第七章 玻尔兹曼统计¶

\[ Z_{1}=\sum\omega_{l}e^{-\beta\varepsilon_{l}} \]

\[ N=e^{-\alpha}z_{1} \]

\[ \begin{aligned}&U=e^{-\alpha}\sum\varepsilon_l\omega_le^{-\beta\varepsilon_l}=e^{-\alpha}\sum-\frac{\partial}{\partial\beta}(\omega_le^{-\beta\varepsilon_l})\\&=e^{-\alpha}(-\frac{\partial}{\partial\beta})\sum(\omega_le^{-\beta\varepsilon_l})=\frac{N}{Z_1}(-\frac{\partial}{\partial\beta})Z_1=-N\frac{\partial}{\partial\beta}\ln Z_1\end{aligned} \]

\[ \begin{aligned}&\mathrm{d}S=\frac1T\mathsf{d}Q=k\beta\mathsf{d}Q=kN\mathsf{d}(\ln Z_1-\beta\frac{\partial\ln Z_1}{\partial\beta})\\&\Rightarrow S=Nk(\ln Z_1-\beta\frac{\partial\ln Z_1}{\partial\beta}).\end{aligned} \]

\[ S=k[N\ln N+\sum\ln\frac{\omega_l}{a_l}a_l]=k\ln\Omega_\mathrm{M}. \]

\[ \begin{array}{l}F=U-TS=-N\frac{\partial\ln Z_1}{\partial\beta}-TNk(\ln Z_1-\beta\frac{\partial\ln Z_1}{\partial\beta})\\=-NkT\ln Z_1.\end{array} \]

\[ Z_1 = (\frac{2\pi m}{h^2\beta})^{\frac32}V \]

\[ p=\frac{N}{\beta}\frac{\partial\operatorname{ln}Z_{1}}{\partial y}=\frac{N}{\beta}\frac{\partial\operatorname{ln}Z_{1}}{\partial V}=\frac{N}{\beta}\frac{\partial\operatorname{ln}[(\frac{2\pi m}{h^{2}\beta})^{\frac{3}{2}}V]}{\partial V}=\frac{N}{\beta}\frac{1}{V}=\frac{NkT}{V} \]

\[ \text{Classical limit condition: }\frac{\omega_l}{a_l}\gg1\text{,or }e^\alpha\gg1 \]

\[ e^\alpha=\frac{Z_1}N=\frac VN(\frac{2\pi m}{h^2\beta})^{\frac32}=\frac VN(\frac{2\pi mkT}{h^2})^{\frac32}\gg1. \]