统计物理的基本概念
1. 研究对象和目标
微观世界 ⇒ 宏观世界
2. 宏观现象 = Avg[微观现象]
3. 经典系统的微观状态
- 离散: 自旋体系
- 连续
4. 量子系统的微观状态
- 离散
- 连续 \(H(\{p\}, \{q\})\rightarrow |p\alpha\rangle |q\beta\rangle \)
4.1. 能量的量子涨落
4.2. 粒子全同性
5. 等机率原理
6. 统计物理与宏观热力学之间的桥梁
7. 统计物理描述的平衡态和熵
8. 微正则系综
9. 正则系综
\[ P_{\gamma} = \frac{e^{-\beta E_{\gamma}}}{Z};\:Z= \sum_{\gamma} e^{-\beta E_{\gamma}}.\]
10. 玻尔兹曼分布
单粒子能级: \(\epsilon_L\), \(N\) 个粒子;
微观状态: \( (j_i, j_2, \ldots, j_N) = \vec{\jmath} \) 能量有 \[ E_{\vec{\jmath}} = \sum_{\alpha = 1}^N \epsilon_{j_{\alpha} = E_{\star}}. \] 配分函数有 \[ \begin{aligned} Z &= \sum_{\vec{\jmath}} e^{-\beta E_{\vec{\jmath}}}\\ &= \sum_{j_1, j_2, \ldots, j_L} e^{-\beta (\epsilon_{j1} + \ldots + \epsilon_{jN}) }\\ &= \prod_{\alpha = 1}^N \left( \sum_{j\alpha} e^{-\beta\epsilon_{j\alpha}} \right) = Z^N.\\ \end{aligned} \] \[ \begin{aligned} \bar{n_i} &= \sum_{\vec{\jmath}} P_{\vec{\jmath}} \left[\sum_{\alpha = 1}^N \delta_{j_{\alpha}i}\right]\\ &= \sum_{\vec{\jmath}} e^{-\beta \sum_{\alpha = 1}^N \epsilon_{j\alpha}} \left[\sum_{\alpha = 1}^N \delta_{j_{\alpha}i}\right] / Z \\ &= \sum_{\vec{\jmath}} \frac{1}{-\beta} \frac{\partial}{\partial \epsilon_i} e^{-\beta \sum_{\alpha = 1}^N \epsilon_{j\alpha}} \left[\sum_{\alpha = 1}^N \delta_{j_{\alpha}i}\right] / Z \\ &= - \frac{N}{\beta} \frac{\partial}{\partial\epsilon_i} \log Z; \frac{\bar{n_i}}{N} &= - \frac{e^{-\beta\epsilon_i}}{\sum_j e^{ -\beta \epsilon_j}} \propto e^{-\beta\epsilon_i}. \end{aligned} \] 上式即玻尔兹曼分布.
11. 正则系综特性函数
\[ F(T, V, N) = E(T, V, N) - TS(T, V, N) \]
12. 巨正则系综
\[ \mathcal{Z} = \sum_r e^{-\beta E_r -\alpha N_r},\: \alpha = -\beta\mu. \]
12.1. 巨正则系统的熵
12.2. 巨正则系综的粒子数涨落
\[ \bar{N^2} - \bar{N}^2 = \frac{\partial}{\partial \alpha} \frac{\partial}{\partial \beta}[ \log Z ] \]
13. Bose-Einstein 分布
- 单粒子能级: \( \epsilon_1\le\epsilon_2\cdots\ge\epsilon_L \)
- 微观状态: \( (n_1, \ldots, n_L),\: \sum_{j=1}^L n_j = N. \)
- 能量: \(E_{n_1, \ldots, n_L} = \sum_{j=1}^L n_j\epsilon_j.\)
- 第 \(j\) 个能级的平均粒子数: \[ \bar{n_j} = 1/\left( e^{\beta(\epsilon - \mu) - 1} \right) \]
14. Fermi-Dirac 分布
15. 相空间
\[ Z_r = \sum_r e^{-\beta E_r} \] 其中微观状态 \(\{r\}\) 构成相空间.
16. 固体热容
\[ C_V = \frac{\partial \bar{E}}{\partial T} = 3Nk_B. \]
16.1. Einstein 固体热容
17. 光子气体
\[ Z = \prod_{\vec{n}} \left[ 1 / (1 - \exp(-\beta \frac{hc}{L}\sqrt{\sum_{i=1}^3 n_i^2}) ) \right] \] \[ \begin{aligned} F(T, V) &= -k_B T\log Z\\ &= 2k_B T \sum_{\vec{n}} \log[1 - e^{-\beta \frac{hc}{L} \sqrt{\vec{n}^2}}]\times\int \mathrm{d}\mu\delta(\mu -\mu(\pi))\\ &=2k_B T\int \mathrm{d}\nu \log [1- e^{-\beta h\mu}]\sum_{\vec{n}} \delta(\nu - \nu(\pi))\\ &=2k_B T\int \mathrm{d}\mu D(\nu)\log [1 - e^{-\beta h\mu}] \end{aligned} \] 态密度 \(D(\mu) = \sum_{\vec{n}} \delta(\mu -\mu(\pi))\) 有 \[ D(\nu)\mathrm{d}\nu = 4\pi \vec{n}^2 d|\vec{n}|, (n \rightarrow \infty). \] \[ \Rightarrow F(T, V) = 2k_B T V \int \mathrm{d}\mu \frac{4\pi}{c^3}\mu^3 \log[1- e^{-\beta h\mu}]\]
计算单位频率, 单位体积的能量 \(u(\mu)\) \[ u(\mu)L^3 \mathrm{d}\mu = \frac{2\times D(\mu)\times(h\mu)}{e^{\beta h\mu} - 1} \] \[ u(\mu) = \frac{8\pi}{c^3}\times \frac{h\mu^3}{e^{\beta h\mu} -1} \]