中心思想
一系统与热源达到平衡，热源温度为 $T$， 化学势为 $\mu $， 那么系统在任意一个包含 $N$ 个粒子，能量 $E$ 的非简并状态的概率为
\begin{equation}
\frac{ \mathrm{e} ^{(\mu N - E)/(kT)}}{\Xi }
\end{equation}
其中，巨配分函数使所有状态的概率之和为一，起到归一化的作用．
\begin{equation}
\Xi = \sum_{N} \sum_i \mathrm{e} ^{(\mu N - E_i)/(kT)}
\end{equation}
推导
“能级导向”
\begin{equation} \begin{aligned}
\Xi & = \sum_{N=1}^\infty \sum_{\{n_i\}}^* \exp\left(N\mu - \sum_{i=1}^\infty n_i \varepsilon_i\right) \beta
= \sum_{N=1}^\infty \sum_{\{n_i\}}^* z^N \prod_{i=0}^\infty \left( \mathrm{e} ^{-\varepsilon_i\beta} \right) ^{n_i}\\
&= \sum_{N=1}^\infty \sum_{\{n_i\}}^* \prod_{i=0}^\infty \left(z \mathrm{e} ^{-\varepsilon_i \beta} \right) ^{n_i}
= \sum_{n_1}^* \sum_{n_2}^* \dots \prod_{i=0}^\infty \left(z \mathrm{e} ^{-\varepsilon_i \beta} \right) ^{n_i}\\
&= \sum_{n_1}^* \left(z \mathrm{e} ^{-\varepsilon_i\beta} \right) ^{n_1} \sum_{n_2}^* \left(z \mathrm{e} ^{ -\varepsilon_i\beta} \right) ^{n_2}\dots
= \prod_i^\infty \sum_{n_i}^* \left(z \mathrm{e} ^{-\varepsilon_i \beta } \right) ^{n_i}
\end{aligned} \end{equation}
系统的热力学性质
由最大概率项假设，
\begin{equation}
1 = \frac{\Omega \mathrm{e} ^{(\mu N - E)/(kT)}}{\Xi}
= \frac{ \mathrm{e} ^{S/k} \mathrm{e} ^{(\mu N - E)/(kT)}}{\Xi}
\end{equation}
\begin{equation}
\mathrm{e} ^{S/k} \mathrm{e} ^{(\mu N - E)/(kT)} = \Xi
\end{equation}
\begin{equation}
E - ST - \mu N = - kT\ln \Xi
\end{equation}
令 $\Phi = - kT\ln \Xi $ 叫做
巨势
\begin{equation}
\Phi = E - ST - \mu N
\end{equation}
\begin{equation}
\Phi = E - ST - \mu N = F - G = E - ST - (E - ST + PV) = - PV
\end{equation}
考虑到 $ \,\mathrm{d}{E} = T \,\mathrm{d}{S} - P \,\mathrm{d}{V} + \mu \,\mathrm{d}{N} $
\begin{equation}
\,\mathrm{d}{\Phi} = -P \,\mathrm{d}{V} - S \,\mathrm{d}{T} - N \,\mathrm{d}{\mu}
\end{equation}
所以
\begin{equation}
S = - \left( \frac{\partial \Phi}{\partial T} \right) _{V,\mu } \qquad
N = - \left( \frac{\partial \Phi}{\partial \mu} \right) _{V,T} \qquad
P = - \left( \frac{\partial \Phi}{\partial V} \right) _{T,\mu}
\end{equation}
另外有一个求能级分布的公式
\begin{equation}
\left\langle n_i \right\rangle = \frac{1}{\Xi} \sum_{N=1}^\infty \sum_{\{n_i\}}^* n_i \exp\left(N\mu - \sum_{i=1}^\infty n_i \varepsilon_i\right) \beta = -\frac{1}{\beta \Xi } \frac{\partial \Xi}{\partial \varepsilon_i} = - kT \frac{\partial}{\partial{\varepsilon_i}} \ln \Xi = \frac{\partial \Phi}{\partial \varepsilon_i}
\end{equation}