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第1,199行：
第1,199行： − 遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。+ − + − − 第1,231行：
第1,229行： − + + 第1,247行：
第1,246行： − + − 第1,267行：
第1,265行： − 这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。+ − 第1,278行：
第1,275行： − − − 第1,297行：
第1,291行： − 在哪里+ − − − 第1,306行：
第1,297行： − − S 是系统的熵 第1,313行：
第1,302行： − 时间到了。+ 第1,329行：
第1,318行： − 平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:+ 第1,349行：
第1,338行： − 用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。+ 第1,368行：
第1,357行： − − 用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math − − 第1,378行：
第1,363行： − + − 给你 − − − 第1,389行：
第1,370行： − 数学 q / math 是进入系统的热流+ + − + − Math 是热量进入系统时的温度。 第1,405行：
第1,386行： − + 第1,424行：
第1,405行： − − 用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math − − 第1,435行：
第1,412行： − 这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。+
→History
The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.
The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.
遍历假设对玻尔兹曼方法也很重要。在很长一段时间内，在具有相同能量的微观态相空间的某些区域所花费的时间与这个区域的体积成正比，即在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和统计集合的平均值是相同的。
克劳修斯还提出了一种传统的学说，他认为熵可以被理解为宏观系统中的分子“无序”，但这种学说已经过时了。
In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:
In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:
1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:
1856年，德国物理学家鲁道夫 • 克劳修斯Rudolf Clausius 阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:
: <math>\int \frac{\delta Q}{T} = -N</math>
: <math>\int \frac{\delta Q}{T} = -N</math>
where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:
where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:
其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:
其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来在1865年，克劳修斯将“等价值”定义为熵。也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后克劳修斯总结道:
This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.
This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.
这句话是第二定律最著名的措辞。由于其语言松散，如万象（经验体系？）与缺乏具体的条件相同，如开放，封闭，或孤立，许多人认为这一简单的陈述意味着热力学第二定律几乎适用于每一个可以想象的学科。这不是真的；这句话只是一个更广泛和更精确的描述的简化版本。
就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:
就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:
where
where
这里S 是系统的熵
S is the entropy of the system and
S is the entropy of the system and
: ''t'' is [[time]].
: ''t'' is [[time]].
t is time.
t is time.
T是时间
The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:
The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:
平衡后用等号。另一种表述孤立系统第二定律的方法是
with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.
with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.
用[math]\dot S{i}[/math]表示系统内所有进程产生熵的速率之和。这个公式的优点是它显示了熵产生的影响。熵产生率是一个非常重要的概念，因为它决定（限制）热机的效率，乘以环境温度[math]T{a}[/math]，得到所谓的耗散能[math]P{diss}=T{a}\dots S{i}[/math]。
<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math>
<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math>
Here
Here
这里
<math>\dot Q</math> is the heat flow into the system
<math>\dot Q</math> is the heat flow into the system
<math>\dot Q</math> 是进入系统的热流。
: <math>T</math> is the temperature at the point where the heat enters the system.
: <math>T</math> is the temperature at the point where the heat enters the system.
<math>T</math> is the temperature at the point where the heat enters the system.
<math>T</math> is the temperature at the point where the heat enters the system.
<math>T</math> 是热量进入系统时的温度。
The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.
The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.
等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。
等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,符号保持不变。如果系统有多处供热，必须求相应项的代数和。
<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math>
<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math>
Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.
Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.
这里<math>\dot S</math> 是进入系统的熵流，与进入系统的物质流有关。它不应该与熵的时间导数混淆。如果物质在几个地方被供给，需要取这些贡献的代数和。
==Statistical mechanics==
==Statistical mechanics==