If the first-order freezing transition occurs over a range of temperatures, and Tg falls within this range, then there is an interesting possibility that the transition is arrested when it is partial and incomplete. Extending these ideas to first-order magnetic transitions being arrested at low temperatures, resulted in the observation of incomplete magnetic transitions, with two magnetic phases coexisting, down to the lowest temperature. First reported in the case of a ferromagnetic to anti-ferromagnetic transition, such persistent phase coexistence has now been reported across a variety of first-order magnetic transitions. These include colossal-magnetoresistance manganite materials,magnetocaloric materials,magnetic shape memory materials,The interesting feature of these observations of Tg falling within the temperature range over which the transition occurs is that the first-order magnetic transition is influenced by magnetic field, just like the structural transition is influenced by pressure. The relative ease with which magnetic fields can be controlled, in contrast to pressure, raises the possibility that one can study the interplay between Tg and Tc in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable the resolution of outstanding issues in understanding glasses.


A disorder-broadened first-order transition occurs over a finite range of temperatures where the fraction of the low-temperature equilibrium phase grows from zero to one (100%) as the temperature is lowered. This continuous variation of the coexisting fractions with temperature raised interesting possibilities. On cooling, some liquids vitrify into a glass rather than transform to the equilibrium crystal phase. This happens if the cooling rate is faster than a critical cooling rate, and is attributed to the molecular motions becoming so slow that the molecules cannot rearrange into the crystal positions. This slowing down happens below a glass-formation temperature Tg, which may depend on the applied pressure. If the first-order freezing transition occurs over a range of temperatures, and Tg falls within this range, then there is an interesting possibility that the transition is arrested when it is partial and incomplete. Extending these ideas to first-order magnetic transitions being arrested at low temperatures, resulted in the observation of incomplete magnetic transitions, with two magnetic phases coexisting, down to the lowest temperature. First reported in the case of a ferromagnetic to anti-ferromagnetic transition, such persistent phase coexistence has now been reported across a variety of first-order magnetic transitions. These include colossal-magnetoresistance manganite materials, magnetocaloric materials, magnetic shape memory materials, and other materials.The interesting feature of these observations of Tg falling within the temperature range over which the transition occurs is that the first-order magnetic transition is influenced by magnetic field, just like the structural transition is influenced by pressure. The relative ease with which magnetic fields can be controlled, in contrast to pressure, raises the possibility that one can study the interplay between Tg and Tc in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable the resolution of outstanding issues in understanding glasses.

在有限的温度范围内，胶体粒子的凝胶化转变已显示为bA紊乱-扩展的一阶转变。随着温度降低，低温平衡相的分数从零增加到一（100％）。随温度变化产生的馏分共存的连续变化带来了许多有趣的可能性。比如在冷却时，一些液体会逐渐玻璃化，而不是转变为平衡晶相。这种情况往往发生在冷却速率比临界冷却速率快的时候，归因于分子运动变得非常缓慢，以至于分子无法重新排列到晶体位置。分子运动的减速通常发生在玻璃的形成温度Tg以下，当然该温度同样可能取决于外在施加压力。如果该一阶冻结相变发生在一定温度范围内，并且Tg恰好落在该范围内，则会发生一种有趣的现象，即当转变是不完整时，该转变会停止。同理可以考虑在低温下被阻止的一阶磁相变，会导致观察到不完全的磁相变，即是说同时存在两个磁相直至最低温度。自首次报道关于铁磁到反铁磁相变以来，现在出现了各种关于一阶磁相变的持久相共存现象。包括了庞磁电阻锰矿材料，磁制冷材料，磁性形状记忆材料和其他材料。当Tg落在发生相变的温度范围内时，观测结果显得非常有趣，其一阶磁相变受到了磁场影响，就像结构相变会受到压力影响一样。与压力相比，控制磁场相对容易，于是提高了研究者们运用穷举法研究Tg和Tc之间相互作用的可能性。一阶磁相变的相位共存将有助于解决理解各种玻璃方面的突出问题。


Critical points 临界点

In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the critical point, at which the transition between liquid and gas becomes a second-order transition. Near the critical point, the fluid is sufficiently hot and compressed that the distinction between the liquid and gaseous phases is almost non-existent. This is associated with the phenomenon of critical opalescence, a milky appearance of the liquid due to density fluctuations at all possible wavelengths (including those of visible light).

In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the critical point, at which the transition between liquid and gas becomes a second-order transition. Near the critical point, the fluid is sufficiently hot and compressed that the distinction between the liquid and gaseous phases is almost non-existent. This is associated with the phenomenon of critical opalescence, a milky appearance of the liquid due to density fluctuations at all possible wavelengths (including those of visible light).

在任何包含液相和气相的系统中，都存在压力和温度的特殊组合，称为临界点，在该临界点处，液相和气相之间的转变即为二级相变。在临界点附近，如果流体足够热并且被压缩，那么几乎不存在液相和气相之间的区别。这与临界乳光现象有关，这是由于液体在所有可能的波长（包括可见光）处的密度波动引起的乳白色表现。


Symmetry 对称性

Phase transitions often involve a symmetry breaking process. For instance, the cooling of a fluid into a crystalline solid breaks continuous translation symmetry: each point in the fluid has the same properties, but each point in a crystal does not have the same properties (unless the points are chosen from the lattice points of the crystal lattice). Typically, the high-temperature phase contains more symmetries than the low-temperature phase due to spontaneous symmetry breaking, with the exception of certain accidental symmetries (e.g. the formation of heavy virtual particles, which only occurs at low temperatures).

Phase transitions often involve a symmetry breaking process. For instance, the cooling of a fluid into a crystalline solid breaks continuous translation symmetry: each point in the fluid has the same properties, but each point in a crystal does not have the same properties (unless the points are chosen from the lattice points of the crystal lattice). Typically, the high-temperature phase contains more symmetries than the low-temperature phase due to spontaneous symmetry breaking, with the exception of certain accidental symmetries (e.g. the formation of heavy virtual particles, which only occurs at low temperatures).

相变通常涉及到对称破坏。例如，将流体冷却至结晶固体会破坏其连续的平移对称性：流体中的每个点都具有相同的属性，但是晶体中的每个点都不具有相同的属性（除非这些点是从晶格点阵的晶格点中选择的）。通常，由于自发对称性破缺，除了某些偶然的对称性（例如，重虚粒子的形成，其仅在低温下发生）外，高温相比低温相具有更多的对称性。


Order parameters 序参数

An order parameter is a measure of the degree of order across the boundaries in a phase transition system; it normally ranges between zero in one phase (usually above the critical point) and nonzero in the other. At the critical point, the order parameter susceptibility will usually diverge.

An order parameter is a measure of the degree of order across the boundaries in a phase transition system; it normally ranges between zero in one phase (usually above the critical point) and nonzero in the other. At the critical point, the order parameter susceptibility will usually diverge.

序参数是相变系统中跨边界的有序/无序度量；它通常在一个为零的阶段（通常在临界点以上）与另一个非零阶段之间。在临界点，序参数的敏感性通常会发散。


An example of an order parameter is the net magnetization in a ferromagnetic system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities.

An example of an order parameter is the net magnetization in a ferromagnetic system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities.

关于序参数的一个示例是，发生相变的铁磁系统中的净磁化强度。对于液/气相变，序参数是它们的密度之差。


From a theoretical perspective, order parameters arise from symmetry breaking. When this happens, one needs to introduce one or more extra variables to describe the state of the system. For example, in the ferromagnetic phase, one must provide the net magnetization, whose direction was spontaneously chosen when the system cooled below the Curie point. However, note that order parameters can also be defined for non-symmetry-breaking transitions.

From a theoretical perspective, order parameters arise from symmetry breaking. When this happens, one needs to introduce one or more extra variables to describe the state of the system. For example, in the ferromagnetic phase, one must provide the net magnetization, whose direction was spontaneously chosen when the system cooled below the Curie point. However, note that order parameters can also be defined for non-symmetry-breaking transitions.

从理论的角度来看，序参数来自对称性破坏。当发生这种情况时，需要引入一个或多个其他变量来描述该系统状态。例如，在铁磁相中，必须提供净磁化强度，因为当系统冷却到居里点以下时，会自动选择其磁化方向。但是，值得注意的是序参数也可以为非对称破坏的相变定义。


Some phase transitions, such as superconducting and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases, the order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition.

Some phase transitions, such as superconducting and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases, the order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition.

某些相变，例如超导和铁磁，可以具有超过一个自由度的多个序参数。在这样的阶段中，序参数可以采用复数，向量甚至张量的形式，其大小在相变时会变为零。


There also exist dual descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as vortex- or defect lines.

There also exist dual descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as vortex- or defect lines.

就无序参数而言，也存在相变的双重描述。 这些表明存在线状激励line-like excitations，例如涡旋线vortex lines或缺陷线defect lines。


Relevance in cosmology 宇宙学的相关性

Symmetry-breaking phase transitions play an important role in cosmology. As the universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the electroweak field into the U(1) symmetry of the present-day electromagnetic field. This transition is important to understanding the asymmetry between the amount of matter and antimatter in the present-day universe (see electroweak baryogenesis).

Symmetry-breaking phase transitions play an important role in cosmology. As the universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the electroweak field into the U(1) symmetry of the present-day electromagnetic field. This transition is important to understanding the asymmetry between the amount of matter and antimatter in the present-day universe (see electroweak baryogenesis).

对称破坏性的相变在宇宙学中起着重要作用。随着宇宙的膨胀和冷却，真空经历了一系列对称破坏的相变。例如，电弱过渡将电弱场的SU（2）×U（1）对称性破坏为当今电磁场的U（1）对称性。这种转变对于理解当今宇宙中物质与反物质之间的不对称性很重要（请参阅弱电重子生成）。


Progressive phase transitions in an expanding universe are implicated in the development of order in the universe, as is illustrated by the work of Eric Chaisson and David Layzer.

Progressive phase transitions in an expanding universe are implicated in the development of order in the universe, as is illustrated by the work of Eric Chaisson and David Layzer.

埃里克·蔡森Eric Chaisson和戴维·莱泽David Layzer的研究表明，正在膨胀的宇宙中的渐进相变与宇宙中的秩序发展有关。


See also relational order theories and order and disorder.

See also relational order theories and order and disorder.

详情另参见关系秩序理论和秩序与无序。


Critical exponents and universality classes 临界指数和通用类型

Continuous phase transitions are easier to study than first-order transitions due to the absence of latent heat, and they have been discovered to have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points.

Continuous phase transitions are easier to study than first-order transitions due to the absence of latent heat, and they have been discovered to have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points.

由于不存在潜伏热Latent heat，连续相变比一阶相变更容易研究，目前发现它们具有许多有趣的性质。与连续相变有关的现象由于与临界点有关而被称为临界现象。


It turns out that continuous phase transitions can be characterized by parameters known as critical exponents. The most important one is perhaps the exponent describing the divergence of the thermal correlation length by approaching the transition. For instance, let us examine the behavior of the heat capacity near such a transition. We vary the temperature T of the system while keeping all the other thermodynamic variables fixed, and find that the transition occurs at some critical temperature Tc . When T is near Tc , the heat capacity C typically has a power law behavior,

It turns out that continuous phase transitions can be characterized by parameters known as critical exponents. The most important one is perhaps the exponent describing the divergence of the thermal correlation length by approaching the transition. For instance, let us examine the behavior of the heat capacity near such a transition. We vary the temperature of the system while keeping all the other thermodynamic variables fixed, and find that the transition occurs at some critical temperature Tc . When is near Tc , the heat capacity typically has a power law behavior,

事实证明，连续相变可以通过称为临界指数的参数来表征。其中最重要的一个参数也许是通过描述逼近相变时热相关长度差异的指数来表示。例如，让我们检测接近发生相变时的热容行为。我们在保持所有其他热力学变量不变的情况下，改变系统的温度T，发现相变发生在某个临界温度Tc处。当T接近Tc时，热容C通常具有幂律行为：

[math]\displaystyle{ C \propto |T_c - T|^{-\alpha}. }[/math]

[math]\displaystyle{ C \propto |T_c - T|^{-\alpha}. }[/math]

数学 c propto | tc-t | ^ {- alpha } . / math


The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent α = 0.59 A similar behavior, but with the exponent ν instead of α, applies for the correlation length.

The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent α = 0.59 A similar behavior, but with the exponent instead of , applies for the correlation length.

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。



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

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

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。



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]\displaystyle{ \gamma }[/math], the exponent of the susceptibility) are not identical.[3]

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]\displaystyle{ \gamma }[/math], the exponent of the susceptibility) are not identical.

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。



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 α = -0.013±0.003.

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.

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。

At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample.[4] This experimental value of α agrees with theoretical predictions based on variational perturbation theory.[5]

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.

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。



For 0 < α < 1, the heat capacity diverges at the transition temperature (though, since α < 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 α ∼ +0.110.

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.

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。



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.

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.

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。



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

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

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

[math]\displaystyle{ \beta=\gamma/(\delta-1) , \qquad \nu=\gamma/(2-\eta) }[/math].

[math]\displaystyle{ \beta=\gamma/(\delta-1) , \qquad \nu=\gamma/(2-\eta) }[/math].

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. ν and η.

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

可以证明只有两个独立的指数，例如。及。



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.

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.

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。



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.

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.

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。



Critical slowing down and other phenomena

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.

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.

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。



Percolation theory

逾渗理论

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.[6] 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.

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。


! -- 相变的平均场理论 --


! -- 波动 --


! -- 相变的重整化群理论 --


! -- om sairam--




Phase transitions in biological systems

生物系统中的相变

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.[7]

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.

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。



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.[8] 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.[9]

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.

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。



It has been proposed that some biological systems might lie near critical points. Examples include neural networks in the salamander retina,[10] bird flocks[11]

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

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

gene expression networks in Drosophila,[12] and protein folding.[13] However, it is not clear whether or not alternative reasons could explain some of the phenomena supporting arguments for criticality.[14] 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.[15]

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.

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。



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.[16]

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.

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。




Experimental

实验性的

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

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

各种方法被用来研究各种效应。选定的例子如下:





  • SQUID (measurement of magnetic transitions)



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





See also

参见























References

参考资料

  1. ↑ 1.0 1.1 Jaeger, Gregg (1 May 1998). "The Ehrenfest Classification of Phase Transitions: Introduction and Evolution". Archive for History of Exact Sciences. 53 (1): 51–81. doi:10.1007/s004070050021.
  2. ↑ 2.0 2.1 Blundell, Stephen J.; Katherine M. Blundell (2008). Concepts in Thermal Physics. Oxford University Press. ISBN 978-0-19-856770-7. 
  3. ↑ Leonard, F.; Delamotte, B. (2015). "Critical exponents can be different on the two sides of a transition". Phys. Rev. Lett. 115 (20): 200601. arXiv:1508.07852. Bibcode:2015PhRvL.115t0601L. doi:10.1103/PhysRevLett.115.200601. PMID 26613426.
  4. ↑ Lipa, J.; Nissen, J.; Stricker, D.; Swanson, D.; Chui, T. (2003). "Specific heat of liquid helium in zero gravity very near the lambda point". Physical Review B. 68 (17): 174518. arXiv:cond-mat/0310163. Bibcode:2003PhRvB..68q4518L. doi:10.1103/PhysRevB.68.174518.
  5. ↑ Kleinert, Hagen (1999). "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions". Physical Review D. 60 (8): 085001. arXiv:hep-th/9812197. Bibcode:1999PhRvD..60h5001K. doi:10.1103/PhysRevD.60.085001.
  6. ↑ Armin Bunde and Shlomo Havlin (1996). Fractals and Disordered Systems. Springer. http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php. 
  7. ↑ D.Y. Lando and V.B. Teif (2000). "Long-range interactions between ligands bound to a DNA molecule give rise to adsorption with the character of phase transition of the first kind". J. Biomol. Struct. Dynam. 17 (5): 903–911. doi:10.1080/07391102.2000.10506578. PMID 10798534.
  8. ↑ YashRoy, R.C. (1987). "13-C NMR studies of lipid fatty acyl chains of chloroplast membranes". Indian Journal of Biochemistry and Biophysics. 24 (6): 177–178.
  9. ↑ YashRoy, R C (1990). "Determination of membrane lipid phase transition temperature from 13-C NMR intensities". Journal of Biochemical and Biophysical Methods. 20 (4): 353–356. doi:10.1016/0165-022X(90)90097-V. PMID 2365951.
  10. ↑ Tkacik, Gasper; Mora, Thierry; Marre, Olivier; Amodei, Dario; Berry II, Michael J.; Bialek, William (2014). "Thermodynamics for a network of neurons: Signatures of criticality". arXiv:1407.5946 [q-bio.NC].
  11. ↑ Bialek, W; Cavagna, A; Giardina, I (2014). "Social interactions dominate speed control in poising natural flocks near criticality". PNAS. 111 (20): 7212–7217. arXiv:1307.5563. Bibcode:2014PNAS..111.7212B. doi:10.1073/pnas.1324045111. PMC 4034227. PMID 24785504.
  12. ↑ Krotov, D; Dubuis, J O; Gregor, T; Bialek, W (2014). "Morphogenesis at criticality". PNAS. 111 (10): 3683–3688. arXiv:1309.2614. Bibcode:2014PNAS..111.3683K. doi:10.1073/pnas.1324186111. PMC 3956198. PMID 24516161.
  13. ↑ Mora, Thierry; Bialek, William (2011). "Are biological systems poised at criticality?". Journal of Statistical Physics. 144 (2): 268–302. arXiv:1012.2242. Bibcode:2011JSP...144..268M. doi:10.1007/s10955-011-0229-4.
  14. ↑ Schwab, David J; Nemenman, Ilya; Mehta, Pankaj (2014). "Zipf's law and criticality in multivariate data without fine-tuning". Physical Review Letters. 113 (6): 068102. arXiv:1310.0448. Bibcode:2014PhRvL.113f8102S. doi:10.1103/PhysRevLett.113.068102. PMC 5142845. PMID 25148352.
  15. ↑ Longo, G.; Montévil, M. (2011-08-01). "From physics to biology by extending criticality and symmetry breakings". Progress in Biophysics and Molecular Biology. Systems Biology and Cancer. 106 (2): 340–347. arXiv:1103.1833. doi:10.1016/j.pbiomolbio.2011.03.005. PMID 21419157.
  16. ↑ Gorban, A.N.; Smirnova, E.V.; Tyukina, T.A. (August 2010). "Correlations, risk and crisis: From physiology to finance". Physica A: Statistical Mechanics and Its Applications. 389 (16): 3193–3217. arXiv:0905.0129. Bibcode:2010PhyA..389.3193G. doi:10.1016/j.physa.2010.03.035.




Further reading

进一步阅读




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



  • 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)[1]


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