这是一份uwa西澳大学PHYS1030的成功案例

$$
p({\sigma})=\frac{e^{-\beta H({\sigma})}}{Z},
$$
where ${\sigma}$ labels a microstate (i.e. a configuration), $H({\sigma})$ is the system Hamiltonian, $\beta$ is the inverse temperature and $Z$ is the partition function. Suppose, now, that we sum over all microstates identifiable as belonging to a certain phase $\gamma$, then
$$
\begin{aligned}
p_{\gamma} &=\sum_{{\sigma} \in \gamma} p({\sigma}) \
&=\frac{1}{Z} \sum_{{\sigma} \in \gamma} e^{-\beta H({\sigma})} \
& \equiv \frac{Z_{\gamma}}{Z},
\end{aligned}
$$

PHYS1030 COURSE NOTES ：

where the last step serves to define the configurational weight $Z_{\gamma}$ of the phase $\gamma$. The relative probabilities of two phases $A$ and $B$ is then
$$
\begin{aligned}
\mathcal{R}{A B} & \equiv \frac{p{A}}{p_{B}}=\frac{Z_{A}}{Z_{B}} \
& \equiv \frac{e^{-\beta F_{A}}}{e^{-\beta F_{B}}}
\end{aligned}
$$
where $F_{\gamma}$ denotes the free energy of phase $\gamma$. It follows that the free energy difference between two phases $A$ and $B$ is simply proportional to the logarithm of the ratio of their a-priori probabilities:
$$
F_{A}-F_{B}=-\frac{1}{\beta} \ln \mathcal{R}{A B} $$ Clearly this equation implies that precisely at coexistence $\left(F{A}=F_{B}\right)$ the system will be found with equal probability in each of the two phases.