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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

我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e  right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 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，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 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 之间能量本征态从 E 以下移动到 E 以上。 因此有

 

  <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math>

  <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math>

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

这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于

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

  <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math>

  <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math>

is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.

is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.

因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e  right) / math 和数学 n { y } left (e + delta e  right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。

因此是 <math>\Omega</math>增长的净贡献。请注意如果 Y dx 大于 <math>\delta E</math>，那么将会有能量本征态从 E 以下移动到<math>E+\delta E</math>以上。它们在<math>N_{Y}\left(E\right)</math> 和 <math>N_{Y}\left(E+\delta E\right)</math>中都有计数，因此上述表达式在这种情况下也是有效的。

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，对 y 的求和得到表达式:

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:

  <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math>

  <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math>

The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:

The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:

关于 x 的欧米茄 / 数学对数导数由以下方式给出:

<math>\Omega</math>的对数关于x的导数由以下方式给出:

  <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math>

  <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math>

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:

第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，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，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:

  <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math>

  <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math>

Combining this with

Combining this with

  <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math>

  <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math>

Gives:

Gives:

  <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math>

  <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math>

===Derivation for systems described by the canonical ensemble===

===Derivation for systems described by the canonical ensemble===