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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

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<font color = 'blue'>都</font>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

这么多的能量本征态。如果数学 <math>Y dx\leq\delta E</math>，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

<font color = 'red'><s>这么多</s></font><font color = 'blue'>这些</font>的能量本征态。如果<math>Y dx\leq\delta E</math>，<font color = 'blue'>则</font>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

Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:

Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:

将上面的表达式表示为对 E 的导数，并且对Y求和得到表达式:

将上面的表达式表示为对 E 的导数，并且对 Y 求和得到表达式:

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，例如它不能根据系统大小进行缩放。相反，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此我们发现:

第一项是集约型的，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>它不能根据系统大小进行缩放。相反，<font color = 'red'><s>最后一项的规模与逆系统的规模一样</s></font><font color = 'blue'>最后一项跟随逆系统的大小而缩放</font>，因此将在热力学极限中消失。因此我们发现:

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canonical ensemble'''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'''<font color = '#ff8000'>正则系综canonical ensemble</font>'''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 (statistical mechanics)|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 (statistical mechanics)|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:

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: