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→Derivation from statistical mechanics
The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.
The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.
'''<font color="#ff8000">气体动力学Kinetic theory of gases</font>'''理论的第一个力学论证由'''<font color="#ff8000">麦克斯韦James Clerk Maxwell</font>'''在1860年给出，指出分子碰撞引起温度均衡<font color = 'blue'>化</font>，因此整体趋向于'''<font color="#ff8000">平衡 Equilibrium </font>'''； '''<font color="#ff8000">玻尔兹曼Ludwig Boltzmann</font>'''在1872年提出的'''<font color="#ff8000"> H 定理H-theorem</font>'''也认为，气体由于碰撞应该随着时间的推移趋向于'''<font color="#ff8000">麦克斯韦-波兹曼分布Maxwell–Boltzmann distribution</font>'''。
'''<font color="#ff8000">气体动力学Kinetic theory of gases</font>'''理论的第一个力学论证由'''<font color="#ff8000">麦克斯韦James Clerk Maxwell</font>'''在1860年给出，指出分子碰撞引起温度均衡化，因此整体趋向于'''<font color="#ff8000">平衡 Equilibrium </font>'''； '''<font color="#ff8000">玻尔兹曼Ludwig Boltzmann</font>'''在1872年提出的'''<font color="#ff8000"> H 定理H-theorem</font>'''也认为，气体由于碰撞应该随着时间的推移趋向于'''<font color="#ff8000">麦克斯韦-波兹曼分布Maxwell–Boltzmann distribution</font>'''。
Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.
Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.
由于'''<font color="#ff8000">洛施密特悖论Loschmidt's paradox</font>'''，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个'''<font color="#ff8000">边界条件boundary condition</font>'''，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始('''<font color="#ff8000">大爆炸 the Big Bang</font>''') ，<font color = 'blue'>尽管</font>也有人提出了其他<font color = 'red'><s>假设</s></font><font color = 'blue'>场景</font>。
由于'''<font color="#ff8000">洛施密特悖论Loschmidt's paradox</font>'''，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个'''<font color="#ff8000">边界条件boundary condition</font>'''，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始('''<font color="#ff8000">大爆炸 the Big Bang</font>''') ，尽管也有人提出了其他场景。
Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:
Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:
基于这些假设，在统计力学中，第二定律不是一个假设，而是'''统计力学基本假设Statistical mechanics#Fundamental postulate|fundamental postulate'''的一个结果，也被称为'''<font color="#ff8000">等先验概率假设 equal prior probability postulate</font>'''。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分即热孤立系统的熵只能增加，是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统且具有能量<math>E</math>的熵表示为：
Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.
Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.
假设我们有一个孤立系统，其宏观状态由许多变量指定。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下到达所有可到达状态的可能性是相等的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能量本征态 不懂翻译是否正确
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能量本征态 不懂翻译是否正确
--～～～～ 我觉得是这样的 --～～～～
We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are
We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are
我们可以把它和由恒定能量 E 下的 x 导出来的熵联系起来。假定我们把 x 改变至 x + dx。然后因为能量本征态依赖于 x， <math>\Omega\left(E\right)</math> 将会改变，这导致能量本征态进入或超出<math>E</math> 和<math>E+\delta E</math> 之间的范围。让我们再次关注<math>\frac{dE_{r}}{dx}</math> 处于 <math>Y</math> 和 <math>Y + \delta Y</math> 之间的能量本征态。由于这些能量本征态的能量增加了 Y dx，所有这些在 E-Y dx 到 E 之间能量本征态<font color = 'blue'>都</font>从 E 以下移动到 E 以上。 因此有
我们可以把它和由恒定能量 E 下的 x 导出来的熵联系起来。假定我们把 x 改变至 x + dx。然后因为能量本征态依赖于 x， <math>\Omega\left(E\right)</math> 将会改变，这导致能量本征态进入或超出<math>E</math> 和<math>E+\delta E</math> 之间的范围。让我们再次关注<math>\frac{dE_{r}}{dx}</math> 处于 <math>Y</math> 和 <math>Y + \delta Y</math> 之间的能量本征态。由于这些能量本征态的能量增加了 Y dx，所有这些在 E-Y dx 到 E 之间能量本征态都从 E 以下移动到 E 以上。 因此有
such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference
such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference
这些的能量本征态。如果<math>Y dx\leq\delta E</math>，则所有这些能量本征态将移动到 <math>E</math> 到 <math>E+\delta E</math>的范围内，使得<math>\Omega</math>增加。从<math>E+\delta E</math>以下移动到<math>E+\delta E</math>以上的能量本征态数目为 <math>N_{Y}\left(E+\delta E\right)</math>。它们的差
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:
第一项是集约型的，即它不能根据系统大小进行缩放。相反，最后一项跟随逆系统的大小而缩放，因此将在热力学极限中消失。因此我们发现:
Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. <font color = 'red'>We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. </font>It follows from the general formula for the entropy:
Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. <font color = 'red'>We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. </font>It follows from the general formula for the entropy:
这里 Z 是一个使所有概率之和<font color = 'red'><s>归一化</s></font><font color = 'blue'>正态化到 1 </font>的因子，这个函数被称为'''<font color = '#ff8000'>配分函数Partition function (statistical mechanics)|partition function</font>'''。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:
这里 Z 是一个使所有概率之和归一化到 1 的因子，这个函数被称为'''<font color = '#ff8000'>配分函数Partition function (statistical mechanics)|partition function</font>'''。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能级依赖温度吗？即“现在我们考虑……的可逆改变”当中，“on which the energy levels depend”修饰（1）in the external parameters，还是（2）in the temperature and in the external parameters
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能级依赖温度吗？即“现在我们考虑……的可逆改变”当中，“on which the energy levels depend”修饰（1）in the external parameters，还是（2）in the temperature and in the external parameters