| 經典微正則系綜 | 經典正則系綜 | 經典巨正則系綜 | ||
| 研究對象 | 孤立系統 | 封閉系統 | 開放系統 | |
| 配分函數 | $Z(T,V,N)=\int_{\Gamma} e^{[-\beta \mathcal{H}(q,p)]}\frac{dqdp}{N!h^{3N}}$ | $\widetilde{Z}(\beta,V,\mu)=\sum_{N>0} \int_{\Gamma} e^{[-\beta (\mathcal{H}-\mu N)]}\frac{dqdp}{N!h^{3N}}$ | ||
| 幾率密度 | $\rho(E,V,N)=C,E \leq \mathcal{H}\leq E+\Delta E$ | $\rho(T,V,N)=\frac{1}{Z}e^{[-\beta \mathcal{H}(q,p)]}$ | $\rho(\beta,V,\mu)=\frac{1}{\widetilde{Z}} e^{[-\beta (\mathcal{H}-\mu N)]}$ | |
| 特征函數 | $S=k_blnW(E,N,V)$ | $F=-\frac{1}{\beta}lnZ(\beta,V,N)$ | $\Omega=-\frac{1}{\beta}ln\widetilde{Z} (\beta,V,\mu)$ | |
| 熱力學量 -- 宏觀微觀 |
熵$S$ | $S=k_blnW$ | $S=-k_B \bar{ln\rho}=k_B\beta^2\frac{\partial F}{\partial \beta}$ | $S=-k_B \bar{ln\rho}=k_B\beta^2\frac{\partial \Omega}{\partial \beta}$ |
| 能量$E$ | $U=E$ | $E=\bar{\mathcal{H}}=-\frac{\partial lnZ}{\partial \beta}$ | $E=\bar{\mathcal{H}}=-\frac{\partial ln\widetilde{Z}}{\partial \beta}+\frac{\mu}{\beta}\frac{\partial ln\widetilde{Z}}{\partial \mu}$ | |
| 壓強$P$ | $P=-(\frac{\partial E}{\partial V})_{S,N}$ | $P=-\bar{(\frac{\partial \mathcal{H}}{\partial V})}=-\frac{\partial lnZ}{\partial \beta}$ | $P=-\bar{(\frac{\partial \mathcal{H}}{\partial V})}=-\frac{\partial \Omega}{\partial V}$ | |
| 化學勢$\mu$ | $\mu=(\frac{\partial E}{\partial N})_{S,V}$ | $\mu=(\frac{\partial F}{\partial N}_{\beta,V})$ | ||
| 粒子數$N$ | $\bar{N}=-\frac{\partial \Omega}{\partial \mu}$ |
| 幾率密度 | 配分函數 | ||
| 經典 | 微正則 | $\rho(E,V,N)=C,E \leq \mathcal{H}\leq E+\Delta E$ | $C=\frac{1}{W}$ |
| 正則 | $\rho(T,V,N)=\frac{1}{Z}e^{[-\beta \mathcal{H}(q,p)]}$ | $Z(T,V,N)=\int_{\Gamma} e^{[-\beta \mathcal{H}(q,p)]}\frac{dqdp}{N!h^{3N}}$ | |
| 巨正則 | $\rho(E_k)=\frac{1}{\widetilde{Z}} e^{[-\beta (\mathcal{H}-\mu N)]}$ | $\widetilde{Z}(\beta,V,\mu)=\sum_{N>0} \int_{\Gamma} e^{[-\beta (\mathcal{H}-\mu N)]}\frac{dqdp}{N!h^{3N}}$ | |
| 量子 | 微正則 | $\rho(E_k)=C,E \leq E_k \leq E+\Delta E$ | $C=\frac{1}{W}$ |
| 正則 | $\rho(T,V,N)=\frac{1}{Z} e^{-\beta E_k}$ | $Z(T,V,N)=\sum_k e^{-\beta E_k}$ | |
| 巨正則 | $\rho(T,V,\mu)=\frac{1}{\widetilde{Z}} e^{-\beta (\widetilde{\mathcal{H}}-\mu \widetilde{N})}$ | $\widetilde{Z}(T,V,\mu)=\sum_{N} e^{\beta \mu N}Z(V,N,T)$ |
常用計算：
\[lnN!=NlnN-N\]