更改

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).

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)。

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。

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'''<font color="#ff8000">H定理H-theorem</font>'''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。

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'''<font color="#ff8000">绝热定理adiabatic theorem</font>'''，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'''<font color = '#ff8000'>能态 energy eigenstate</font>'''，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：

对于外部变量 x 可以定义'''<font color="#ff8000">广义力generalized force</font>''' 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：

 

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