平衡热力学

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Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermodynamic equilibrium there are no net macroscopic flows of matter or of energy, either within a system or between systems.

Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermodynamic equilibrium there are no net macroscopic flows of matter or of energy, either within a system or between systems.

热力学平衡是热力学 Thermodynamics的一个不言自明的 Axiomatic概念。它是单个热力学系统 Thermodynamic System的内部状态 State，或者是几个热力学系统之间通过或多或少的渗透或不渗透的壁 Wall连接的关系。，无论是在一个系统内还是在系统之间，在热力学平衡中不存在物质 Matter或能量 Energy的净宏观 Macroscopic 流动 Flow。

In a system that is in its own state of internal thermodynamic equilibrium, no macroscopic change occurs.

在一个处于内部热力学平衡状态的系统中，不会发生宏观 Macroscopic变化。

Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative equilibria. Systems can be in one kind of mutual equilibrium, though not in others. In thermodynamic equilibrium, all kinds of equilibrium hold at once and indefinitely, until disturbed by a thermodynamic operation. In a macroscopic equilibrium, perfectly or almost perfectly balanced microscopic exchanges occur; this is the physical explanation of the notion of macroscopic equilibrium.

互热力学平衡体系同时处于热 Thermal平衡、机械 Mechanical平衡、化学 Chemical平衡和辐射 Radiative平衡。系统处于一种相互平衡状态，就不能处于其他状态。在热力学平衡中所有的平衡同时并且无限期地保持，直到被热力学操作 Thermodynamic Operation打破。在一个宏观平衡中，完全或几乎完全平衡的微观交换发生了; 这是对宏观平衡概念的物理解释。

A thermodynamic system in a state of internal thermodynamic equilibrium has a spatially uniform temperature. Its intensive properties, other than temperature, may be driven to spatial inhomogeneity by an unchanging long-range force field imposed on it by its surroundings.

一个处于内部热力学系统状态的热力学平衡有一个空间均匀的温度 Temperature。除了温度以外，它的强度性质 Intensive Properties是由于周围环境施加的不变的长程力场而导致空间不均匀性。

In systems that are at a state of non-equilibrium there are, by contrast, net flows of matter or energy. If such changes can be triggered to occur in a system in which they are not already occurring, the system is said to be in a meta-stable equilibrium.

In systems that are at a state of non-equilibrium there are, by contrast, net flows of matter or energy. If such changes can be triggered to occur in a system in which they are not already occurring, the system is said to be in a meta-stable equilibrium.

相比之下，处于非平衡状态 non-equilibrium的系统中有物质或能量的净流动。如果这些变化可以在一个还没有发生的系统中被触发，那么这个系统就被称为处于一个亚稳定的平衡状态。

Though not a widely named a "law," it is an axiom of thermodynamics that there exist states of thermodynamic equilibrium. The second law of thermodynamics states that when a body of material starts from an equilibrium state, in which, portions of it are held at different states by more or less permeable or impermeable partitions, and a thermodynamic operation removes or makes the partitions more permeable and it is isolated, then it spontaneously reaches its own, new state of internal thermodynamic equilibrium, and this is accompanied by an increase in the sum of the entropies of the portions.

虽然不是一个广泛命名的“定律” ，但存在热力学平衡状态是一个热力学公理 Axiom。热力学第二定律 second law of thermodynamics指出，当一个物质体从一个平衡状态开始，在这个状态中，它的一部分被或多或少渗透或不渗透的分区保持在不同的状态，并且是孤立的，热力学操作移除分区或使分区更具渗透性，然后它自发地达到自己内部热力学平衡的新状态，并伴随着部分熵 Entropy的总和增加。

Overview

概览

模板:Thermodynamics

Classical thermodynamics deals with states of dynamic equilibrium. The state of a system at thermodynamic equilibrium is the one for which some thermodynamic potential is minimized, or for which the entropy (S) is maximized, for specified conditions. One such potential is the Helmholtz free energy (A), for a system with surroundings at controlled constant temperature and volume:

经典热力学研究动态平衡 Dynamic Equilibrium的状态。系统的热力学平衡状态是对于特定的条件，一些热动力位能 Thermodynamic Potential被最小化，或者熵 Entropy(S)被最大化。对于一个周围环境温度和体积可控的系统，有一个亥姆霍兹自由能 Helmholtz Free Energy(A)这样的位能:

[math]\displaystyle{ A = U - TS }[/math]

A = U-TS

Another potential, the Gibbs free energy (G), is minimized at thermodynamic equilibrium in a system with surroundings at controlled constant temperature and pressure:

另一个潜在的吉布斯自由能 Gibbs Free Energy(G)，在恒定温度和压力的系统中，热力学平衡最小:

[math]\displaystyle{ G = U - TS + PV }[/math]

where T denotes the absolute thermodynamic temperature, P the pressure, S the entropy, V the volume, and U the internal energy of the system.

其中 T 表示绝对热力学温度，P 表示压强，S 表示熵，V 表示体积，U 表示体系的内能。

Thermodynamic equilibrium is the unique stable stationary state that is approached or eventually reached as the system interacts with its surroundings over a long time. The above-mentioned potentials are mathematically constructed to be the thermodynamic quantities that are minimized under the particular conditions in the specified surroundings.

热力学平衡是一种独特的稳定定态，当系统长时间与周围环境相互作用时，它可以被接近或最终到达。上述位能是数学构造的热力学量，在特定的环境条件下最小化。

Conditions

情况

For a completely isolated system, S is maximum at thermodynamic equilibrium. 对于一个完全孤立的系统，S是热力学平衡中最大值。

For a system with controlled constant temperature and volume, A is minimum at thermodynamic equilibrium. 对于一个温度和体积可控的系统来说，A是热力学平衡的最小值。

For a system with controlled constant temperature and pressure, G is minimum at thermodynamic equilibrium. 对于一个恒温恒压的系统，G是热力学平衡的最小值。

The various types of equilibriums are achieved as follows:

实现各种类型的平衡的方法如下:

Two systems are in thermal equilibrium when their temperatures are the same. 当两个系统的温度 Temperature相同时，它们就处于热平衡状态

Two systems are in mechanical equilibrium when their pressures are the same. 当两个体系的压力 Pressure相同时，它们就处于力学平衡

Two systems are in diffusive equilibrium when their chemical potentials are the same. 当两个题体系的化学势 Chemical Potential相同时，它们就处于扩散平衡

All forces are balanced and there is no significant external driving force.

所有的力 Force都是平衡的，没有明显的外部驱动力

Relation of exchange equilibrium between systems

系统之间的交换均衡关系

Often the surroundings of a thermodynamic system may also be regarded as another thermodynamic system. In this view, one may consider the system and its surroundings as two systems in mutual contact, with long-range forces also linking them. The enclosure of the system is the surface of contiguity or boundary between the two systems. In the thermodynamic formalism, that surface is regarded as having specific properties of permeability. For example, the surface of contiguity may be supposed to be permeable only to heat, allowing energy to transfer only as heat. Then the two systems are said to be in thermal equilibrium when the long-range forces are unchanging in time and the transfer of energy as heat between them has slowed and eventually stopped permanently; this is an example of a contact equilibrium. Other kinds of contact equilibrium are defined by other kinds of specific permeability.^[1] When two systems are in contact equilibrium with respect to a particular kind of permeability, they have common values of the intensive variable that belongs to that particular kind of permeability. Examples of such intensive variables are temperature, pressure, chemical potential.

Often the surroundings of a thermodynamic system may also be regarded as another thermodynamic system. In this view, one may consider the system and its surroundings as two systems in mutual contact, with long-range forces also linking them. The enclosure of the system is the surface of contiguity or boundary between the two systems. In the thermodynamic formalism, that surface is regarded as having specific properties of permeability. For example, the surface of contiguity may be supposed to be permeable only to heat, allowing energy to transfer only as heat. Then the two systems are said to be in thermal equilibrium when the long-range forces are unchanging in time and the transfer of energy as heat between them has slowed and eventually stopped permanently; this is an example of a contact equilibrium. Other kinds of contact equilibrium are defined by other kinds of specific permeability. When two systems are in contact equilibrium with respect to a particular kind of permeability, they have common values of the intensive variable that belongs to that particular kind of permeability. Examples of such intensive variables are temperature, pressure, chemical potential.

通常，热力学系统的周围环境也可以被看作是另一个热力学系统。在这种观点中，我们可以把系统及其周围环境看作是相互接触的两个系统，远程作用力也将它们联系在一起。系统的包围物是两个系统之间的接触面或边界。在热力学公式中，该表面被认为具有特定的渗透性质。例如，接触的表面可能被认为只能透热，使能量只能作为热传递。当远程力在时间上不发生变化，两个系统之间的热量传递减慢并最终永久停止时，这两个系统被称为热平衡; 这就是接触平衡的一个例子。其它类型的接触平衡可用其它类型的比渗透率来定义。当两个系统对于某一特定类型的渗透率处于接触平衡时，它们具有属于该特定类型渗透率的强度变量的共同值。这种强度变量的例子有温度、压力、化学势。

A contact equilibrium may be regarded also as an exchange equilibrium. There is a zero balance of rate of transfer of some quantity between the two systems in contact equilibrium. For example, for a wall permeable only to heat, the rates of diffusion of internal energy as heat between the two systems are equal and opposite. An adiabatic wall between the two systems is 'permeable' only to energy transferred as work; at mechanical equilibrium the rates of transfer of energy as work between them are equal and opposite. If the wall is a simple wall, then the rates of transfer of volume across it are also equal and opposite; and the pressures on either side of it are equal. If the adiabatic wall is more complicated, with a sort of leverage, having an area-ratio, then the pressures of the two systems in exchange equilibrium are in the inverse ratio of the volume exchange ratio; this keeps the zero balance of rates of transfer as work.

接触平衡也可视为交换平衡。在接触平衡状态下，两系统之间某些量的传递速率存在零平衡。例如，对于只能透热的壁，内能作为热在两个系统之间的扩散速率是相等并相反的。两个系统之间的绝热壁只对作为功传递的能量有渗透作用; 在力学平衡，两个系统之间作为功的能量传递速率相等且相反。如果是一个简单的壁，那么通过它的体积转移率也是相等且相反的; 即它两边的压力是相等的。如果绝热壁比较复杂，有一种杠杆，有一个面积比，那么两个体系在交换平衡中的压力与体积交换比成反比，这使得转移率的零平衡作功。

A radiative exchange can occur between two otherwise separate systems. Radiative exchange equilibrium prevails when the two systems have the same temperature.^[2]

A radiative exchange can occur between two otherwise separate systems. Radiative exchange equilibrium prevails when the two systems have the same temperature.

辐射交换可以发生在两个不同的系统之间。当两个体系温度相同时，辐射交换平衡占优势。

Thermodynamic state of internal equilibrium of a system

系统内部平衡的热力学状态

A collection of matter may be entirely isolated from its surroundings. If it has been left undisturbed for an indefinitely long time, classical thermodynamics postulates that it is in a state in which no changes occur within it, and there are no flows within it. This is a thermodynamic state of internal equilibrium.^[3]^[4] (This postulate is sometimes, but not often, called the "minus first" law of thermodynamics.^[5] One textbook^[6] calls it the "zeroth law", remarking that the authors think this more befitting that title than its more customary definition, which apparently was suggested by Fowler.)

A collection of matter may be entirely isolated from its surroundings. If it has been left undisturbed for an indefinitely long time, classical thermodynamics postulates that it is in a state in which no changes occur within it, and there are no flows within it. This is a thermodynamic state of internal equilibrium. (This postulate is sometimes, but not often, called the "minus first" law of thermodynamics. One textbook calls it the "zeroth law", remarking that the authors think this more befitting that title than its more customary definition, which apparently was suggested by Fowler.)

物质的集合可能与其周围的环境完全孤立 Isolated。如果它在无限长的时间内一直保持不受干扰，按照经典热力学假定，它处于一个没有发生任何变化，没有流动的状态，即内部平衡的热力学状态。(这种假设有时被称为“负第一”热力学定律，但并不常见。有教科书称之为“第零定律” ，作者福勒 Fowler认为这个名称是更符合惯例的定义 More Customary Definition。)

Such states are a principal concern in what is known as classical or equilibrium thermodynamics, for they are the only states of the system that are regarded as well defined in that subject. A system in contact equilibrium with another system can by a thermodynamic operation be isolated, and upon the event of isolation, no change occurs in it. A system in a relation of contact equilibrium with another system may thus also be regarded as being in its own state of internal thermodynamic equilibrium.

这种状态是所谓的经典热力学或平衡态热力学的主要关注点，因为它们是系统中被认为在这门学科中得到很好定义的唯一状态。一个与另一个系统处于接触平衡状态可以被一个热力学操作 Thermodynamic Operation隔离，在隔离发生时，其内部不会发生任何变化。因此，一个与另一个系统处于接触平衡状态时也可以被视为处于其自身的内部热力学平衡状态。

Multiple contact equilibrium

多点接触平衡

The thermodynamic formalism allows that a system may have contact with several other systems at once, which may or may not also have mutual contact, the contacts having respectively different permeabilities. If these systems are all jointly isolated from the rest of the world those of them that are in contact then reach respective contact equilibria with one another.

热力学形式论允许一个系统同时与其他多个系统接触，这些系统可能也可能没有相互接触，且这些接触具有不同的渗透性。如果这些系统都与世界其他部分相互隔离，那么它们彼此之间就会达到各自的接触平衡。

If several systems are free of adiabatic walls between each other, but are jointly isolated from the rest of the world, then they reach a state of multiple contact equilibrium, and they have a common temperature, a total internal energy, and a total entropy.^[7]^[8]^[9]^[10] Amongst intensive variables, this is a unique property of temperature. It holds even in the presence of long-range forces. (That is, there is no "force" that can maintain temperature discrepancies.) For example, in a system in thermodynamic equilibrium in a vertical gravitational field, the pressure on the top wall is less than that on the bottom wall, but the temperature is the same everywhere.

If several systems are free of adiabatic walls between each other, but are jointly isolated from the rest of the world, then they reach a state of multiple contact equilibrium, and they have a common temperature, a total internal energy, and a total entropy. Amongst intensive variables, this is a unique property of temperature. It holds even in the presence of long-range forces. (That is, there is no "force" that can maintain temperature discrepancies.) For example, in a system in thermodynamic equilibrium in a vertical gravitational field, the pressure on the top wall is less than that on the bottom wall, but the temperature is the same everywhere.

如果几个系统彼此之间没有绝热壁，但是它们与世界其他部分共同隔离，那么它们就会达到多重接触平衡状态，且有共同的温度，内能和熵。在众多变量中，这是温度的一个独特性质。即使在远距离作用力存在的情况下，它也是有效的。(也就是说，没有“力”可以维持温度的差异。)举个例子，在热力学平衡的一个垂直的引力场系统中，顶部壁面的压力比底部壁面的压力小，但是各处的温度都是一样的。

A thermodynamic operation may occur as an event restricted to the walls that are within the surroundings, directly affecting neither the walls of contact of the system of interest with its surroundings, nor its interior, and occurring within a definitely limited time. For example, an immovable adiabatic wall may be placed or removed within the surroundings. Consequent upon such an operation restricted to the surroundings, the system may be for a time driven away from its own initial internal state of thermodynamic equilibrium. Then, according to the second law of thermodynamics, the whole undergoes changes and eventually reaches a new and final equilibrium with the surroundings. Following Planck, this consequent train of events is called a natural thermodynamic process.^[11] It is allowed in equilibrium thermodynamics just because the initial and final states are of thermodynamic equilibrium, even though during the process there is transient departure from thermodynamic equilibrium, when neither the system nor its surroundings are in well defined states of internal equilibrium. A natural process proceeds at a finite rate for the main part of its course. It is thereby radically different from a fictive quasi-static 'process' that proceeds infinitely slowly throughout its course, and is fictively 'reversible'. Classical thermodynamics allows that even though a process may take a very long time to settle to thermodynamic equilibrium, if the main part of its course is at a finite rate, then it is considered to be natural, and to be subject to the second law of thermodynamics, and thereby irreversible. Engineered machines and artificial devices and manipulations are permitted within the surroundings.^[12]^[13] The allowance of such operations and devices in the surroundings but not in the system is the reason why Kelvin in one of his statements of the second law of thermodynamics spoke of "inanimate" agency; a system in thermodynamic equilibrium is inanimate.^[14]

A thermodynamic operation may occur as an event restricted to the walls that are within the surroundings, directly affecting neither the walls of contact of the system of interest with its surroundings, nor its interior, and occurring within a definitely limited time. For example, an immovable adiabatic wall may be placed or removed within the surroundings. Consequent upon such an operation restricted to the surroundings, the system may be for a time driven away from its own initial internal state of thermodynamic equilibrium. Then, according to the second law of thermodynamics, the whole undergoes changes and eventually reaches a new and final equilibrium with the surroundings. Following Planck, this consequent train of events is called a natural thermodynamic process. It is allowed in equilibrium thermodynamics just because the initial and final states are of thermodynamic equilibrium, even though during the process there is transient departure from thermodynamic equilibrium, when neither the system nor its surroundings are in well defined states of internal equilibrium. A natural process proceeds at a finite rate for the main part of its course. It is thereby radically different from a fictive quasi-static 'process' that proceeds infinitely slowly throughout its course, and is fictively 'reversible'. Classical thermodynamics allows that even though a process may take a very long time to settle to thermodynamic equilibrium, if the main part of its course is at a finite rate, then it is considered to be natural, and to be subject to the second law of thermodynamics, and thereby irreversible. Engineered machines and artificial devices and manipulations are permitted within the surroundings. The allowance of such operations and devices in the surroundings but not in the system is the reason why Kelvin in one of his statements of the second law of thermodynamics spoke of "inanimate" agency; a system in thermodynamic equilibrium is inanimate.

热力学操作可能作为一个事件发生在周围环境的墙壁上，既不直接影响与周围环境联系的墙壁，也不直接影响其内部，且发生在一个明确的有限的时间内。例如，一个固定的绝热壁可以在环境中被放置或拆除。由于这种操作仅限于周围环境，系统可能会在一段时间内远离自身最初的内部状态---- 热力学平衡。根据热力学第二定律的说法，整体经历了变化并最终与周围环境达到了新的平衡。继Planck之后，这一连串的事件被称为自然热力学过程 Thermodynamic Process。即使在这个过程中，当系统和周围环境都不处于明确的内部平衡状态时，存在着暂时偏离热力学平衡的现象，由于初始状态和最终状态都是热力学平衡的，这在平衡热力学中也是允许的。一个自然过程在其主要过程中以有限的速率进行，因此它从根本上不同于虚构的准静态“过程”，后者在整个过程中无限缓慢地进行，且虚构的“可逆”。经典热力学允许，即使一个过程可能需要很长的时间才能达到热力学平衡，如果主要部分的过程是在一个有限的比率中，那么它被认为是自然的，并受制于热力学第二定律，即不可逆的。工程设计的机器、人工设备和操作是允许在周围环境中进行的。允许在环境中而不是在系统中进行此类操作和设备，是开尔文在他的一个热力学第二定律的陈述中提到无生命机构 Inanimate Agency的原因; 在热力学平衡的系统是无生命的。

Otherwise, a thermodynamic operation may directly affect a wall of the system.

否则，热力学操作可能会直接影响系统的壁。

It is often convenient to suppose that some of the surrounding subsystems are so much larger than the system that the process can affect the intensive variables only of the surrounding subsystems, and they are then called reservoirs for relevant intensive variables.

通常可以很方便地假设周围的某些子系统比系统大得多，以至于这个过程只能影响周围子系统的强度变量，然后将这些子系统称为相关强度变量的储备库。

Local and global equilibrium

局部和全局均衡

It is useful to distinguish between global and local thermodynamic equilibrium. In thermodynamics, exchanges within a system and between the system and the outside are controlled by intensive parameters. As an example, temperature controls heat exchanges. Global thermodynamic equilibrium (GTE) means that those intensive parameters are homogeneous throughout the whole system, while local thermodynamic equilibrium (LTE) means that those intensive parameters are varying in space and time, but are varying so slowly that, for any point, one can assume thermodynamic equilibrium in some neighborhood about that point.

It is useful to distinguish between global and local thermodynamic equilibrium. In thermodynamics, exchanges within a system and between the system and the outside are controlled by intensive parameters. As an example, temperature controls heat exchanges. Global thermodynamic equilibrium (GTE) means that those intensive parameters are homogeneous throughout the whole system, while local thermodynamic equilibrium (LTE) means that those intensive parameters are varying in space and time, but are varying so slowly that, for any point, one can assume thermodynamic equilibrium in some neighborhood about that point.

区分全局性和局部性热力学平衡是很有用的。在热力学中，系统内部和系统与外部之间的交换是由强度 Intensive参数控制的。例如，温度 Temperature控制热交换 Heat Equation。全局热力学平衡(GTE)意味着这些强度 Intensive参数在整个系统中是均匀的，而局部热力学平衡(LTE)意味着这些强度参数在空间和时间上是变化的，但变化如此缓慢，以至于对于任何一点，人们都可以假设在某个邻近的某个热力学平衡。

If the description of the system requires variations in the intensive parameters that are too large, the very assumptions upon which the definitions of these intensive parameters are based will break down, and the system will be in neither global nor local equilibrium. For example, it takes a certain number of collisions for a particle to equilibrate to its surroundings. If the average distance it has moved during these collisions removes it from the neighborhood it is equilibrating to, it will never equilibrate, and there will be no LTE. Temperature is, by definition, proportional to the average internal energy of an equilibrated neighborhood. Since there is no equilibrated neighborhood, the concept of temperature doesn't hold, and the temperature becomes undefined.

如果对系统的描述要求强度参数的变化过大，那么这些强度参数的定义所依据的假设本身就会破裂，系统将既不处于全局平衡，也不处于局部平衡。例如，粒子需要一定数量的碰撞来平衡其周围环境。如果在这些碰撞中移动的平均距离使它离开平衡邻域，它将永远不会平衡，也就不会有 LTE。根据定义，温度与平衡邻域的平均内部能量成正比。由于没有达到平衡邻域，温度的概念就不成立，温度也就变得不确定。

It is important to note that this local equilibrium may apply only to a certain subset of particles in the system. For example, LTE is usually applied only to massive particles. In a radiating gas, the photons being emitted and absorbed by the gas doesn't need to be in a thermodynamic equilibrium with each other or with the massive particles of the gas in order for LTE to exist. In some cases, it is not considered necessary for free electrons to be in equilibrium with the much more massive atoms or molecules for LTE to exist.

值得注意的是，这种局部平衡可能只适用于系统中的某个粒子子集。例如，LTE 通常只适用于大 Massive质量粒子。在辐射 Radiation气体中，被气体发射和吸收的光子 Photon不需要彼此在一个热力学平衡内，或与气体中的大量粒子在一起，LTE 才能存在。在某些情况下，自由电子并不需要与大得多的原子或分子处于平衡状态，LTE 才能存在。

As an example, LTE will exist in a glass of water that contains a melting ice cube. The temperature inside the glass can be defined at any point, but it is colder near the ice cube than far away from it. If energies of the molecules located near a given point are observed, they will be distributed according to the Maxwell–Boltzmann distribution for a certain temperature. If the energies of the molecules located near another point are observed, they will be distributed according to the Maxwell–Boltzmann distribution for another temperature.

例如，LTE 将存在于一个装有融化的冰块 Ice Cube的水杯中。玻璃杯内的温度可以在任何时候定义，但是离冰块近的地方要比离冰块远的地方冷。如果观察到位于给定点附近的分子的能量，它们将按照麦克斯韦-波兹曼分布在一定温度下的分布。如果观察到位于另一点附近的分子的能量，它们将按照麦克斯韦-波兹曼分布 Maxwell–Boltzmann distribution分布到另一个温度。

Local thermodynamic equilibrium does not require either local or global stationarity. In other words, each small locality need not have a constant temperature. However, it does require that each small locality change slowly enough to practically sustain its local Maxwell–Boltzmann distribution of molecular velocities. A global non-equilibrium state can be stably stationary only if it is maintained by exchanges between the system and the outside. For example, a globally-stable stationary state could be maintained inside the glass of water by continuously adding finely powdered ice into it in order to compensate for the melting, and continuously draining off the meltwater. Natural transport phenomena may lead a system from local to global thermodynamic equilibrium. Going back to our example, the diffusion of heat will lead our glass of water toward global thermodynamic equilibrium, a state in which the temperature of the glass is completely homogeneous.^[15]

Local thermodynamic equilibrium does not require either local or global stationarity. In other words, each small locality need not have a constant temperature. However, it does require that each small locality change slowly enough to practically sustain its local Maxwell–Boltzmann distribution of molecular velocities. A global non-equilibrium state can be stably stationary only if it is maintained by exchanges between the system and the outside. For example, a globally-stable stationary state could be maintained inside the glass of water by continuously adding finely powdered ice into it in order to compensate for the melting, and continuously draining off the meltwater. Natural transport phenomena may lead a system from local to global thermodynamic equilibrium. Going back to our example, the diffusion of heat will lead our glass of water toward global thermodynamic equilibrium, a state in which the temperature of the glass is completely homogeneous.

局部热力学平衡不需要局部或全局的平稳性。换句话说，每一个小区域不需要有一个恒定的温度。然而，它确实要求每个小的局部变化缓慢到足以维持其局部麦克斯韦-波兹曼分布的分子速度。全局非平衡态只有通过系统与外界的交换才能保持稳定。例如，通过在水杯中不断添加细粉冰来补偿熔化，并持续排出融水，可以保持全球稳定的静止状态。自然迁移现象 Transport Phenomena可能导致一个从局部到全局的热力学平衡系统。回顾我们的例子，热量的扩散将导致我们的玻璃杯水流向全局热力学平衡，在这种状态下，玻璃杯的温度是完全均匀的。

Reservations

保留意见

Careful and well informed writers about thermodynamics, in their accounts of thermodynamic equilibrium, often enough make provisos or reservations to their statements. Some writers leave such reservations merely implied or more or less unstated.

仔细和消息灵通的热力学作家，在他们对热力学平衡的描述中，经常对他们的陈述提出附带条件或保留意见。有些笔者只是含蓄地或多或少地保留了这种意见。

For example, one widely cited writer, H. B. Callen writes in this context: "In actuality, few systems are in absolute and true equilibrium." He refers to radioactive processes and remarks that they may take "cosmic times to complete, [and] generally can be ignored". He adds "In practice, the criterion for equilibrium is circular. Operationally, a system is in an equilibrium state if its properties are consistently described by thermodynamic theory!"^[16]

For example, one widely cited writer, H. B. Callen writes in this context: "In actuality, few systems are in absolute and true equilibrium." He refers to radioactive processes and remarks that they may take "cosmic times to complete, [and] generally can be ignored". He adds "In practice, the criterion for equilibrium is circular. Operationally, a system is in an equilibrium state if its properties are consistently described by thermodynamic theory!"

例如，一位被广泛引用的作家H.B.卡伦 H.B.Callen在这里写道: “实际上，很少有系统处于绝对和真正的平衡状态。”他提到了放射性过程，认为它们可能需要“宇宙时间才能完成，因此通常可以忽略”。他补充道: “在实践中，平衡的标准是循环的。在操作上，如果一个系统的性质一致地用热力学理论来描述，那么它就处于平衡状态! ”

J.A. Beattie and I. Oppenheim write: "Insistence on a strict interpretation of the definition of equilibrium would rule out the application of thermodynamics to practically all states of real systems."^[17]

J.A. Beattie and I. Oppenheim write: "Insistence on a strict interpretation of the definition of equilibrium would rule out the application of thermodynamics to practically all states of real systems."

J.A.贝蒂和 I.奥本海姆写道: “坚持对平衡定义的严格解释，将排除热力学应用于实际系统的所有状态。”

Another author, cited by Callen as giving a "scholarly and rigorous treatment",^[18] and cited by Adkins as having written a "classic text",^[19] A.B. Pippard writes in that text: "Given long enough a supercooled vapour will eventually condense, ... . The time involved may be so enormous, however, perhaps 10¹⁰⁰ years or more, ... . For most purposes, provided the rapid change is not artificially stimulated, the systems may be regarded as being in equilibrium."^[20]

Another author, cited by Callen as giving a "scholarly and rigorous treatment", and cited by Adkins as having written a "classic text", A.B. Pippard writes in that text: "Given long enough a supercooled vapour will eventually condense, ... . The time involved may be so enormous, however, perhaps 10¹⁰⁰ years or more, ... . For most purposes, provided the rapid change is not artificially stimulated, the systems may be regarded as being in equilibrium."

Callen引用另一位作者的话说，他给出了“学术且严谨的论述” ，Adkins引用他的话说，他写了一本“经典著作”—— A.B.皮帕德 A.B.Pippard在文中写道: “只要时间足够长，过冷水蒸汽最终会凝结，... ..。时间可能是巨大的，也许长达10年或者更长。就大多数目的而言，只要这种迅速的变化不是人为地刺激，这些系统就可以被视为处于平衡状态。”

Another author, A. Münster, writes in this context. He observes that thermonuclear processes often occur so slowly that they can be ignored in thermodynamics. He comments: "The concept 'absolute equilibrium' or 'equilibrium with respect to all imaginable processes', has therefore, no physical significance." He therefore states that: "... we can consider an equilibrium only with respect to specified processes and defined experimental conditions." ^[21]

Another author, A. Münster, writes in this context. He observes that thermonuclear processes often occur so slowly that they can be ignored in thermodynamics. He comments: "The concept 'absolute equilibrium' or 'equilibrium with respect to all imaginable processes', has therefore, no physical significance." He therefore states that: "... we can consider an equilibrium only with respect to specified processes and defined experimental conditions."

另一位作者A.Münster，写在这种情况下。他观察到热核反应发生的速度非常缓慢，以至于在热力学中可以忽略不计。他评论道: “‘绝对平衡’或‘所有可想象过程的平衡’的概念没有物理意义。”他说: “ ... 我们只能考虑特定过程和确定的实验条件下的平衡。”

According to L. Tisza: "... in the discussion of phenomena near absolute zero. The absolute predictions of the classical theory become particularly vague because the occurrence of frozen-in nonequilibrium states is very common."^[22]

According to L. Tisza: "... in the discussion of phenomena near absolute zero. The absolute predictions of the classical theory become particularly vague because the occurrence of frozen-in nonequilibrium states is very common."

根据L.缇莎 L.Tisza的说法: “ ... 在讨论接近绝对零度的现象时。经典理论的绝对预测变得特别模糊，因为在非平衡状态下发生冻结是非常普遍的。”

Definitions

定义

The most general kind of thermodynamic equilibrium of a system is through contact with the surroundings that allows simultaneous passages of all chemical substances and all kinds of energy. A system in thermodynamic equilibrium may move with uniform acceleration through space but must not change its shape or size while doing so; thus it is defined by a rigid volume in space. It may lie within external fields of force, determined by external factors of far greater extent than the system itself, so that events within the system cannot in an appreciable amount affect the external fields of force. The system can be in thermodynamic equilibrium only if the external force fields are uniform, and are determining its uniform acceleration, or if it lies in a non-uniform force field but is held stationary there by local forces, such as mechanical pressures, on its surface.

一个系统最普遍的热力学平衡是通过与周围环境的接触，允许所有化学物质和各种能量同时通过。热力学平衡中的系统可以以均匀加速度在空间中运动，但在这样做时不能改变其形状或大小; 因此它是由空间中的刚性体积来定义的。它可能存在于外力场中，由远远大于系统本身的外部因素决定，因此系统内的事件不会对外力场产生相当大的影响。只有当外力场是均匀的，并且确定了它的均匀加速度，或者它处于一个非均匀力场中，但是由于表面的局部力，例如机械压力，使它保持静止时，这个系统才能处于热力学平衡。

Thermodynamic equilibrium is a primitive notion of the theory of thermodynamics. According to P.M. Morse: "It should be emphasized that the fact that there are thermodynamic states, ..., and the fact that there are thermodynamic variables which are uniquely specified by the equilibrium state ... are not conclusions deduced logically from some philosophical first principles. They are conclusions ineluctably drawn from more than two centuries of experiments."^[23] This means that thermodynamic equilibrium is not to be defined solely in terms of other theoretical concepts of thermodynamics. M. Bailyn proposes a fundamental law of thermodynamics that defines and postulates the existence of states of thermodynamic equilibrium.^[24]

Thermodynamic equilibrium is a primitive notion of the theory of thermodynamics. According to P.M. Morse: "It should be emphasized that the fact that there are thermodynamic states, ..., and the fact that there are thermodynamic variables which are uniquely specified by the equilibrium state ... are not conclusions deduced logically from some philosophical first principles. They are conclusions ineluctably drawn from more than two centuries of experiments." This means that thermodynamic equilibrium is not to be defined solely in terms of other theoretical concepts of thermodynamics. M. Bailyn proposes a fundamental law of thermodynamics that defines and postulates the existence of states of thermodynamic equilibrium.

热力学平衡是热力学理论的一个基本概念 Primitive Notion。P.M.莫尔斯 P.M.Morse说: “应该强调的是，存在热力学状态这一事实，以及存在由平衡态唯一指定的热力学变量这一事实，并不是从某些哲学第一原理得出的逻辑结论。这些结论不可避免地来自两个多世纪的实验。”这意味着热力学平衡不能仅仅用热力学的其他理论概念来定义。提出了一个基本的热力学定律理论，它定义并假设了热力学平衡的存在。

Textbook definitions of thermodynamic equilibrium are often stated carefully, with some reservation or other.

教科书上对热力学平衡的定义经常被小心地陈述，有些保留或其他。

For example, A. Münster writes: "An isolated system is in thermodynamic equilibrium when, in the system, no changes of state are occurring at a measurable rate." There are two reservations stated here; the system is isolated; any changes of state are immeasurably slow. He discusses the second proviso by giving an account of a mixture oxygen and hydrogen at room temperature in the absence of a catalyst. Münster points out that a thermodynamic equilibrium state is described by fewer macroscopic variables than is any other state of a given system. This is partly, but not entirely, because all flows within and through the system are zero.^[25]

For example, A. Münster writes: "An isolated system is in thermodynamic equilibrium when, in the system, no changes of state are occurring at a measurable rate." There are two reservations stated here; the system is isolated; any changes of state are immeasurably slow. He discusses the second proviso by giving an account of a mixture oxygen and hydrogen at room temperature in the absence of a catalyst. Münster points out that a thermodynamic equilibrium state is described by fewer macroscopic variables than is any other state of a given system. This is partly, but not entirely, because all flows within and through the system are zero.

例如，A.Münster 写道: “一个孤立的系统处于热力学平衡时期，在这个系统中，状态没有发生可测量的变化。”这里陈述了两个保留意见; 系统是隔离的; 状态的任何更改都极其缓慢。他讨论了第二个限制性条款，给出了在没有催化剂的情况下室温下氧和氢的混合物。明斯特指出，热力学平衡状态比给定系统的其他状态用更少的宏观变量来描述。这是部分的，但不是完全的，因为系统内部和通过系统的所有流都是零。

R. Haase's presentation of thermodynamics does not start with a restriction to thermodynamic equilibrium because he intends to allow for non-equilibrium thermodynamics. He considers an arbitrary system with time invariant properties. He tests it for thermodynamic equilibrium by cutting it off from all external influences, except external force fields. If after insulation, nothing changes, he says that the system was in equilibrium.^[26]

R. Haase's presentation of thermodynamics does not start with a restriction to thermodynamic equilibrium because he intends to allow for non-equilibrium thermodynamics. He considers an arbitrary system with time invariant properties. He tests it for thermodynamic equilibrium by cutting it off from all external influences, except external force fields. If after insulation, nothing changes, he says that the system was in equilibrium.

R. Haase's的热力学演示并不以热力学平衡为开始，因为他打算考虑非平衡态热力学。他考虑一个具有时不变性质的任意系统。他通过切断除外力场以外的所有外部影响来测试它的热力学平衡。如果在绝缘之后，没有任何变化，他说，系统处于平衡状态。

In a section headed "Thermodynamic equilibrium", H.B. Callen defines equilibrium states in a paragraph. He points out that they "are determined by intrinsic factors" within the system. They are "terminal states", towards which the systems evolve, over time, which may occur with "glacial slowness".^[27] This statement does not explicitly say that for thermodynamic equilibrium, the system must be isolated; Callen does not spell out what he means by the words "intrinsic factors".

In a section headed "Thermodynamic equilibrium", H.B. Callen defines equilibrium states in a paragraph. He points out that they "are determined by intrinsic factors" within the system. They are "terminal states", towards which the systems evolve, over time, which may occur with "glacial slowness". This statement does not explicitly say that for thermodynamic equilibrium, the system must be isolated; Callen does not spell out what he means by the words "intrinsic factors".

在一个标题为“热力学平衡”的章节中，H.B. Callen在一段话中定义平衡状态.他指出，它们“是由系统内部的内在因素决定的”。它们是“终端状态” ，随着时间的推移，系统会朝着这个终端状态演化，这种演化可能会以“冰川般缓慢”的速度发生。这个声明并没有明确说明对于热力学平衡来说，系统必须是孤立的; Callen也没有说明他所说的“内在因素”是什么意思。

Another textbook writer, C.J. Adkins, explicitly allows thermodynamic equilibrium to occur in a system which is not isolated. His system is, however, closed with respect to transfer of matter. He writes: "In general, the approach to thermodynamic equilibrium will involve both thermal and work-like interactions with the surroundings." He distinguishes such thermodynamic equilibrium from thermal equilibrium, in which only thermal contact is mediating transfer of energy.^[28]

Another textbook writer, C.J. Adkins, explicitly allows thermodynamic equilibrium to occur in a system which is not isolated. His system is, however, closed with respect to transfer of matter. He writes: "In general, the approach to thermodynamic equilibrium will involve both thermal and work-like interactions with the surroundings." He distinguishes such thermodynamic equilibrium from thermal equilibrium, in which only thermal contact is mediating transfer of energy.

另一位教科书作者，C.J.Adkins，明确地允许热力学平衡在非孤立的系统中发生。然而，他的系统在物质转移方面是封闭的。他写道: “一般来说，热力学平衡的方法将包括与周围环境的热和类似工作的相互作用。”他将这种热力学平衡与只有热接触才能调解能量传递的热平衡相区别。

Another textbook author, J.R. Partington, writes: "(i) An equilibrium state is one which is independent of time." But, referring to systems "which are only apparently in equilibrium", he adds : "Such systems are in states of ″false equilibrium.″" Partington's statement does not explicitly state that the equilibrium refers to an isolated system. Like Münster, Partington also refers to the mixture of oxygen and hydrogen. He adds a proviso that "In a true equilibrium state, the smallest change of any external condition which influences the state will produce a small change of state ..."^[29] This proviso means that thermodynamic equilibrium must be stable against small perturbations; this requirement is essential for the strict meaning of thermodynamic equilibrium.

Another textbook author, J.R. Partington, writes: "(i) An equilibrium state is one which is independent of time." But, referring to systems "which are only apparently in equilibrium", he adds : "Such systems are in states of ″false equilibrium.″" Partington's statement does not explicitly state that the equilibrium refers to an isolated system. Like Münster, Partington also refers to the mixture of oxygen and hydrogen. He adds a proviso that "In a true equilibrium state, the smallest change of any external condition which influences the state will produce a small change of state ..." This proviso means that thermodynamic equilibrium must be stable against small perturbations; this requirement is essential for the strict meaning of thermodynamic equilibrium.

另一位教科书作者J.R.帕廷顿 J.R.Partington写道: “(i)平衡状态是独立于时间的状态。”但是，在提到“只是明显处于平衡状态”的系统时，他补充说: “这样的系统处于‘虚假平衡’状态。帕廷顿的陈述没有明确指出平衡是指一个孤立的系统。和Münster一样，Partington也指的是氧和氢的混合物。他补充了一个条件，“在一个真正的平衡状态，任何影响状态的外部条件的最小变化都会产生一个微小的状态变化... ... ”这个条件意味着热力学平衡必须在小的扰动下保持稳定; 这个要求对于热力学平衡的严格意义是必不可少的。

A student textbook by F.H. Crawford has a section headed "Thermodynamic Equilibrium". It distinguishes several drivers of flows, and then says: "These are examples of the apparently universal tendency of isolated systems toward a state of complete mechanical, thermal, chemical, and electrical—or, in a single word, thermodynamic—equilibrium."^[30]

A student textbook by F.H. Crawford has a section headed "Thermodynamic Equilibrium". It distinguishes several drivers of flows, and then says: "These are examples of the apparently universal tendency of isolated systems toward a state of complete mechanical, thermal, chemical, and electrical—or, in a single word, thermodynamic—equilibrium."

在F.H.Crawford的一本学生教科书中，有一个标题为“热力学平衡”的章节。它区分了几种流动的驱动因素，然后说: “这些是孤立系统明显普遍趋向于完全机械、热、化学和电力状态的例子——或者简单地说，热力学平衡状态。”

A monograph on classical thermodynamics by H.A. Buchdahl considers the "equilibrium of a thermodynamic system", without actually writing the phrase "thermodynamic equilibrium". Referring to systems closed to exchange of matter, Buchdahl writes: "If a system is in a terminal condition which is properly static, it will be said to be in equilibrium."^[31] Buchdahl's monograph also discusses amorphous glass, for the purposes of thermodynamic description. It states: "More precisely, the glass may be regarded as being in equilibrium so long as experimental tests show that 'slow' transitions are in effect reversible."^[32] It is not customary to make this proviso part of the definition of thermodynamic equilibrium, but the converse is usually assumed: that if a body in thermodynamic equilibrium is subject to a sufficiently slow process, that process may be considered to be sufficiently nearly reversible, and the body remains sufficiently nearly in thermodynamic equilibrium during the process.^[33]

A monograph on classical thermodynamics by H.A. Buchdahl considers the "equilibrium of a thermodynamic system", without actually writing the phrase "thermodynamic equilibrium". Referring to systems closed to exchange of matter, Buchdahl writes: "If a system is in a terminal condition which is properly static, it will be said to be in equilibrium." Buchdahl's monograph also discusses amorphous glass, for the purposes of thermodynamic description. It states: "More precisely, the glass may be regarded as being in equilibrium so long as experimental tests show that 'slow' transitions are in effect reversible." It is not customary to make this proviso part of the definition of thermodynamic equilibrium, but the converse is usually assumed: that if a body in thermodynamic equilibrium is subject to a sufficiently slow process, that process may be considered to be sufficiently nearly reversible, and the body remains sufficiently nearly in thermodynamic equilibrium during the process.

经典热力学专著作者H.A.Buchdahl认为“热力学系统的平衡”，并没有写下词语“热力学平衡”。Buchdahl在提到封闭的物质交换系统时写道: “如果一个系统处于一种终端状态，而这种终端状态恰好是静态的，那么它将被称为处于平衡状态。”为了热力学描述的目的，Buchdahl的专著也讨论了非晶态玻璃。它说: “更准确地说，只要实验测试表明‘慢’跃迁实际上是可逆的，玻璃就可以被认为处于平衡状态。”将这个但书作为热力学平衡定义的一部分并不是惯例，但是通常的假设是相反的: 如果一个热力学平衡中的物体经历了一个足够缓慢的过程，那么这个过程可以被认为是足够接近可逆的，而且在这个过程中，这个物体足够接近于热力学平衡。

A. Münster carefully extends his definition of thermodynamic equilibrium for isolated systems by introducing a concept of contact equilibrium. This specifies particular processes that are allowed when considering thermodynamic equilibrium for non-isolated systems, with special concern for open systems, which may gain or lose matter from or to their surroundings. A contact equilibrium is between the system of interest and a system in the surroundings, brought into contact with the system of interest, the contact being through a special kind of wall; for the rest, the whole joint system is isolated. Walls of this special kind were also considered by C. Carathéodory, and are mentioned by other writers also. They are selectively permeable. They may be permeable only to mechanical work, or only to heat, or only to some particular chemical substance. Each contact equilibrium defines an intensive parameter; for example, a wall permeable only to heat defines an empirical temperature. A contact equilibrium can exist for each chemical constituent of the system of interest. In a contact equilibrium, despite the possible exchange through the selectively permeable wall, the system of interest is changeless, as if it were in isolated thermodynamic equilibrium. This scheme follows the general rule that "... we can consider an equilibrium only with respect to specified processes and defined experimental conditions." ^[21] Thermodynamic equilibrium for an open system means that, with respect to every relevant kind of selectively permeable wall, contact equilibrium exists when the respective intensive parameters of the system and surroundings are equal.^[1] This definition does not consider the most general kind of thermodynamic equilibrium, which is through unselective contacts. This definition does not simply state that no current of matter or energy exists in the interior or at the boundaries; but it is compatible with the following definition, which does so state.

A. Münster carefully extends his definition of thermodynamic equilibrium for isolated systems by introducing a concept of contact equilibrium. This specifies particular processes that are allowed when considering thermodynamic equilibrium for non-isolated systems, with special concern for open systems, which may gain or lose matter from or to their surroundings. A contact equilibrium is between the system of interest and a system in the surroundings, brought into contact with the system of interest, the contact being through a special kind of wall; for the rest, the whole joint system is isolated. Walls of this special kind were also considered by C. Carathéodory, and are mentioned by other writers also. They are selectively permeable. They may be permeable only to mechanical work, or only to heat, or only to some particular chemical substance. Each contact equilibrium defines an intensive parameter; for example, a wall permeable only to heat defines an empirical temperature. A contact equilibrium can exist for each chemical constituent of the system of interest. In a contact equilibrium, despite the possible exchange through the selectively permeable wall, the system of interest is changeless, as if it were in isolated thermodynamic equilibrium. This scheme follows the general rule that "... we can consider an equilibrium only with respect to specified processes and defined experimental conditions."

通过引入接触平衡的概念，A. Münster小心地扩展了孤立系统的热力学平衡定义。对于非孤立系统，特别关注开放系统，开放系统可能从周围环境中获得或失去物质，在考虑非孤立系统时允许的特殊过程，这些过程特别关注开放热力学平衡。利益系统和周围系统之间的接触平衡，通过一种特殊的墙与利益系统接触，其余的连接系统是孤立的。这种特殊类型的墙也被C.喀喇西奥多里 C.Carathéodory考虑过，其他作家也提到过。它们具有选择性渗透性。它们可能只对机械工作有渗透性，或者只对热有渗透性，或者只对某种特定的化学物质有渗透性。每个接触平衡定义了一个强度参数; 例如，只能透热的壁定义了一个经验温度。对于感兴趣的体系的每一种化学成分，都可以存在接触平衡。在接触平衡中，尽管有可能通过选择性渗透壁进行交换，感兴趣的系统是不变的，好像它是在孤立的热力学平衡。这个方案遵循的一般规则是: “ ... ... 我们只能考虑特定过程和特定实验条件下的平衡。”

M. Zemansky also distinguishes mechanical, chemical, and thermal equilibrium. He then writes: "When the conditions for all three types of equilibrium are satisfied, the system is said to be in a state of thermodynamic equilibrium".^[34]

M.泽曼斯基 M.Zemansky还区分了机械、化学和热平衡。他接着写道: “当这三种均衡的条件都满足时，系统就处于热力学平衡状态。”

P.M. Morse writes that thermodynamics is concerned with "states of thermodynamic equilibrium". He also uses the phrase "thermal equilibrium" while discussing transfer of energy as heat between a body and a heat reservoir in its surroundings, though not explicitly defining a special term 'thermal equilibrium'.

P.M.Morse写道，热力学关注的是“热力学平衡状态”。在讨论物体与周围热源之间的热量传递时，他也使用了“热平衡”这个短语，尽管没有明确定义一个特殊的术语“热平衡”。

P.M. Morse writes that thermodynamics is concerned with "states of thermodynamic equilibrium". He also uses the phrase "thermal equilibrium" while discussing transfer of energy as heat between a body and a heat reservoir in its surroundings, though not explicitly defining a special term 'thermal equilibrium'.^[35]

P.M.莫尔斯 P.M.Morse写道，热力学关注的是“热力学平衡状态”。在讨论物体与周围热源之间的热量传递时，他也使用了“热平衡”这个短语，尽管没有明确定义一个特殊的术语“热平衡”

J.R. Waldram writes of "a definite thermodynamic state". He defines the term "thermal equilibrium" for a system "when its observables have ceased to change over time". But shortly below that definition he writes of a piece of glass that has not yet reached its "full thermodynamic equilibrium state".

J.R. Waldram写到了“一个明确的热力学状态”。他将一个系统定义为“当其观测量随时间停止变化时”的“热平衡”。但是在这个定义之下不久，他写到一块玻璃还没有达到“完全的热力学平衡状态”。

J.R. Waldram writes of "a definite thermodynamic state". He defines the term "thermal equilibrium" for a system "when its observables have ceased to change over time". But shortly below that definition he writes of a piece of glass that has not yet reached its "full thermodynamic equilibrium state".^[36]

J.R. Waldram写到了“一个明确的热力学状态”。他将一个系统定义为“当其观测量随时间停止变化时”的“热平衡”。但是在这个定义之下不久，他写到一块玻璃还没有达到“完全的热力学平衡状态”。

Considering equilibrium states, M. Bailyn writes: "Each intensive variable has its own type of equilibrium." He then defines thermal equilibrium, mechanical equilibrium, and material equilibrium. Accordingly, he writes: "If all the intensive variables become uniform, thermodynamic equilibrium is said to exist." He is not here considering the presence of an external force field.

考虑到平衡状态，M.Bailyn写道: “每个密集型变量都有自己的平衡类型。”然后他定义了热平衡、力学平衡和物质平衡。因此，他写道: “如果所有的密集变量都是一致的，那么热力学平衡就是存在的。”他在这里没有考虑外力场的存在。

Considering equilibrium states, M. Bailyn writes: "Each intensive variable has its own type of equilibrium." He then defines thermal equilibrium, mechanical equilibrium, and material equilibrium. Accordingly, he writes: "If all the intensive variables become uniform, thermodynamic equilibrium is said to exist." He is not here considering the presence of an external force field.^[37]

考虑到平衡状态，M.Bailyn写道: “每个密集型变量都有自己的平衡类型。”然后他定义了热平衡、力学平衡和物质平衡。因此，他写道: “如果所有的密集变量都是一致的，那么热力学平衡就是存在的。”他在这里没有考虑外力场的存在。

J.G. Kirkwood and I. Oppenheim define thermodynamic equilibrium as follows: "A system is in a state of thermodynamic equilibrium if, during the time period allotted for experimentation, (a) its intensive properties are independent of time and (b) no current of matter or energy exists in its interior or at its boundaries with the surroundings." It is evident that they are not restricting the definition to isolated or to closed systems. They do not discuss the possibility of changes that occur with "glacial slowness", and proceed beyond the time period allotted for experimentation. They note that for two systems in contact, there exists a small subclass of intensive properties such that if all those of that small subclass are respectively equal, then all respective intensive properties are equal. States of thermodynamic equilibrium may be defined by this subclass, provided some other conditions are satisfied.

J.G.Kirkwood 和 I.Oppenheim 将热力学平衡定义为: “一个系统处于热力学平衡状态，如果，在分配给实验的时间内，(a)它的密集特性与时间无关，(b)它的内部或与周围环境的边界处没有物质或能量流。”显然，他们没有把定义限制在孤立的或封闭的系统。它们不讨论“缓慢”发生变化的可能性，并且超出了分配给实验的时间范围。他们注意到，对于两个相接触的系统，存在一个密集性质的小子类，如果这个小子类的所有子类都相等，那么所有各自的密集性质都相等。热力学平衡状态可以由这个子类定义，只要满足其他一些条件。

J.G. Kirkwood and I. Oppenheim define thermodynamic equilibrium as follows: "A system is in a state of thermodynamic equilibrium if, during the time period allotted for experimentation, (a) its intensive properties are independent of time and (b) no current of matter or energy exists in its interior or at its boundaries with the surroundings." It is evident that they are not restricting the definition to isolated or to closed systems. They do not discuss the possibility of changes that occur with "glacial slowness", and proceed beyond the time period allotted for experimentation. They note that for two systems in contact, there exists a small subclass of intensive properties such that if all those of that small subclass are respectively equal, then all respective intensive properties are equal. States of thermodynamic equilibrium may be defined by this subclass, provided some other conditions are satisfied.^[38]

J.G.柯克伍德 J.G.Kirkwood和 I.Oppenheim 将热力学平衡定义为: “一个系统处于热力学平衡状态，如果，在分配给实验的时间内，(a)它的密集特性与时间无关，(b)它的内部或与周围环境的边界处没有物质或能量流。”显然，他们没有把定义限制在孤立的或封闭的系统。它们不讨论“缓慢”发生变化的可能性，并且超出了分配给实验的时间范围。他们注意到，对于两个相接触的系统，存在一个密集性质的小子类，如果这个小子类的所有子类都相等，那么所有各自的密集性质都相等。热力学平衡状态可以由这个子类定义，只要满足其他一些条件。

Characteristics of a state of internal thermodynamic equilibrium

内部热力学平衡状态的特征

A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous. Planck introduces his treatise with a brief account of heat and temperature and thermal equilibrium, and then announces: "In the following we shall deal chiefly with homogeneous, isotropic bodies of any form, possessing throughout their substance the same temperature and density, and subject to a uniform pressure acting everywhere perpendicular to the surface." As did Carathéodory, Planck was setting aside surface effects and external fields and anisotropic crystals. Though referring to temperature, Planck did not there explicitly refer to the concept of thermodynamic equilibrium. In contrast, Carathéodory's scheme of presentation of classical thermodynamics for closed systems postulates the concept of an "equilibrium state" following Gibbs (Gibbs speaks routinely of a "thermodynamic state"), though not explicitly using the phrase 'thermodynamic equilibrium', nor explicitly postulating the existence of a temperature to define it.

在没有外力的情况下，由单一相组成的热力学系统，在其自身的内部热力学平衡中，是均匀的。普朗克在论文中简要介绍了热量、温度和热平衡，然后宣布: “在下文中，我们将主要讨论任何形式的均匀、各向同性的物体，它们在整个物质中具有相同的温度和密度，并受到到处垂直于表面的均匀压力作用。”和 Carathéodory 一样，Planck 将表面效应、外场和各向异性晶体排除在外。虽然普朗克提到了温度，但并没有明确提到热力学平衡的概念。相比之下，Carathéodory 关于封闭系统的经典热力学演示方案假设了一个遵循 Gibbs 的“平衡态”的概念(Gibbs 经常提到一个“热力学状态”) ，虽然没有明确地使用短语‘热力学平衡’ ，也没有明确地假设存在一个温度来定义它。

Homogeneity in the absence of external forces

在没有外力的情况下的同质性

A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous.^[39] This means that the material in any small volume element of the system can be interchanged with the material of any other geometrically congruent volume element of the system, and the effect is to leave the system thermodynamically unchanged. In general, a strong external force field makes a system of a single phase in its own internal thermodynamic equilibrium inhomogeneous with respect to some intensive variables. For example, a relatively dense component of a mixture can be concentrated by centrifugation. 在没有外力的情况下，由单一相组成的热力学系统，在其自身的内部热力学平衡中是均匀的。这意味着系统中任何小体积单元中的材料可以与系统中任何其他几何相等的体积单元中的材料互换，其效果是使系统在热力学上保持不变。一般来说，一个强外力场使得一个单相系统在其自身的内部热力学平衡中对于一些强变量 Intensive Variable是不均匀的。例如，可以通过离心来浓缩混合物中相对密度较大的组分。

The temperature within a system in thermodynamic equilibrium is uniform in space as well as in time. In a system in its own state of internal thermodynamic equilibrium, there are no net internal macroscopic flows. In particular, this means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform. Considerations of kinetic theory or statistical mechanics also support this statement.

热力学平衡系统内的温度在时间和空间上都是均匀的。在一个处于内部热力学平衡的系统中，不存在净的内部宏观流动。特别是，这意味着系统的所有局部都处于相互辐射交换平衡。这意味着系统的温度在空间上是均匀的。对动力学理论或统计力学的考虑也支持这种说法。

Uniform temperature

均匀温度

In order that a system may be in its own internal state of thermodynamic equilibrium, it is of course necessary, but not sufficient, that it be in its own internal state of thermal equilibrium; it is possible for a system to reach internal mechanical equilibrium before it reaches internal thermal equilibrium. As noted above, according to A. Münster, the number of variables needed to define a thermodynamic equilibrium is the least for any state of a given isolated system. As noted above, J.G. Kirkwood and I. Oppenheim point out that a state of thermodynamic equilibrium may be defined by a special subclass of intensive variables, with a definite number of members in that subclass.

为了使一个系统处于它自己的内部热力学平衡状态，它当然是必要的，但不是充分的，它必须处于它自己的内部热平衡状态; 一个系统在到达内部热平衡之前到达内部力学平衡是可能的。如上所述，根据 A.Münster的说法，对于给定的孤立系统的任何状态，定义一个热力学平衡所需的变量数是最少的。如上所述，J.G.Kirkwood 和 I.Oppenheim 指出，热力学平衡状态可以由一个特殊的子类的强变量定义，该子类中有一定数量的成员。

Such equilibrium inhomogeneity, induced by external forces, does not occur for the intensive variable temperature. According to E.A. Guggenheim, "The most important conception of thermodynamics is temperature."^[40] Planck introduces his treatise with a brief account of heat and temperature and thermal equilibrium, and then announces: "In the following we shall deal chiefly with homogeneous, isotropic bodies of any form, possessing throughout their substance the same temperature and density, and subject to a uniform pressure acting everywhere perpendicular to the surface."^[39] As did Carathéodory, Planck was setting aside surface effects and external fields and anisotropic crystals. Though referring to temperature, Planck did not there explicitly refer to the concept of thermodynamic equilibrium. In contrast, Carathéodory's scheme of presentation of classical thermodynamics for closed systems postulates the concept of an "equilibrium state" following Gibbs (Gibbs speaks routinely of a "thermodynamic state"), though not explicitly using the phrase 'thermodynamic equilibrium', nor explicitly postulating the existence of a temperature to define it.

这种由外力引起的平衡不均匀性，在强烈变化的温度 Temperature下不会发生。E.A.古根海姆 E.A. Guggenheim认为，“热力学最重要的概念是温度。“Planck在介绍他的论文时，简要叙述了热、温度和热平衡，然后宣布: ”在下文中，我们将主要讨论任何形式的均匀、各向同性的物体，它们的物质具有相同的温度和密度，并受到到处垂直于表面的均匀压力的作用和Carathéodory 一样，Planck将表面效应、外场和各向异性晶体排除在外。虽然Planck提到了温度，但并没有明确提到热力学平衡的概念。相比之下，Carathéodory关于封闭系统的经典热力学演示方案假设了一个遵循 Gibbs 的“平衡态”的概念(Gibbs 经常提到一个“热力学状态”) ，虽然没有明确地使用短语‘热力学平衡’ ，也没有明确地假设存在一个温度来定义它。

If the thermodynamic equilibrium lies in an external force field, it is only the temperature that can in general be expected to be spatially uniform. Intensive variables other than temperature will in general be non-uniform if the external force field is non-zero. In such a case, in general, additional variables are needed to describe the spatial non-uniformity.

如果热力学平衡位于一个外力场中，那么通常只有温度在空间上是均匀的。如果外力场非零，温度以外的强度变量通常是不均匀的。在这种情况下，一般需要附加变量来描述空间非均匀性。

The temperature within a system in thermodynamic equilibrium is uniform in space as well as in time. In a system in its own state of internal thermodynamic equilibrium, there are no net internal macroscopic flows. In particular, this means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform.^[2] This is so in all cases, including those of non-uniform external force fields. For an externally imposed gravitational field, this may be proved in macroscopic thermodynamic terms, by the calculus of variations, using the method of Langrangian multipliers.^[41]^[42]^[43]^[44]^[45]^[46] Considerations of kinetic theory or statistical mechanics also support this statement.^[47]^[48]^[49]^[50]^[51]^[52]^[53]

热力学平衡系统内的温度在时间和空间上都是均匀的。在一个处于内部热力学平衡的系统中，不存在净的内部宏观流动。特别是，这意味着系统的所有局部都处于相互辐射交换平衡。这意味着系统的温度在空间上是均匀的。这在所有情况下都是如此，包括那些非均匀外力场。对于外加的引力场，这可以用宏观热力学条件来证明，用变分法乘数法可以证明。对动力学理论或统计力学的考虑也支持这种说法。

In order that a system may be in its own internal state of thermodynamic equilibrium, it is of course necessary, but not sufficient, that it be in its own internal state of thermal equilibrium; it is possible for a system to reach internal mechanical equilibrium before it reaches internal thermal equilibrium.^[54]

为了使一个系统处于它自己的内部热力学平衡状态，它当然是必要的，但不是充分的，它必须处于它自己的内部热平衡状态; 一个系统在到达内部热平衡之前到达内部力学平衡是可能的。

As noted above, J.R. Partington points out that a state of thermodynamic equilibrium is stable against small transient perturbations. Without this condition, in general, experiments intended to study systems in thermodynamic equilibrium are in severe difficulties.

如上所述，j.r. Partington 指出热力学平衡状态在小的瞬态扰动下是稳定的。如果没有这个条件，一般来说，在21热力学平衡用于研究系统的实验就会遇到严重的困难。

Number of real variables needed for specification

规范所需的实变量数目

In his exposition of his scheme of closed system equilibrium thermodynamics, C. Carathéodory initially postulates that experiment reveals that a definite number of real variables define the states that are the points of the manifold of equilibria.^[7] In the words of Prigogine and Defay (1945): "It is a matter of experience that when we have specified a certain number of macroscopic properties of a system, then all the other properties are fixed."^[55]^[56] As noted above, according to A. Münster, the number of variables needed to define a thermodynamic equilibrium is the least for any state of a given isolated system. As noted above, J.G. Kirkwood and I. Oppenheim point out that a state of thermodynamic equilibrium may be defined by a special subclass of intensive variables, with a definite number of members in that subclass.

在他关于封闭系统平衡态热力学方案的论述中，C.Carathéodory 最初假定实验揭示了一定数量的实变量定义了作为平衡态流形点的状态。用 Prigogine 和 Defay (1945)的话说: “这是一个经验问题，当我们确定了一个系统的一定数量的宏观属性，那么所有其他属性都是固定的。如上所述，A. Münster认为，对于给定孤立系统的任何状态来说，定义热力学平衡所需的变量数量是最少的。如上所述，J.G. Kirkwood 和 I. Oppenheim 指出，热力学平衡状态可以由一个特殊的子类的集约变量来定义，该子类中有一定数量的成员。

When a body of material starts from a non-equilibrium state of inhomogeneity or chemical non-equilibrium, and is then isolated, it spontaneously evolves towards its own internal state of thermodynamic equilibrium. It is not necessary that all aspects of internal thermodynamic equilibrium be reached simultaneously; some can be established before others. For example, in many cases of such evolution, internal mechanical equilibrium is established much more rapidly than the other aspects of the eventual thermodynamic equilibrium. Another example is that, in many cases of such evolution, thermal equilibrium is reached much more rapidly than chemical equilibrium.

当一个物质体从不均匀的非平衡状态或化学非平衡状态开始，然后被孤立，它自发地发展到自己的内部热力学平衡状态。没有必要同时达到内部热力学平衡的所有方面; 有些方面可以先于其他方面建立起来。例如，在这种演变的许多情况下，内部力学平衡的建立比最终热力学平衡的其他方面要快得多。另一个例子是，在这种演变的许多情况下，热平衡的发展要比化学平衡快得多。

If the thermodynamic equilibrium lies in an external force field, it is only the temperature that can in general be expected to be spatially uniform. Intensive variables other than temperature will in general be non-uniform if the external force field is non-zero. In such a case, in general, additional variables are needed to describe the spatial non-uniformity. 如果热力学平衡位于一个外力场中，那么通常只有温度在空间上是均匀的。如果外力场非零，温度以外的强度变量通常是不均匀的。在这种情况下，一般需要附加变量来描述空间非均匀性。

Stability against small perturbations

对小扰动的稳定性

In an isolated system, thermodynamic equilibrium by definition persists over an indefinitely long time. In classical physics it is often convenient to ignore the effects of measurement and this is assumed in the present account.

在一个孤立的系统中，根据定义，热力学平衡可以持续无限长的时间。在经典物理学中，忽略测量的影响通常是很方便的，现在我们假设这一点。

As noted above, J.R. Partington points out that a state of thermodynamic equilibrium is stable against small transient perturbations. Without this condition, in general, experiments intended to study systems in thermodynamic equilibrium are in severe difficulties. 如上所述，J.R. Partington 指出热力学平衡状态在小的瞬态扰动下是稳定的。如果没有这个条件，一般来说，在热力学平衡用于研究系统的实验就会遇到严重的困难。

To consider the notion of fluctuations in an isolated thermodynamic system, a convenient example is a system specified by its extensive state variables, internal energy, volume, and mass composition. By definition they are time-invariant. By definition, they combine with time-invariant nominal values of their conjugate intensive functions of state, inverse temperature, pressure divided by temperature, and the chemical potentials divided by temperature, so as to exactly obey the laws of thermodynamics. But the laws of thermodynamics, combined with the values of the specifying extensive variables of state, are not sufficient to provide knowledge of those nominal values. Further information is needed, namely, of the constitutive properties of the system.

考虑孤立热力学系统中的涨落概念，一个方便的例子是由其广泛的状态变量、内能、体积和质量组成指定的系统。根据定义，它们是时不变的。根据定义，它们与它们的共轭强度函数的时不变名义值相结合，即状态函数、反温度函数、压力除以温度函数、化学势除以温度函数，从而精确服从热力学定律。但是热力学定律，加上指定广泛的状态变量的值，不足以提供这些名义值的知识。需要进一步的信息，即系统的构成性质。

Approach to thermodynamic equilibrium within an isolated system

孤立系统中的热力学平衡

It may be admitted that on repeated measurement of those conjugate intensive functions of state, they are found to have slightly different values from time to time. Such variability is regarded as due to internal fluctuations. The different measured values average to their nominal values.

可以承认，在重复测量这些共轭密集态函数时，发现它们的值随时间略有不同。这种可变性被认为是由于内部波动。不同测量值与其名义值的平均值。

When a body of material starts from a non-equilibrium state of inhomogeneity or chemical non-equilibrium, and is then isolated, it spontaneously evolves towards its own internal state of thermodynamic equilibrium. It is not necessary that all aspects of internal thermodynamic equilibrium be reached simultaneously; some can be established before others. For example, in many cases of such evolution, internal mechanical equilibrium is established much more rapidly than the other aspects of the eventual thermodynamic equilibrium.^[54] Another example is that, in many cases of such evolution, thermal equilibrium is reached much more rapidly than chemical equilibrium.^[57]

当一个物质体从不均匀的非平衡状态或化学非平衡状态开始，然后被孤立，它自发地朝着自己的内部热力学平衡状态演化。没有必要同时达到内部热力学平衡的所有方面; 有些方面可以先于其他方面建立起来。例如，在这种演变的许多情况下，内部力学平衡的建立比最终热力学平衡的其他方面要快得多。另一个例子是，在这种演变的许多情况下，热平衡的发展要比化学平衡快得多

If the system is truly macroscopic as postulated by classical thermodynamics, then the fluctuations are too small to detect macroscopically. This is called the thermodynamic limit. In effect, the molecular nature of matter and the quantal nature of momentum transfer have vanished from sight, too small to see. According to Buchdahl: "... there is no place within the strictly phenomenological theory for the idea of fluctuations about equilibrium (see, however, Section 76)."

如果这个系统真的像经典热力学所假定的那样是宏观的，那么这个系统的涨落太小了，宏观上无法检测到。这就是所谓的热力学极限。实际上，物质的分子性质和动量转移的量子性质已经从我们的视线中消失，因为它们太小而看不见。根据布赫达尔: “ ... 在严格的现象学理论中，平衡的涨落概念是没有位置的

Fluctuations within an isolated system in its own internal thermodynamic equilibrium

孤立系统内部热力学平衡的波动

If the system is repeatedly subdivided, eventually a system is produced that is small enough to exhibit obvious fluctuations. This is a mesoscopic level of investigation. The fluctuations are then directly dependent on the natures of the various walls of the system. The precise choice of independent state variables is then important. At this stage, statistical features of the laws of thermodynamics become apparent.

如果系统被重复细分，最终会产生一个足够小以显示明显波动的系统。这是一个介观层面的研究。波动则直接取决于系统各墙的性质。因此，精确地选择独立状态变量是很重要的。在这个阶段，热力学定律的统计特征变得明显。

In an isolated system, thermodynamic equilibrium by definition persists over an indefinitely long time. In classical physics it is often convenient to ignore the effects of measurement and this is assumed in the present account. 在一个孤立的系统中，根据定义，热力学平衡可以持续无限长的时间。在经典物理学中，忽略测量的影响通常是很方便的，现在我们假设这一点。

If the mesoscopic system is further repeatedly divided, eventually a microscopic system is produced. Then the molecular character of matter and the quantal nature of momentum transfer become important in the processes of fluctuation. One has left the realm of classical or macroscopic thermodynamics, and one needs quantum statistical mechanics. The fluctuations can become relatively dominant, and questions of measurement become important.

如果介观系统进一步重复分裂，最终产生一个微观系统。物质的分子性质和动量传递的量子性质在涨落过程中起着重要作用。这已经离开了经典热力学或宏观热力学的领域，即需要量子统计力学。波动可以变得相对占主导地位，测量问题变得重要。

To consider the notion of fluctuations in an isolated thermodynamic system, a convenient example is a system specified by its extensive state variables, internal energy, volume, and mass composition. By definition they are time-invariant. By definition, they combine with time-invariant nominal values of their conjugate intensive functions of state, inverse temperature, pressure divided by temperature, and the chemical potentials divided by temperature, so as to exactly obey the laws of thermodynamics.^[58] But the laws of thermodynamics, combined with the values of the specifying extensive variables of state, are not sufficient to provide knowledge of those nominal values. Further information is needed, namely, of the constitutive properties of the system.

考虑孤立热力学系统中的涨落概念，一个方便的例子是由其广泛的状态变量、内能、体积和质量组成指定的系统。根据定义，它们是时不变的。根据定义，它们与它们的共轭强度函数的时不变名义值相结合，即状态函数、反温度函数、压力除以温度函数、化学势除以温度函数，从而精确服从热力学定律。但是热力学定律，加上指定广泛的状态变量的值，不足以提供这些名义值的知识。需要进一步的信息，即系统的构成性质。

The statement that 'the system is its own internal thermodynamic equilibrium' may be taken to mean that 'indefinitely many such measurements have been taken from time to time, with no trend in time in the various measured values'. Thus the statement, that 'a system is in its own internal thermodynamic equilibrium, with stated nominal values of its functions of state conjugate to its specifying state variables', is far far more informative than a statement that 'a set of single simultaneous measurements of those functions of state have those same values'. This is because the single measurements might have been made during a slight fluctuation, away from another set of nominal values of those conjugate intensive functions of state, that is due to unknown and different constitutive properties. A single measurement cannot tell whether that might be so, unless there is also knowledge of the nominal values that belong to the equilibrium state.

系统是它自己的内部热力学平衡的说法可能被理解为无限期地多次这样的测量已经不时地被采取，在不同的测量值中没有时间的趋势。因此，一个系统处于它自己的内部热力学平衡，它的状态变量与它的状态变量共轭的函数的标称值相对应，这种说法远比一个状态函数的一组单一的同时测量值具有相同的值的说法信息量大得多。这是因为单次测量可能是由于未知和不同的组成性质，在轻微的波动中，远离那些共轭的状态密集函数的另一组名义值。一个单一的测量不能告诉是否可能是这样，除非还有属于平衡状态的名义值的知识。

It may be admitted that on repeated measurement of those conjugate intensive functions of state, they are found to have slightly different values from time to time. Such variability is regarded as due to internal fluctuations. The different measured values average to their nominal values. 可以承认，在重复测量这些共轭密集态函数时，发现它们的值随时间略有不同。这种可变性被认为是由于内部波动。不同测量值与其名义值的平均值。

If the system is truly macroscopic as postulated by classical thermodynamics, then the fluctuations are too small to detect macroscopically. This is called the thermodynamic limit. In effect, the molecular nature of matter and the quantal nature of momentum transfer have vanished from sight, too small to see. According to Buchdahl: "... there is no place within the strictly phenomenological theory for the idea of fluctuations about equilibrium (see, however, Section 76)."^[59] 如果这个系统真的像经典热力学所假定的那样是宏观的，那么这个系统的涨落太小了，宏观上无法检测到。这就是所谓的热力学极限。实际上，物质的分子性质和动量转移的量子性质已经从我们的视线中消失，因为它们太小而看不见。根据Buchdahl: “ ... ... 在严格的现象学理论中，平衡的涨落概念是没有位置的

If the system is repeatedly subdivided, eventually a system is produced that is small enough to exhibit obvious fluctuations. This is a mesoscopic level of investigation. The fluctuations are then directly dependent on the natures of the various walls of the system. The precise choice of independent state variables is then important. At this stage, statistical features of the laws of thermodynamics become apparent. 如果系统被重复细分，最终会产生一个足够小以显示明显波动的系统。这是一个介观层面的研究。波动则直接取决于系统各墙的性质。因此，精确地选择独立状态变量是很重要的。在这个阶段，热力学定律的统计特征变得明显。

An explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by B. C. Eu. He considers two systems in thermal contact, one a thermometer, the other a system in which there are occurring several irreversible processes, entailing non-zero fluxes; the two systems are separated by a wall permeable only to heat. He considers the case in which, over the time scale of interest, it happens that both the thermometer reading and the irreversible processes are steady. Then there is thermal equilibrium without thermodynamic equilibrium. Eu proposes consequently that the zeroth law of thermodynamics can be considered to apply even when thermodynamic equilibrium is not present; also he proposes that if changes are occurring so fast that a steady temperature cannot be defined, then "it is no longer possible to describe the process by means of a thermodynamic formalism. In other words, thermodynamics has no meaning for such a process." This illustrates the importance for thermodynamics of the concept of temperature.

热平衡和热力学平衡之间的明确区分是由 B.C.Eu 提出的。他认为两个系统在热接触，一个是温度计，另一个是一个系统，其中有几个不可逆过程，产生非零通量; 这两个系统被一个只透热的壁隔开。他考虑了这样一种情况，在有兴趣的时间尺度上，温度计读数和不可逆过程都是稳定的。然后是没有热平衡的热力学平衡。因此，欧盟提出，即使在没有热力学第零定律的情况下，也可以考虑应用热力学平衡; 他还提出，如果变化发生得太快，以至于无法确定一个稳定的温度，那么“用热力学形式主义来描述这一过程就不再可能了。换句话说，热力学对这样一个过程没有意义。”这说明了温度概念对热力学的重要性。

If the mesoscopic system is further repeatedly divided, eventually a microscopic system is produced. Then the molecular character of matter and the quantal nature of momentum transfer become important in the processes of fluctuation. One has left the realm of classical or macroscopic thermodynamics, and one needs quantum statistical mechanics. The fluctuations can become relatively dominant, and questions of measurement become important. 如果介观系统进一步重复分裂，最终产生一个微观系统。物质的分子性质和动量传递的量子性质在涨落过程中起着重要作用。一个人已经离开了经典热力学或宏观热力学的领域，他需要量子统计力学。波动可以变得相对占主导地位，测量问题变得重要。

Thermal equilibrium is achieved when two systems in thermal contact with each other cease to have a net exchange of energy. It follows that if two systems are in thermal equilibrium, then their temperatures are the same.

当两个相互热接触的系统不再有净能量交换时，就会产生热平衡。因此，如果两个系统处于热平衡，那么它们的温度是相同的。

The statement that 'the system is its own internal thermodynamic equilibrium' may be taken to mean that 'indefinitely many such measurements have been taken from time to time, with no trend in time in the various measured values'. Thus the statement, that 'a system is in its own internal thermodynamic equilibrium, with stated nominal values of its functions of state conjugate to its specifying state variables', is far far more informative than a statement that 'a set of single simultaneous measurements of those functions of state have those same values'. This is because the single measurements might have been made during a slight fluctuation, away from another set of nominal values of those conjugate intensive functions of state, that is due to unknown and different constitutive properties. A single measurement cannot tell whether that might be so, unless there is also knowledge of the nominal values that belong to the equilibrium state. 系统是它自己的内部热力学平衡的说法可能被理解为无限期地多次这样的测量已经不时地被采取，在不同的测量值中没有时间的趋势。因此，一个系统处于它自己的内部热力学平衡，它的状态变量与它的状态变量共轭的函数的标称值相对应，这种说法远比一个状态函数的一组单一的同时测量值具有相同的值的说法信息量大得多。这是因为单次测量可能是由于未知和不同的组成性质，在轻微的波动中，远离那些共轭的状态密集函数的另一组名义值。一个单一的测量不能告诉是否可能是这样，除非还有属于平衡状态的名义值的知识。

Thermal equilibrium occurs when a system's macroscopic thermal observables have ceased to change with time. For example, an ideal gas whose distribution function has stabilised to a specific Maxwell–Boltzmann distribution would be in thermal equilibrium. This outcome allows a single temperature and pressure to be attributed to the whole system. For an isolated body, it is quite possible for mechanical equilibrium to be reached before thermal equilibrium is reached, but eventually, all aspects of equilibrium, including thermal equilibrium, are necessary for thermodynamic equilibrium.

当一个系统的宏观热观测量不再随时间变化时，就会出现热平衡。例如，一种分布函数稳定到一个特定的麦克斯韦-波兹曼分布的理想气体将在热平衡。这个结果可以将单一的温度和压力归因于整个系统。对于一个孤立的物体来说，在达到力学平衡之前达到热平衡是很有可能的，但是最终，所有方面的平衡，包括热平衡，对于热力学平衡来说都是必要的。

Thermal equilibrium

热平衡

An explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by B. C. Eu. He considers two systems in thermal contact, one a thermometer, the other a system in which there are occurring several irreversible processes, entailing non-zero fluxes; the two systems are separated by a wall permeable only to heat. He considers the case in which, over the time scale of interest, it happens that both the thermometer reading and the irreversible processes are steady. Then there is thermal equilibrium without thermodynamic equilibrium. Eu proposes consequently that the zeroth law of thermodynamics can be considered to apply even when thermodynamic equilibrium is not present; also he proposes that if changes are occurring so fast that a steady temperature cannot be defined, then "it is no longer possible to describe the process by means of a thermodynamic formalism. In other words, thermodynamics has no meaning for such a process."^[60] This illustrates the importance for thermodynamics of the concept of temperature.

热平衡和热力学平衡之间的明确区分是由 B.C. Eu 提出的。他认为两个系统在热接触，一个是温度计，另一个是一个系统，其中有几个不可逆过程，产生非零通量; 这两个系统被一个只透热的壁隔开。他考虑了这样一种情况，在有兴趣的时间尺度上，温度计读数和不可逆过程都是稳定的。然后是没有热平衡的热力学平衡。因此，欧盟提出，即使在没有热力学第零定律的情况下，也可以考虑应用热力学平衡; 他还提出，如果变化发生得太快，以至于无法确定一个稳定的温度，那么“用热力学形式主义来描述这一过程就不再可能了。换句话说，热力学对这样一个过程没有意义。“这说明了温度概念对热力学的重要性。

A system's internal state of thermodynamic equilibrium should be distinguished from a "stationary state" in which thermodynamic parameters are unchanging in time but the system is not isolated, so that there are, into and out of the system, non-zero macroscopic fluxes which are constant in time.

一个系统的热力学平衡内部状态应该区别于一个热力学参数在时间上是不变的，但系统不是孤立的“定态” ，因此在系统内外有非零的宏观流动，这些流动在时间上是常数。

Thermal equilibrium is achieved when two systems in thermal contact with each other cease to have a net exchange of energy. It follows that if two systems are in thermal equilibrium, then their temperatures are the same.^[61]

当两个相互热接触 Thermal Contact的系统不再有净能量交换时，就会产生热平衡 Thermal Equilibrium。因此，如果两个系统处于热平衡，那么它们的温度是相同的

Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium because they are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems. The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. Many natural systems still today remain beyond the scope of currently known macroscopic thermodynamic methods.

非平衡态热力学是热力学的一个分支，研究的是非热力学平衡系统。大多数在自然界中发现的系统并不是在21热力学平衡，因为它们正在变化或者可以被触发随着时间而变化，并且不断地和间断地受到来自其他系统的物质和能量流动的影响。非平衡体系的热力学研究比平衡态热力学研究需要更多的一般概念。许多自然系统今天仍然超出了目前已知的宏观热力学方法的范围。

Thermal equilibrium occurs when a system's macroscopic thermal observables have ceased to change with time. For example, an ideal gas whose distribution function has stabilised to a specific Maxwell–Boltzmann distribution would be in thermal equilibrium. This outcome allows a single temperature and pressure to be attributed to the whole system. For an isolated body, it is quite possible for mechanical equilibrium to be reached before thermal equilibrium is reached, but eventually, all aspects of equilibrium, including thermal equilibrium, are necessary for thermodynamic equilibrium.^[62]

当一个系统的宏观 Macroscopic热观测量不再随时间变化时，就会出现热平衡。例如，一种分布函数 Distribution Function稳定到一个特定的麦克斯韦-波兹曼分布 Maxwell–Boltzmann distribution的理想气体 Ideal Gas将在热平衡。这个结果可以将单一的温度 Temperature和压力 Pressure归因于整个系统。对于一个孤立的物体来说，在达到力学平衡之前达到热平衡是很有可能的，但是最终，所有方面的平衡，包括热平衡，对于热力学平衡来说都是必要的。

Laws governing systems which are far from equilibrium are also debatable. One of the guiding principles for these systems is the maximum entropy production principle. It states that a non-equilibrium system evolves such as to maximize its entropy production.

定律规定远离平衡的系统也是有争议的。这些系统的指导原则之一就是最大产生熵原则。它指出，非平衡系统可以最大化其产生熵进行演化。

Non-equilibrium

非平衡

Thermodynamic models

热力学模型

A system's internal state of thermodynamic equilibrium should be distinguished from a "stationary state" in which thermodynamic parameters are unchanging in time but the system is not isolated, so that there are, into and out of the system, non-zero macroscopic fluxes which are constant in time.^[63] 一个系统的内部热力学平衡状态应该区别于一个热力学参数在时间上是不变的，但是系统不是孤立的“定态” ，因此在系统内外有非零的宏观流动，这些流动在时间上是常数。

Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium because they are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems. The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. Many natural systems still today remain beyond the scope of currently known macroscopic thermodynamic methods. 非平衡态热力学是热力学的一个分支，主要研究非热力学平衡系统。大多数在自然界中发现的系统并不是在21热力学平衡，因为它们正在变化或者可以随着时间的推移而被触发变化，并且不断地和间断地受制于物质和能量进出其他系统的流动。非平衡体系的热力学研究比平衡态热力学研究需要更多的一般概念。许多自然系统今天仍然超出目前已知的宏观热力学方法的范围。

Laws governing systems which are far from equilibrium are also debatable. One of the guiding principles for these systems is the maximum entropy production principle.^[64]^[65] It states that a non-equilibrium system evolves such as to maximize its entropy production.^[66]^[67]

管理系统的定律远离平衡也是有争议的。这些系统的指导原则之一是最大产生熵原理，它指出一个非平衡系统的演化，如最大化其产生熵

Topics in control theory

控制理论主题

编者推荐

集智文章推荐

什么是非平衡态热力学 | 集智百科

非平衡态热力学 Non-equilibrium thermodynamics 是热力学的一个分支，研究某些不处于热力学平衡中的物理系统

General references

Cesare Barbieri (2007) Fundamentals of Astronomy. First Edition (QB43.3.B37 2006) CRC Press

,

Hans R. Griem (2005) Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics), Cambridge University Press, New York

C. Michael Hogan, Leda C. Patmore and Harry Seidman (1973) Statistical Prediction of Dynamic Thermal Equilibrium Temperatures using Standard Meteorological Data Bases, Second Edition (EPA-660/2-73-003 2006) United States Environmental Protection Agency Office of Research and Development, Washington, D.C. [1]

F. Mandl (1988) Statistical Physics, Second Edition, John Wiley & Sons

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↑ Chapman, S., Cowling, T.G. (1939/1970), Section 4.14, pp. 75–78.
↑ Partington, J.R. (1949), pp. 275–278.
↑ Coombes, C.A.; Laue, H. (1985). "A paradox concerning the temperature distribution of a gas in a gravitational field". Am. J. Phys. 53 (3): 272–273. Bibcode:1985AmJPh..53..272C. doi:10.1119/1.14138.
↑ Román, F.L.; White, J.A.; Velasco, S. (1995). "Microcanonical single-particle distributions for an ideal gas in a gravitational field". Eur. J. Phys. 16 (2): 83–90. Bibcode:1995EJPh...16...83R. doi:10.1088/0143-0807/16/2/008.
↑ Velasco, S.; Román, F.L.; White, J.A. (1996). "On a paradox concerning the temperature distribution of an ideal gas in a gravitational field". Eur. J. Phys. 17: 43–44. doi:10.1088/0143-0807/17/1/008.
↑ ^54.0 ^54.1 Fitts, D.D. (1962), p. 43.
↑ Prigogine, I., Defay, R. (1950/1954), p. 1.
↑ Silbey, R.J., Alberty, R.A., Bawendi, M.G. (1955/2005), p. 4.
↑ Denbigh, K.G. (1951), p. 42.

↑ Tschoegl, N.W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam,

, p. 21.

↑ Buchdahl, H.A. (1966), p. 16.
↑ Eu, B.C. (2002), page 13.
↑ R. K. Pathria, 1996
↑ de Groot, S.R., Mazur, P. (1962), p. 44.
↑ de Groot, S.R., Mazur, P. (1962), p. 43.
↑ Ziegler, H. (1983). An Introduction to Thermomechanics.. North Holland, Amsterdam..
↑ Onsager, Lars (1931). "Reciprocal Relations in Irreversible Processes". Phys. Rev. 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405.
↑ Kleidon, A.; et., al. (2005). Non-equilibrium Thermodynamics and the Production of Entropy. (Heidelberg: Springer. ed.).
↑ Belkin, Andrey; et., al. (2015). "Self-Assembled Wiggling Nano-Structures and the Principle of Maximum Entropy Production". Sci. Rep. 5: 8323. Bibcode:2015NatSR...5E8323B. doi:10.1038/srep08323. PMC 4321171. PMID 25662746.

Cited bibliography

Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, third edition, McGraw-Hill, London,

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Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York,

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Beattie, J.A., Oppenheim, I. (1979). Principles of Thermodynamics, Elsevier Scientific Publishing, Amsterdam,

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Boltzmann, L. (1896/1964). Lectures on Gas Theory, translated by S.G. Brush, University of California Press, Berkeley.

Buchdahl, H.A. (1966). The Concepts of Classical Thermodynamics, Cambridge University Press, Cambridge UK.

Callen, H.B. (1960/1985). Thermodynamics and an Introduction to Thermostatistics, (1st edition 1960) 2nd edition 1985, Wiley, New York,

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Carathéodory, C. (1909). Untersuchungen über die Grundlagen der Thermodynamik, Mathematische Annalen, 67: 355–386. A translation may be found here. Also a mostly reliable translation is to be found at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.

Chapman, S., Cowling, T.G. (1939/1970). The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, third edition 1970, Cambridge University Press, London.

Crawford, F.H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert Hart-Davis, London, Harcourt, Brace & World, Inc.

de Groot, S.R., Mazur, P. (1962). Non-equilibrium Thermodynamics, North-Holland, Amsterdam. Reprinted (1984), Dover Publications Inc., New York,

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Denbigh, K.G. (1951). Thermodynamics of the Steady State, Methuen, London.

Eu, B.C. (2002). Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics, Kluwer Academic Publishers, Dordrecht,

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Fitts, D.D. (1962). Nonequilibrium thermodynamics. A Phenomenological Theory of Irreversible Processes in Fluid Systems, McGraw-Hill, New York.

Gibbs, J.W. (1876/1878). On the equilibrium of heterogeneous substances, Trans. Conn. Acad., 3: 108–248, 343–524, reprinted in The Collected Works of J. Willard Gibbs, Ph.D, LL. D., edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp. 55–353.

Griem, H.R. (2005). Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics), Cambridge University Press, New York

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Guggenheim, E.A. (1949/1967). Thermodynamics. An Advanced Treatment for Chemists and Physicists, fifth revised edition, North-Holland, Amsterdam.

Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081.

Kirkwood, J.G., Oppenheim, I. (1961). Chemical Thermodynamics, McGraw-Hill Book Company, New York.

Landsberg, P.T. (1961). Thermodynamics with Quantum Statistical Illustrations, Interscience, New York.

Lieb, E. H.; Yngvason, J. (1999). "The Physics and Mathematics of the Second Law of Thermodynamics". Phys. Rep. 310 (1): 1–96. arXiv:cond-mat/9708200. Bibcode:1999PhR...310....1L. doi:10.1016/S0370-1573(98)00082-9. S2CID 119620408.

Levine, I.N. (1983), Physical Chemistry, second edition, McGraw-Hill, New York,

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Maxwell, J.C. (1867). "On the dynamical theory of gases". Phil. Trans. Roy. Soc. London. 157: 49–88.

Morse, P.M. (1969). Thermal Physics, second edition, W.A. Benjamin, Inc, New York.

Münster, A. (1970). Classical Thermodynamics, translated by E.S. Halberstadt, Wiley–Interscience, London.

Partington, J.R. (1949). An Advanced Treatise on Physical Chemistry, volume 1, Fundamental Principles. The Properties of Gases, Longmans, Green and Co., London.

Pippard, A.B. (1957/1966). The Elements of Classical Thermodynamics, reprinted with corrections 1966, Cambridge University Press, London.

Planck. M. (1914). The Theory of Heat Radiation, a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.

Prigogine, I. (1947). Étude Thermodynamique des Phénomènes irréversibles, Dunod, Paris, and Desoers, Liège.

Prigogine, I., Defay, R. (1950/1954). Chemical Thermodynamics, Longmans, Green & Co, London.

Silbey, R.J., Alberty, R.A., Bawendi, M.G. (1955/2005). Physical Chemistry, fourth edition, Wiley, Hoboken NJ.

ter Haar, D., Wergeland, H. (1966). Elements of Thermodynamics, Addison-Wesley Publishing, Reading MA.

Thomson, W. (March 1851). "On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam". Transactions of the Royal Society of Edinburgh. XX (part II): 261–268, 289–298. Also published in Thomson, W. (December 1852). "On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam". Phil. Mag. 4. IV (22): 8–21. Retrieved 25 June 2012.

Tisza, L. (1966). Generalized Thermodynamics, M.I.T Press, Cambridge MA.

Uhlenbeck, G.E., Ford, G.W. (1963). Lectures in Statistical Mechanics, American Mathematical Society, Providence RI.

Waldram, J.R. (1985). The Theory of Thermodynamics, Cambridge University Press, Cambridge UK,

.

Zemansky, M. (1937/1968). Heat and Thermodynamics. An Intermediate Textbook, fifth edition 1967, McGraw–Hill Book Company, New York.

External links

Category:Equilibrium chemistry

类别: 平衡化学

Breakdown of Local Thermodynamic Equilibrium George W. Collins, The Fundamentals of Stellar Astrophysics, Chapter 15

Category:Thermodynamic cycles

类别: 热力循环

Thermodynamic Equilibrium, Local and otherwise lecture by Michael Richmond

Category:Thermodynamic processes

类别: 热力学过程

Non-Local Thermodynamic Equilibrium in Cloudy Planetary Atmospheres Paper by R. E. Samueison quantifying the effects due to non-LTE in an atmosphere

Category:Thermodynamic systems

类别: 热力学系统

Local Thermodynamic Equilibrium

Category:Thermodynamics

分类: 热力学

This page was moved from wikipedia:en:Thermodynamic equilibrium. Its edit history can be viewed at 平衡热力学/edithistory

[Nster_a-1] 1.0 ^1.1 Münster, A. (1970), p. 49.

[Planck_1914_40-2] 2.0 ^2.1 Planck. M. (1914), p. 40.

[3] Haase, R. (1971), p. 4.

[4] Callen, H.B. (1960/1985), p. 26.

[5] Marsland, Robert; Brown, Harvey R.; Valente, Giovanni (2015). "Time and irreversibility in axiomatic thermodynamics". American Journal of Physics. 83 (7): 628–634. Bibcode:2015AmJPh..83..628M. doi:10.1119/1.4914528.

[6] Uhlenbeck, G.E., Ford, G.W. (1963), p. 5.

[Caratheodory-7] 7.0 ^7.1 Carathéodory, C. (1909).

[8] Prigogine, I. (1947), p. 48.

[9] Landsberg, P. T. (1961), pp. 128–142.

[10] Tisza, L. (1966), p. 108.

[11] Guggenheim, E.A. (1949/1967), § 1.12.

[12] Levine, I.N. (1983), p. 40.

[13] Lieb, E.H., Yngvason, J. (1999), pp. 17–18.

[14] Thomson, W. (1851).

[15] H.R. Griem, 2005

[16] Callen, H.B. (1960/1985), p. 15.

[17] Beattie, J.A., Oppenheim, I. (1979), p. 3.

[18] Callen, H.B. (1960/1985), p. 485.

[19] Adkins, C.J. (1968/1983), p. xiii.

[20] Pippard, A.B. (1957/1966), p. 6.

[Nster-21] 21.0 ^21.1 Münster, A. (1970), p. 53.

[22] Tisza, L. (1966), p. 119.

[23] Morse, P.M. (1969), p. 7.

[24] Bailyn, M. (1994), p. 20.

[25] Münster, A. (1970), p. 52.

[26] Haase, R. (1971), pp. 3–4.

[27] Callen, H.B. (1960/1985), p. 13.

[28] Adkins, C.J. (1968/1983), p. 7.

[29] Partington, J.R. (1949), p. 161.

[30] Crawford, F.H. (1963), p. 5.

[31] Buchdahl, H.A. (1966), p. 8.

[32] Buchdahl, H.A. (1966), p. 111.

[33] Adkins, C.J. (1968/1983), p. 8.

[34] Zemansky, M. (1937/1968), p. 27.

[35] Morse, P.M. (1969), pp. 6, 37.

[36] Waldram, J.R. (1985), p. 5.

[37] Bailyn, M. (1994), p. 21.

[38] Kirkwood, J.G., Oppenheim, I. (1961), p. 2

[Planck_1903_3-39] 39.0 ^39.1 Planck, M. (1897/1927), p.3.

[40] Guggenheim, E.A. (1949/1967), p.5.

[41] Gibbs, J.W. (1876/1878), pp. 144-150.

[42] ter Haar, D., Wergeland, H. (1966), pp. 127–130.

[43] Münster, A. (1970), pp. 309–310.

[44] Bailyn, M. (1994), pp. 254-256.

[45] Verkley, W.T.M.; Gerkema, T. (2004). "On maximum entropy profiles". J. Atmos. Sci. 61 (8): 931–936. Bibcode:2004JAtS...61..931V. doi:10.1175/1520-0469(2004)061<0931:omep>2.0.co;2.

[46] Akmaev, R.A. (2008). "On the energetics of maximum-entropy temperature profiles". Q. J. R. Meteorol. Soc. 134 (630): 187–197. Bibcode:2008QJRMS.134..187A. doi:10.1002/qj.209.

[47] Maxwell, J.C. (1867).

[48] Boltzmann, L. (1896/1964), p. 143.

[49] Chapman, S., Cowling, T.G. (1939/1970), Section 4.14, pp. 75–78.

[50] Partington, J.R. (1949), pp. 275–278.

[51] Coombes, C.A.; Laue, H. (1985). "A paradox concerning the temperature distribution of a gas in a gravitational field". Am. J. Phys. 53 (3): 272–273. Bibcode:1985AmJPh..53..272C. doi:10.1119/1.14138.

[52] Román, F.L.; White, J.A.; Velasco, S. (1995). "Microcanonical single-particle distributions for an ideal gas in a gravitational field". Eur. J. Phys. 16 (2): 83–90. Bibcode:1995EJPh...16...83R. doi:10.1088/0143-0807/16/2/008.

[53] Velasco, S.; Román, F.L.; White, J.A. (1996). "On a paradox concerning the temperature distribution of an ideal gas in a gravitational field". Eur. J. Phys. 17: 43–44. doi:10.1088/0143-0807/17/1/008.

[Fitts_43-54] 54.0 ^54.1 Fitts, D.D. (1962), p. 43.

[55] Prigogine, I., Defay, R. (1950/1954), p. 1.

[56] Silbey, R.J., Alberty, R.A., Bawendi, M.G. (1955/2005), p. 4.

[57] Denbigh, K.G. (1951), p. 42.

[58] Tschoegl, N.W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam,
v
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, p. 21.

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[59] Buchdahl, H.A. (1966), p. 16.

[60] Eu, B.C. (2002), page 13.

[61] R. K. Pathria, 1996

[62] Groot, S.R., Mazur, P. (1962), p. 44.

[63] Groot, S.R., Mazur, P. (1962), p. 43.

[64] Ziegler, H. (1983). An Introduction to Thermomechanics.. North Holland, Amsterdam..

[65] Onsager, Lars (1931). "Reciprocal Relations in Irreversible Processes". Phys. Rev. 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405.

[66] Kleidon, A.; et., al. (2005). Non-equilibrium Thermodynamics and the Production of Entropy. (Heidelberg: Springer. ed.).

[67] Belkin, Andrey; et., al. (2015). "Self-Assembled Wiggling Nano-Structures and the Principle of Maximum Entropy Production". Sci. Rep. 5: 8323. Bibcode:2015NatSR...5E8323B. doi:10.1038/srep08323. PMC 4321171. PMID 25662746.

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