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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.

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，一位被广泛引用的作家 h · b · 卡伦在这里写道: “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和评论，认为它们可能需要“宇宙时间才能完成，而且通常可以忽略”。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! ”

例如，一位被广泛引用的作家'''<font color="#ff8000">H.B.卡伦 H.B.Callen</font>'''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和评论，认为它们可能需要“宇宙时间才能完成，而且通常可以忽略”。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."

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。”

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", 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."

卡伦引用另一位作者的话说，他给出了“学术性和严谨的论述” ，阿德金斯引用他的话说，他写了一本“经典著作”—— a.b。皮帕德在文中写道: “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，然而，也许10年或者更长的时间。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。”

Callen引用另一位作者的话说，他给出了“学术性和严谨的论述” ，Adkins引用他的话说，他写了一本“经典著作”—— '''<font color="#ff8000">A.B.皮帕德 A.B.Pippard</font>'''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，然而，也许10年或者更长的时间。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."  

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. m ü nster 就是在这种背景下写作的。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，因此没有物理意义。”因此他说: “ ... 我们只能考虑特定过程和确定的实验条件下的平衡。”

另一位作者A. Münster就是在这种背景下写作的。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，因此没有物理意义。”因此他说: “ ... 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."

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. 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。”

根据'''<font color="#ff8000">L.缇莎 L.Tisza</font>'''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==

==Definitions==

 

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。莫尔斯说: “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。

热力学平衡是热力学理论的一个'''<font color="#ff8000">基本概念 Primitive Notion</font>'''。'''<font color="#ff8000">P.M.莫尔斯 P.M.Morse</font>'''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。

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，a. m ü nster 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。

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.

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.

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".

在一个标题为“热力学平衡”的章节中，h.b。Callen defines equilibrium states in a paragraph.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。

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.

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，茜洁。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。

另一位教科书作者，'''<font color="#ff8000">C.J.阿德金斯 C.J. Adkins</font>'''，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 ..." 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.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 m ü nster 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。

另一位教科书作者 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."

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."

一本 f.h. 的学生教科书。克劳福德有一个标题为“热力学平衡”的部门。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。”

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." 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.

一部经典热力学的专著。布赫达尔考虑的是“热力学系统的平衡” ，实际上并没有写下短语“热力学平衡”。布赫达尔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，布赫达尔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。

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。