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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。在热平衡，一个孤立系统的熵包含数学 e / math 的能量是:

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：

  <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math>

  <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math>

 

where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.

where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.

其中 math Omega left (e  right) / math 是 math e / math 和 math e + delta e / math 之间一个小区间内的量子状态数。这里的 math  delta e / math 是一个宏观上很小的能量区间，并且保持不变。严格地说，这意味着熵取决于对 math delta e / math 的选择。然而，在香港特别行政区热力学极限。在无穷大系统的极限条件下，特定熵(单位体积或单位质量的熵)不依赖于数学上的增量 e / math。

其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。

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，例如，参考总体积，活塞在系统中的位置等。然后 math Omega / math 将取决于这些变量的值。如果一个变量不是固定的，例如。我们不会在某个位置夹住活塞) ，那么因为所有可达状态在平衡状态下的可能性是相等的，平衡状态下的自由变量是这样的，因为这是平衡状态下最可能的情况。

假设我们有一个孤立系统，其宏观状态由许多变量描述。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所有可到达状态的等几率到达的，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。

If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).

If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).

如果变量最初固定到某个值，然后在释放时，当达到新的平衡时，变量将自我调整，使 math Omega / math 最大化，这意味着熵将增加或保持不变(如果变量固定的值恰好是平衡值)。

如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。

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 {{math|''H''}} 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 {{math|''H''}} 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.

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.

假设我们从一个平衡状态出发，突然移除了对一个变量的约束。然后在我们做完这个之后，有一些关于可达到的微观状态的数学 / Omega / math，但是还没有达到平衡，所以系统处于可达到的状态的实际概率还不等于 math 1 / Omega / math 的先验概率。我们已经看到，在最终的平衡状态，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 定理证明了在中间离开平衡态时，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。