热力学平衡

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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Thermodynamic equilibrium是热力学的一个公理概念。它是一个单一的热力学系统的内部状态，或者是由或多或少的渗透性或不渗透性的壁连接的几个热力学系统之间的关系。在热力学平衡状态下，无论是系统内部还是系统之间，都没有物质或能量的宏观净流动。

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

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

在一个自身处于内部热力学平衡状态的系统中，不会发生宏观变化。

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.

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.

处于相互热力学平衡的系统同时处于相互热、力学、化学和辐射平衡。系统可以处于一种相互平衡状态之中，但在另一种相互平衡状态中则不然。在热力学平衡中，所有种类的平衡都同时并无限期地保持着，直到被热力学操作所打破。在宏观平衡中，会发生完全或几乎完全平衡的微观交换；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.

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平衡状态的热力学系统在一个空间上存在均匀的温度。除了温度以外，它的强度性质可能被周围环境强加给它的一个不变的长程力场而导致空间上的不均匀性。

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.

相反，在处于非平衡状态的系统中，存在着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，那么这个系统就被认为处于一个亚稳定的平衡状态meta-stable equilibrium。

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.

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.

虽然没有被公认为 "定律"，但它是热力学的一个公理，即存在着热力学平衡状态。热力学第二定律指出，当一个物体从平衡状态开始时，它的一部分被或多或少的、渗透性或不渗透性的壁保持在不同的状态。当一个热力学操作移出或使壁更具渗透性时，它会由孤立的状态自发地达到自己的、新的内部热力学平衡状态，这个过程同时伴随着各部分熵的总和的增加。

Overview 概述

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:

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的状态。处于热力学平衡状态的系统是指在特定条件下，某些热力学势达到最小，或熵（S）达到最大的状态。一个这样的势是亥姆霍兹自由能（A），用于一个周围环境温度和体积可控的系统来说，有：

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

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

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

另一个势，是吉布斯自由能（G），在热力学平衡时，在恒温和恒压的系统中，它取得最小值：

[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.

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 表示热力学温度，p 表示压强，S 表示熵，V 表示体积，U 表示系统的内能。

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.

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.

热力学平衡是系统与周围环境长期相互作用的过程中接近或最终达到的、独特的稳定静止状态。上述势在数学上被构造为在特定条件下、特定环境中取得最小值的热力学量。

Conditions 条件

• For a completely isolated system, S is maximum at thermodynamic equilibrium.

• For a system with controlled constant temperature and volume, A is minimum at thermodynamic equilibrium.

• For a system with controlled constant temperature and pressure, G is minimum at thermodynamic equilibrium.

对于一个完全被隔绝的系统，S在热力学平衡时是最大的。

对于一个恒定温度和体积的系统，A在热力学平衡时是最小的。

对于一个恒定温度和压强的系统，G在热力学平衡时是最小的。

The various types of equilibriums are achieved as follows:

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.

• Two systems are in mechanical equilibrium when their pressures are the same.

• Two systems are in diffusive equilibrium when their chemical potentials are the same.

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

当两个系统的温度相同时，它们处于热平衡状态。

当两个系统的压强相同时，它们处于力学平衡状态。

当两个系统的化学势相同时，它们处于化学平衡状态。

所有的力都是平衡的，没有明显的外部驱动力。

Relation of exchange equilibrium between systems 系统间的交换平衡关系

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. 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，一个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，我们可以把系统和它的周围环境看作是两个相互接触的系统，长程力可以把它们联系起来。系统的表面是两个系统之间的接触面或边界。在热力学中，这个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进行热渗透，只允许能量以热的形式传递。那么，当长程力在时间上是不变的，并且能量以热的形式在它们之间的传递已经减慢并最终永久停止时，这两个系统就被认为处于热平衡状态；其它类型的接触平衡可以由其他种类的特定渗透率定义。 两个系统相对于特定类型的渗透率处于接触平衡状态时，它们具有属于这个特定类型渗透率的强度量的共同值。属于这种强度量的例子有温度、压强、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.

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contact equilibrium也可以被看作是交换平衡。在接触平衡中，两个系统之间某些量的传递率是零平衡的。例如，对于一个只能透热的壁来说，两个系统之间作为热量的内能传递的速率是相等的，而且是相反的。两个系统之间的绝热壁只对以功形式转移的能量具有 "渗透性"；在力学平衡中，两个系统之间以功形式传递能量的速率是相等和相反的。如果壁是简单的，那么穿过它的体积传递率也是相等和相反的；而且它两边的压强是相等的。如果绝热壁比较复杂，会存在一种杠杆作用，这时会有一个面积比，两个系统在交换平衡时的压力与体积交换比会成反比；这就保持了功的传递率的零平衡。

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

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原本独立的系统之间。当两个系统具有相同的温度时，辐射交换平衡占据优势。

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. (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.)

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.)

一个物质的集合可能完全与周围环境隔绝。如果它被无限期地长期不受干扰，经典热力学认为，它处于一种状态。在这种状态下，它内部不发生任何变化，也不存在任何流动，这是一种内部平衡的热力学状态。(这个假设有时被称为热力学“负一”定律，但并不常见。 (This postulate is sometimes, but not often, called the "minus first" law of thermodynamics. One textbook 一本教科书称之为“热力学第零定律” ，并指出作者认为这比其更习惯的定义更适合这个假设，这个观点显然是由福勒提出的。)

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.

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.

这样的状态是所谓的经典热力学或平衡热力学的主要关注点，因为它们是系统中唯一被认为在这个学科中明确定义的状态。一个与另一个系统处于接触平衡的系统可以通过热力学操作被隔绝，在被隔绝的事件发生时，它不会发生任何变化。因此，一个与另一个系统处于接触平衡关系的系统，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.

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.

热力学形式上允许一个系统可以同时与其他几个系统接触。这些系统可能也可能没有相互接触，这些接触分别具有不同的渗透性。如果这些系统都与世界其他部分相互隔绝，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. 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绝开来，那么它们就会达到一个多接触平衡状态multiple contact equilibrium，并且它们有一个共同的温度，一个共同的内能，和一个共同的熵。 在强度量中，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，它也能维持。(也就是说，没有“力”可以维持温度的差异。)举个例子，在一个垂直引力场中处于热力学平衡的系统中，顶壁上的压力小于底壁上的压力，但各处的温度是相同的。

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. 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.

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.

热力学操作可以作为“限制”在周围环境中的壁上发生，既不直接影响系统与周围环境的接触壁，也不影响其内部，并且在绝对有限的时间内发生。例如，一个固定的绝热壁可以在环境中被添加或去除。由于这种仅限于周围环境的操作，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。然后，根据热力学第二定律，整个系统经历了变化，并最终与周围环境达到了新的、最终的平衡。按照普朗克的说法，这一连串的事件被称为自然热力学过程。 这在平衡热力学中是被允许的，只是因为当系统和它周围的环境都明确不处于内部平衡状态时，初始和最终状态处于热力学平衡，即使在这个过程中存在着对热力学平衡的短暂偏离。一个自然过程在其过程的主要部分以有限的速度进行。因此，它与虚构的准静态 "过程 "完全不同，后者在整个过程中无限缓慢地进行，并且 "可逆"是虚构的。经典热力学认为，即使一个过程可能需要很长的时间才能达到热力学平衡，但如果它的主要部分是以有限的速度进行的，那么它就被认为是自然的，并且受到热力学第二定律的约束，因此是不可逆的。热力学允许在周围环境中，使用工程机械和人工装置及操作，但仅仅只是允许在周围环境中进行这种操作和装置，而不允许在系统中进行这种操作和运行这些装置。这就是开尔文在其关于热力学第二定律的论述中谈到这些“无生命”结构的原因；一个处于热力学平衡的系统是无生命的。 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.

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

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.

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.

通常简便的做法是，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量，然后将这些子系统称为相关强度量的储备库。

Local and global equilibrium 局部与全局平衡

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.

区分全局和局部热力学平衡是有用的。在热力学中，系统内和系统与外界的交换是由强度量控制的。例如，温度控制热交换。全局热力学平衡（GTE）global thermodynamic equilibrium意味着这些强度量在整个系统中是均匀的，而局部热力学平衡（LTE）local thermodynamic equilibrium意味着这些强度量在空间和时间上是变化的，但变化得很慢，以至于对于任何一点，我们可以假设在关于该点的某个邻域中实现热力学平衡。

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.

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量的变化太大，那么这些强度量的定义所依据的假设就会被推翻，系统将既不处于全局平衡，也不处于局部平衡。例如，一个粒子需要一定数量的碰撞才能与其周围环境达到平衡。如果它在这些碰撞中移动的平均距离，使它离开了与之平衡的邻域，它将永远不会平衡，也就不会有 LTE。根据定义，温度与与之达到平衡的邻域的平均内能成正比。由于没有达到平衡的邻域，温度的概念就不成立了，温度也就变得没有定义。

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.

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 通常只适用于大质量粒子。在辐射气体中，被气体发射和吸收的光子不需要彼此或与气体的大质量粒子处于热力学平衡状态，以使 LTE 得以存在。在某些情况下，人们认为自由电子不需要与质量大得多的原子或分子处于平衡状态，以使 LTE 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.

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.

举个例子，LTE 将存在于一个装有融化的冰块的水杯中。玻璃杯内的温度可以在任意一点定义，但在冰块附近比远离冰块的地方更冷。如果观察到位于给定某一点附近的分子的能量，它们将服从一定温度下的麦克斯韦-玻尔兹曼分布Maxwell–Boltzmann distribution。如果观察到位于另一点附近的分子的能量，它们将服从另一个温度下的麦克斯韦-玻尔兹曼分布。

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.

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.

局部热力学平衡并不要求局部或全局的静止性。换句话说，每一个小的局部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足够缓慢，以实际上维持其局部服从麦克斯韦-玻尔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，一个全局稳定的静止状态可以通过在水杯中不断加入细小的冰粉来补偿融化，并不断排掉融化的水来维持。自然输运现象可能促进一个系统从局部热力学平衡到全局的热力学平衡。回到我们的例子，热量的扩散将引导我们的水杯走向全局热力学平衡，即杯子的温度完全均匀的状态。

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.

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.

细心的和见多识广的热力学研究者，在他们对热力学平衡的描述中，经常对他们的陈述提出限制或保留意见。有些人只是含蓄地或多或少地提供了这种保留意见。

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!"

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，一位工作被广泛引用的研究者赫伯特·卡伦（H.B. Callen）曾写道 "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，并说它们可能需要 "宇宙时间来完成，并且通常可以被忽略"。他补充说："在实践中，平衡的标准是循环的。从操作上讲，如果一个系统的特性被热力学理论一致地描述，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."

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."

贝蒂（J.A. Beattie）和奥本海姆（I. Oppenheim）写道："坚持对平衡定义的严格解释将排除热力学对实际系统的几乎所有状态的应用"。

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."

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."

另一位被赫伯特·卡伦（H.B. Callen）引用认为其给予了 "学术性的严格处理"， 并被阿德金斯（Adkins）引用为写了一篇 "经典文章 "的作者布莱恩·皮帕尔德（A.B Pippard）写道："只要时间足够长，过冷的蒸汽最终会凝结，......。然而，所涉及的时间可能是如此巨大，也许是10¹⁰⁰ 年或更久，......。在大多数情况下，只要不人为地刺激快速变化，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."

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."

另一位作者明斯特（A.Münster）在这样的背景下指出，热核过程往往发生得很慢，以至于在热力学中可以被忽略。他评论说 "因此，'绝对平衡'或'与所有可想象的过程有关的平衡'的概念没有物理意义"。因此，他指出 "......我们只能考虑与特定过程和确定的实验条件有关的平衡。"

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."

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."

根据拉斯洛·蒂萨（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，因为冻结在非平衡状态的发生是非常普遍的"。

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.

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其与周围环境的接触，允许所有化学物质和各种能量同时通过。一个处于热力学平衡状态的系统可以以均匀的加速度在空间中移动，但在移动过程中不得改变其形状或大小；因此，它被定义为空间中的一个刚性体积。它可能存在于外部力场中，由远大于系统本身的外部因素决定，因此系统内的事件不能明显地影响外部力场。只有当外部力场是均匀的，并决定其均匀加速度时，系统才能处于热力学平衡状态，或者如果它位于一个非均匀的力场中，但被其表面的局部力，如机械压力，保持在那里不动时。

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.

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。根据莫尔斯（P.M.Morse）的说法: “应该强调的是，存在热力学状态的事实，......，以及存在由平衡状态唯一指定的热力学变量的事实......不是从某些哲学第一原理中逻辑地推导出来的结论。它们是从两个多世纪的实验中不可避免地得出的结论"。这意味着，热力学平衡不能仅仅根据热力学的其他理论概念来定义。贝林恩（M. Bailyn）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状态的存在。

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

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往往是小心翼翼地陈述的，并有一些保留意见或其他看法。

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.

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，明斯特（A. Münster）写道："一个孤立的系统处于热力学平衡状态，当该系统中没有以可测量的速度发生状态变化时"。这里有两个限制：系统是孤立的；任何状态的变化都是不可估量的缓慢。他讨论了第二个限制性条件，给出了室温下没有催化剂的氧气和氢气混合物的描述。明斯特指出，热力学平衡状态是由较少的宏观变量描述的，而不是一个特定系统的任何其他状态。这一个部分是，但全局就不是，因为系统内和通过系统的所有流量都是零。

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.

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.

哈斯（R. Haase）对热力学的介绍并没有从限制热力学平衡开始，因为他允许非平衡热力学的存在。他考虑了一个具有时间不变性的任意系统。他通过切断除了外部力场外所有外部影响来测试它的热力学平衡。如果绝缘后，没有任何变化，他认为该系统是处于平衡状态。

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". 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".

在题为 "热力学平衡 "的一节中，赫伯特·卡伦（H.B. Callen）用一段话定义了平衡状态。他指出，它们 "是由系统内的内在因素决定的"。它们是 "终端状态"，随着时间的推移，系统朝着这个方向发展，这可能以 "冰河般的缓慢 "发生。这段话并没有明确指出，为了实现热力学平衡，系统必须是孤立的；卡伦并没有阐明他所说的 "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.

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.

另一位教科书作者阿德金斯（C.J. Adkins）明确允许热力学平衡发生在一个非孤立的系统中。然而，他的系统在物质转移方面是封闭的。他写道："一般来说，达到热力学平衡的方法将涉及与周围环境的热作用和类似工作的相互作用"。他将这种热力学平衡与热平衡区分开来，在热平衡中，只有热接触在调节能量的转移。

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.

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.

另一位教科书作者帕廷顿（J.R. Partington）写道："（i）平衡状态是一个独立于时间的状态"。但是，在提到 "只是表面上处于平衡状态 "的系统时，他补充说："这种系统处于″假平衡″状态。" 帕廷顿的声明并没有明确指出平衡是指一个孤立的系统。像明斯特一样，帕廷顿也提到了氧气和氢气的混合物。他补充了一个条件："在真正的平衡状态下，任何影响状态的外部条件的最小变化都会产生状态的微小变化......" 这意味着热力学平衡必须对微小的扰动保持稳定；这一要求对于热力学平衡的严格含义至关重要。

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."

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."

克劳福德（F.H. Crawford）的一本给写给学生的课本中的一节标题为 "热力学平衡"。它区分了几种流动的驱动力，然后说："这些是孤立系统走向完全的机械、热、化学和电的状态的明显普遍趋势的例子，或者用一个词来形容，就是热力学平衡"

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.

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.

布赫达尔（H.A. Buchdahl）的经典热力学专著考虑了 "热力学系统的平衡"，但实际上没有写 "热力学平衡 "这个词。在提到封闭于物质交换的系统时，布赫达尔写道："如果一个系统处于适当静态的终端状态，它将被说成是处于平衡状态。"布赫达尔的专著还讨论了无定形玻璃，以用于热力学描述。它指出 "更准确地说，只要实验测试表明'缓慢'的转变实际上是可逆的，就可以认为玻璃处于平衡状态。"将这一条件作为热力学平衡定义的一部分是不习惯的，但通常会有相反的假设：如果一个处于热力学平衡状态的物体受到一个足够缓慢的过程，该过程可以被认为是足够接近可逆的，并且该物体在该过程中保持足够接近热力学平衡的状态。

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." 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. 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."

明斯特（A. Münster）通过引入接触平衡的概念，仔细地扩展了他对孤立系统的热力学平衡的定义。这限定了在考虑非孤立系统的热力学平衡时允许的特殊过程，特别是开放系统，这些系统可能从其周围环境获得或失去物质。接触平衡是在系统和周围环境之间，与其他系统接触，接触是通过一种特殊的壁；对于其余部分，整个联合系统是孤立的。康斯坦丁·卡拉西奥多里（C. Carathéodory）也考虑过这种特殊的壁，其他人也提到过。它们是有选择的、可渗透的。它们可能只对机械功渗透，或只对热渗透，或只对某些特定化学物质渗透。每一个接触平衡都定义了一个强度量；例如，一个只对热渗透的墙壁定义了一个经验温标。对于系统的每个化学成分，都可以存在一个接触平衡。在接触平衡中，尽管有可能通过选择性渗透的壁进行交换，但系统是不变的，就像它处于孤立的热力学平衡中一样。这个方案遵循的一般规则是："......我们只能针对特定的过程和确定的实验条件来考虑平衡。"开放系统的热力学平衡意味着，就每一种相关的选择性渗透壁而言，当系统和周围环境的各自强度量相等时，接触平衡就存在。这个定义并没有简单地说明在内部或边界不存在任何物质或能量的流动；但它与下面的定义是兼容的，后者也确实是这样说的。

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".

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'.

泽曼斯基（M. Zemansky）还区分了力学平衡、化学平衡和热平衡。他接着写道："当所有三种平衡的条件都得到满足时，系统就被说成是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'.

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".

莫尔斯（P.M. Morse）写道，热力学关注的是 "热力学平衡状态"。他还使用了 "热平衡 "这一短语，同时讨论了能量作为热量在身体和周围环境中的储热体之间的转移，但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".

瓦尔德姆（J.R. Waldram）写道， "一个明确的热力学状态"。他为一个系统定义了 "热平衡 "一词，即 "当它的观察指标不再随时间变化时"。但在这个定义之后的文章中，他写道，一块玻璃还没有达到其 "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.

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，贝林恩（M. Bailyn）写道："每个强度量都有自己的平衡类型"。然后他定义了热平衡、机械平衡和物质平衡。因此，他写道："如果所有的强度量变得均匀，那么热力学平衡就被说成是存在的"。他在这里没有考虑外部力场的存在。

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 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.

柯克伍德（J.G. Kirkwood）和奥本海姆（I. Oppenheim）将热力学平衡定义如下。"一个系统处于热力学平衡状态，如果在分配给实验的时间段内，(a)其强度量与时间无关，(b)在其内部或与周围环境的边界不存在物质或能量的流动。" 很明显，他们没有把定义限制在孤立的或封闭的系统上。他们没有讨论以 "冰河般的缓慢 "发生变化的可能性，并且超出了分配给实验的时间段。他们指出，对于两个接触的系统，存在着一个拥有强度量的小亚类，如果该小亚类中的所有属性都分别相等，那么所有各自的强度量都相等。只要满足其他一些条件，热力学平衡状态就可以由这个子类来定义。

Characteristics of a state of internal thermodynamic equilibrium 内部热力学平衡状态的特征

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. 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元素都可以与系统中任何其他几何上一致的体积元素互换，其效果是使系统在热力学上保持不变。一般来说，一个强大的外部力场使一个处于自身内部热力学平衡的单相系统在某些强度量方面不均匀。例如，混合物中相对密集的成分可以通过离心法进行浓缩。

Uniform temperature 均匀温度

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." 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.

这种由外力引起的平衡不均匀性，对于强度量温度来说不会发生。根据古根海姆（E.A. Guggenheim）的说法，"热力学最重要的概念是温度"。普朗克（Planck）在介绍他的论文时简要介绍了热和温度以及热平衡，然后声称： "在下文中，我们将主要讨论任何形式的均质、各向同性的物体，它们在整个物质中拥有相同的温度和密度，并在垂直于表面的各处受到均匀的压力作用。" 和康斯坦丁·卡拉西奥多里（C. Carathéodory）一样，普朗克把表面效应、外部场和各向异性的晶体排除在外。虽然提到了温度，但普朗克并没有明确提及热力学平衡的概念。相比之下，卡拉西奥多里在吉布斯（Gibbs）之后提出了 "平衡状态 "的概念（吉布斯常规地谈到了 "热力学状态"），这是一种对封闭系统的经典热力学的表述方法，尽管没有明确使用 "热力学平衡"这一短语，也没有明确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. 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. 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. Considerations of kinetic theory or statistical mechanics also support this statement.

如果热力学平衡位于一个外部力场中，一般来说，只有温度可以认为在空间上是均匀的。如果外部力场不为零，温度以外的强度量一般将是不均匀的。在这种情况下，一般来说，需要额外的变量来描述空间的不均匀性。在热力学平衡的系统内，温度在空间和时间上都是均匀的。在一个自身处于内部热力学平衡状态的系统中，不存在净的内部宏观流动。特别是，这意味着系统的所有部分都处于相互辐射交换平衡状态。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，包括那些非均匀的外部力场。对于一个外部强加的引力场，这可以在宏观热力学范畴中证明，通过微积分变化，并使用Langrangian乘数的方法。对动力学理论或统计力学的思想也支持这一说法。 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. Considerations of kinetic theory or statistical mechanics also support this statement.

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.

为了使一个系统可能处于其自身的内部热力学平衡状态，它当然需要，但还不够，它必须处于其自身的内部热平衡状态；一个系统有可能在达到内部热平衡之前达到内部力学平衡。

A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous. 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.

在没有外力的情况下，一个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。为了使一个系统可能处于它自己的内部热力学平衡状态，它当然有必要处于它自己的内部热平衡状态，但这还不够；一个系统有可能在达到内部热平衡之前达到内部力学平衡。

Number of real variables needed for specification 规范所需的实际变量的数量

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. 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." 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.

在阐述他的封闭系统平衡热力学方案时，康斯坦丁·卡拉西奥多里（C. Carathéodory）最初的假设实验表明，一定数量的实际变量定义了作为平衡流形的点的状态。用普利高津（Prigogine）和德法伊（Defay）在1945年的说法："这是一个经验问题，当我们规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，根据明斯特（A.Münster）的看法，对于一个给定的孤立系统的任何状态，定义热力学平衡所需的变量数量是最少的。如上所述，柯克伍德（J.G. Kirkwood）和奥本海姆（I. Oppenheim）指出，热力学平衡状态可以由一个特殊的强度量子类来定义，该子类中的成员数量是确定的。

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部力场中，一般来说，只有温度可以认为在空间上是均匀的。如果外部力场不为零，温度以外的强度量一般将是不均匀的。在这种情况下，一般来说，需要额外的变量来描述空间的不均匀性。

Stability against small perturbations 对小扰动的稳定性

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，帕廷顿（J.R. Partington）指出，热力学平衡状态对小的瞬态扰动是稳定的。如果没有这个条件，一般来说，旨在研究热力学平衡系统的实验会遇到严重困难。

Approach to thermodynamic equilibrium within an isolated system 孤立系统热力学平衡的方法

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.

当一个物质从非均匀性或化学非平衡状态开始，然后被隔绝，它会自发地朝着自己的内部热力学平衡状态演变。但内部热力学平衡不会在所有地方同时达到；有些平衡可以在其他平衡之前达到。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，热平衡的达成比化学平衡的达成要快得多。

Fluctuations within an isolated system in its own internal thermodynamic equilibrium 孤立系统内部热力学平衡中的涨落

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，热力学平衡持续了无限长的时间。在经典物理学中，忽略测量的影响往往是很方便的，在本文中也是这样假设的。

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.

为了考虑一个孤立的热力学系统中的涨落fluctuations概念，一个方便的例子是由其广延量、内能、体积和质量组成指定的系统。根据定义，它们是时间不变的。根据定义，它们与它们的共轭状态强化函数的时间不变的标称值相结合，反温度、压力除以温度、化学势除以温度，以便完全遵守热力学定律。但是，热力学定律，结合指定的广延量的值，并不足以提供关于这些标称值的知识。还需要进一步的信息，即系统的构成属性。

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，在重复测量这些状态的共轭密集函数时，发现它们在不同时间有轻微的不同值。这种变化被认为是由于内部涨落造成的。不同的测量值平均到它们的标称值。

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)."

如果系统是真正的宏观系统，正如经典热力学所假设的那样，那么涨落就太小了，无法从宏观上检测。这被称为热力学极限。实际上，物质的分子性质和动量传递的量子性质已经从人们的视线中消失了，小到无法看到。布赫达尔（Buchdahl）认为："......在严格的现象学理论中，没有关于平衡的涨落的想法的位置（不过，可以见第76节）"。

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.

如果该系统被反复细分，最终会产生一个小到足以表现出明显涨落的系统。这是一个介观层面的研究。然后，涨落直接取决于系统的各个壁的性质。那么，独立状态变量的精确选择就很重要了。在这个阶段，热力学定律的统计特征变得明显。

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.

如果中观系统被进一步反复分割，最终会产生一个微观系统。那么，物质的分子特性和动量传递的量子性质在波动过程中变得很重要。人们已经离开了经典或宏观热力学的领域，而需要量子统计力学。涨落可以成为相对的主导，而测量的问题也变得重要。

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.

“系统处于其自身的内部热力学平衡状态”的说法可以被理解为 "许多这样的不确定的测量已经在各种测量值中没有时间趋势"。因此，"一个系统处于它自己的内部热力学平衡状态，其状态函数的标称值与其指定的状态变量共轭" 的说法，远比 "这些状态函数的一组单一的同时测量值具有相同的值 "的说法更有参考价值。这是因为单一的测量可能是在轻微的涨落中进行的，偏离了那些共轭密集状态函数的另一组名义值，那是由于未知的和不同的构成属性。除非对属于平衡状态的标称值也有了解，否则单次测量无法判断是否可能如此。

Thermal equilibrium 热平衡

Main article: Thermal equilibrium

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 several occurring 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.

欧（B. C. Eu）明确区分了 "热平衡 "和 "热力学平衡"。他考虑了两个热接触的系统，一个是温度计，另一个是有几个不可逆过程发生的系统，需要非零通量；这两个系统被一堵只对热渗透的墙分开。他考虑的情况是，在有限的时间范围内，温度计读数和不可逆过程都是稳定的。那么就有了热平衡而没有热力学平衡。因此，欧提出，即使在不存在热力学平衡的情况下，也可以考虑适用热力学第三定律；他还提出，如果变化发生得如此之快，以至于无法定义稳定的温度，那么 "就不再可能通过热力学形式来描述这一过程。换句话说，热力学对这样的过程没有意义"。这说明了温度的概念对热力学的重要性。

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.

当相互之间有热接触的两个系统不再有能量的净交换时，就达到了热平衡。由此可见，如果两个系统处于热平衡状态，那么它们的温度是相同的。

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指标不再随时间变化时，就会出现热平衡。例如，一个理想气体的分布函数已经稳定在一个特定的麦克斯韦-波尔兹曼分布上，就处于热平衡状态。这种结果允许将单一的温度和压强作用域整个系统。对于一个孤立的系统，很可能在达到热平衡之前就达到了力学平衡，但最终，平衡的所有方面，包括热平衡，都是热力学平衡的必要条件。

Non-equilibrium 非平衡

Main article: Non-equilibrium thermodynamics

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.

一个系统的内部热力学平衡状态应与 "静止状态 "区分开来，在这种状态下，热力学参数在时间上是不变的，但系统不是孤立的，因此，在进入和离开系统时，存在着在时间上不变的非零宏观通量。

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.

非平衡热力学是热力学的一个分支，处理不处于热力学平衡的系统。在自然界中发现的大多数系统都不处于热力学平衡状态，因为它们正在发生变化或可以随着时间的推移而引发变化，并且不断地和不连续地受到其他系统的物质和能量流动的影响。对非平衡系统的热力学研究需要比平衡热力学所处理的更普遍的概念。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. It states that a non-equilibrium system evolves such as to maximize its entropy production.

研究远离平衡的系统的规律也是值得商榷的。这些系统的指导原则之一是最大熵产原则。它指出，一个非平衡系统的演变是为了使其熵的产生最大化。 It states that a non-equilibrium system evolves such as to maximize its entropy production.

See also 参见

Thermodynamic models 热力学模型

• Non-random two-liquid model (NRTL model) - Phase equilibrium calculations 非随机双液体模型——相平衡计算

• UNIQUAC model - Phase equilibrium calculations UNIQUAC 模型——相平衡计算

• Time crystal 时间晶体

Topics in control theory 控制理论主题

• Steady state 稳态

• Transient state 瞬态

• Coefficient diagram method 系数图法

• Control reconfiguration 控制重配

• Cut-insertion theorem 切割-插入定理

• Feedback 反馈

• H infinity H-无穷

• Hankel singular value 汉克尔奇异值

• Krener's theorem 克雷纳定理

• Lead-lag compensator 前导滞后补偿器

• Minor loop feedback 小环反馈

• Multi-loop feedback 多环反馈

• Positive systems 阳性系统

• Radial basis function 径向基函数

• Root locus 根基位置

• Signal-flow graphs 信号流图

• Stable polynomial 稳定多项式

• State space representation 状态空间表示

• Underactuation 不充分作用

• Youla–Kucera parametrization 尤拉-库塞拉参数化

• Markov chain approximation method 马尔科夫链近似方法

Other related topics 其他相关主题

• Automation and remote control 自动化和远程控制

• Bond graph 建合图

• Control engineering 控制工程

• Control–feedback–abort loop 控制-反馈-中止循环

• Controller (control theory) 控制器（控制理论）

• Cybernetics 控制论

• Intelligent control 智能控制

• Mathematical system theory 数学系统理论

• Negative feedback amplifier 负反馈放大器

• People in systems and control 系统和控制论中做出贡献的人

• Perceptual control theory 感知控制理论

• Systems theory 系统理论

• Time scale calculus 时标微积分

General references 一般参考文献

• 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

Cited bibliography 引用参考书目

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      • 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.
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              • Guggenheim, E.A. (1949/1967). Thermodynamics. An Advanced Treatment for Chemists and Physicists, fifth revised edition, North-Holland, Amsterdam.
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              • 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.
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                • Planck. M. (1914). The Theory of Heat Radiation, a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.
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                • 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.
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                  .
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                  External links 外部链接

                  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 热力学平衡/edithistory