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→Derivation of the entropy change for reversible processes
The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:
The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:
热力学第二定律的第二部分指出，经历可逆过程的系统的熵变为:
<math>dS =\frac{\delta Q}{T}</math>
<math>dS =\frac{\delta Q}{T}</math>
<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math>
<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math>
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。
The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:
The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:
对于外部变量 x 可以定义广义力 X ，因此如果 x 增加 dx，那么<math>X dx</math> 就是系统所做的功。例如，如果 x 是体积，那么 X就 是压强。一个已知处于能量本征态<math>E_{r}</math>的系统的广义力为：
<math>X = -\frac{dE_{r}}{dx}</math>
<math>X = -\frac{dE_{r}}{dx}</math>
Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:
Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:
由于系统可以在<math>\delta E</math> 区间内的任意能量本征态内，我们将系统的广义力定义为上述表达式的期望值:
<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math>
<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math>
To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:
To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:
为了计算平均值，我们按照能量本征态来划分<math>\Omega\left(E\right)</math>，通过计算有多少能量本征态的<math>\frac{dE_{r}}{dx}</math>值在区间在<math>Y</math> 到 <math>Y + \delta Y</math>之间。把这个数字叫做为 <math>\Omega_{Y}\left(E\right)</math>，我们有:
<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math>
<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math>
The average defining the generalized force can now be written:
The average defining the generalized force can now be written:
根据平均值定义的广义力现在可以写成:
<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math>
<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math>
数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math
数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math
===Derivation for systems described by the canonical ensemble===
===Derivation for systems described by the canonical ensemble===