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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.

气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。

'''气体动力学Kinetic theory of gases'''理论的第一个力学论证由'''詹姆斯·克拉克·麦克斯韦James Clerk Maxwell'''在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； '''路德维希·玻尔兹曼Ludwig Boltzmann'''在1872年提出的''' H 定理H-theorem'''也认为，气体由于碰撞应该随着时间的推移趋向于'''麦克斯韦-波兹曼分布Maxwell–Boltzmann distribution'''。

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.

由于洛施密特悖论，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边界条件，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(大爆炸) 。也有人提出了其他假设。

由于'''洛施密特悖论Loschmidt's paradox'''，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Correlation and dependence|uncorrelated'''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'''边界条件boundary condition'''，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('''大爆炸Big Bang''') 。也有人提出了其他假设。

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:

考虑到这些假设，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的一个结果，也被称为等先验概率假设。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。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。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包含能量<math>E</math>的熵表示为：

考虑到这些假设，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，也被称为等先验概率假设。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。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。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包含能量<math>E</math>的熵表示为：

Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity  increases monotonically as a function of time during the intermediate out of equilibrium state.

Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity  increases monotonically as a function of time during the intermediate out of equilibrium state.

假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。

假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的'''H定理H-theorem'''证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。

See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.

See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.

请参阅此处查看该定义的正当性。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的能量本征态，因此系统的能量会随着其所在能量本征态的能量变化而变化。

请参阅'''此处Microcanonical ensemble|here'''查看该定义的正当性。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的'''绝热定理adiabatic theorem'''，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的能量本征态，因此系统的能量会随着其所在能量本征态的能量变化而变化。

If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:

If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:

如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:

如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由'''正则系综canonical ensemble'''给出:

Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. 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. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:

这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。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的因子，这个函数被称为'''配分函数Partition function (statistical mechanics)|partition function'''。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: