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Edit 1 (done)
文章 Boltzmann distribution, 删除下面这句话:

The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of entropy (the Gibbs entropy formula $S = -k_{\mathrm{B}}\sum_i p_i \log p_i$ ) and the thermodynamic definition of entropy ($\mathrm{d} S = \frac{\delta Q_\text{rev}}{T}$, and the fundamental thermodynamic relation).

然后增加下面这个章节:

Generalized Boltzmann distribution
Distribution of the form
 * $$\Pr\left(\omega\right)\propto\exp\left[\sum_{\eta=1}^{n}\frac{X_{\eta}x_{\eta}^{\left(\omega\right)}}{k_{B}T}-\frac{E^{\left(\omega\right)}}{k_{B}T}\right]$$

is called generalized Boltzmann distribution. Although equations of this type has been widely used for a long time, the name "Generalized Boltzmann distribution" was only introduced in 2019.

The Boltzmann distribution is a special case of the generalized Boltzmann distribution. The generalized Boltzmann distribution is used in statistical mechanics to describe canonical ensemble, grand canonical ensemble and isothermal–isobaric ensemble. The generalized Boltzmann distribution is usually derived from principle of maximum entropy, but there are other derivations.

The generalized Boltzmann distribution has the following properties:
 * It is the only distribution for which the entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics.
 * It is the only distribution that is mathematically consistent with the fundamental thermodynamic relation where state functions are described by ensemble average.

Edit 2 (done)
文章 statistical mechanics, 把下面的这句话

Other fundamental postulates for statistical mechanics have also been proposed.

扩充成

Other fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates : 1. The probability density function is proportional to some function of the ensemble parameters and random variables.

2. Thermodynamic state functions are described by ensemble averages of random variables.

3. The entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics.

where the third postulate can be replaced by the following : 1. At infinite temperature, all the microstates have the same probability.

Edit 3 (done)
文章 Fundamental thermodynamic relation 新增以下章节

Relationship to statistical mechanics
The fundamental thermodynamic relation and statistical mechanical principles can be derived from one another.

Derivation of the fundamental thermodynamic relation from statistical mechanical principles
保持原有的内容不变

Derivation of statistical mechanical principles from the fundamental thermodynamic relation
It has been shown that the fundamental thermodynamic relation together with the following three postulates 1. The probability density function is proportional to some function of the ensemble parameters and random variables.

2. Thermodynamic state functions are described by ensemble averages of random variables.

3. The entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics. is sufficient to build the theory of statistical mechanics without the equal a priori probability postulate.

For example, in order to derive the Boltzmann distribution, we assume the probability density of microstate $i$ satisfies $\Pr(i)\propto f(E_i,T)$. The normalization factor (partition function) is therefore

$$ Z = \sum_i f(E_i, T). $$

The entropy is therefore given by

$$ S = k_B \sum_i \frac{f(E_i, T)}{Z} \log\left(\frac{f(E_i, T)}{Z}\right). $$

If we change the temperature $T$ by $dT$ while keeping the volume of the system constant, the change of entropy satisfies

$$ dS=\left(\frac{\partial S}{\partial T}\right)_V dT $$

where

$$ \left(\frac{\partial S}{\partial T}\right)_V = -k_B \sum_i\frac{Z\cdot\frac{\partial f(E_i, T)}{\partial T}\cdot\log f(E_i, T)-\frac{\partial Z}{\partial T}\cdot f(E_i, T)\cdot\log f(E_i, T)}{Z^2} = -k_B \sum_i \frac{\partial}{\partial T} \left(\frac{f(E_i, T)}{Z}\right)\cdot\log f(E_i, T) $$

Considering that

$$ \left\langle E\right\rangle = \sum_i \frac{f(E_i, T)}{Z}\cdot E_i $$

we have

$$ d\left\langle E\right\rangle = \sum_i \frac{\partial}{\partial T} \left(\frac{f(E_i, T)}{Z}\right)\cdot E_i \cdot dT $$

From the fundamental thermodynamic relation, we have

$$ -\frac{dS}{k_B}+\frac{d\left\langle E\right\rangle}{k_B T} + \frac{P}{k_B T}dV = 0 $$

Since we kept $V$ constant when perturbing $T$, we have $dV=0$. Combining the equations above, we have

$$ \sum_i \frac{\partial}{\partial T} \left(\frac{f(E_i, T)}{Z}\right)\cdot \left[\log f(E_i, T)+\frac{E_i}{k_B T}\right]\cdot dT = 0 $$

Physics laws should be universal, i.e., the above equation must hold for arbitrary systems, and the only way for this to happen is

$$ \log f(E_i, T)+\frac{E_i}{k_B T} = 0 $$

That is

$$ f(E_i, T)=\exp\left(-\frac{E_i}{k_B T}\right). $$

It has been shown that the third postulate in the above formalism can be replaced by the following : 1. At infinite temperature, all the microstates have the same probability. However, the mathematical derivation will be much more complicated.

Edit 4 (done)
文章Entropy_(statistical_thermodynamics)这句话

It has been shown that the Gibbs Entropy is equal to the classical "heat engine" entropy characterized by $$dS = \frac{\delta Q}{T} \!$$

扩充成

It has been shown that the Gibbs Entropy is equal to the classical "heat engine" entropy characterized by $$dS = \frac{\delta Q}{T} \!$$, and the generalized Boltzmann distribution is a sufficient and necessary condition for this equivalence. Furthermore, the Gibbs Entropy is the only entropy that is equivalent to the classical "heat engine" entropy under the following postulates : 1. The probability density function is proportional to some function of the ensemble parameters and random variables.

2. Thermodynamic state functions are described by ensemble averages of random variables.

3. At infinite temperature, all the microstates have the same probability.

Edit 5 (done)
文章Canonical ensemble这句话

The canonical ensemble is uniquely determined for a given physical system at a given temperature, and does not depend on arbitrary choices such as choice of coordinate system (classical mechanics), or basis (quantum mechanics), or of the zero of energy.

扩充成

The canonical ensemble is uniquely determined for a given physical system at a given temperature, and does not depend on arbitrary choices such as choice of coordinate system (classical mechanics), or basis (quantum mechanics), or of the zero of energy. The canonical ensemble is the only ensemble with constant $N$, $V$, and $T$ that reproduces the fundamental thermodynamic relation.