平衡态速度分布律

\[f(v) = \frac{\mathrm{d}N}{N \mathrm{d}v}\] 三维空间中 \[f(\vec{v}) = f(v_x)\mathrm{d}v_x f(v_y)\mathrm{d}v_y f(v_z)\mathrm{d}v_z\] \[\Rightarrow f(v_x, v_y, v_z) = \left( \frac{m}{2\pi kT} \right)^{3/2} \mathrm{e}^{- \frac{m v^2}{2 kT}}\] \[f(v_i) = \sqrt{\frac{m}{2\pi k T}} \mathrm{e}^{- \frac{m v_i^2}{2kT}}\] \[\Rightarrow F(v)\mathrm{d}v = 4\pi v^2 f(\vec{v}) = 4\pi v^2 \left( \frac{m}{2\pi kT} \right)^{3/2} e^{- \frac{mv^2}{2kT}}\]

三种特征速率:

1. 最概然速率 \(v_p\): \[v_p = \sqrt{\frac{2k_B T}{m}} = \sqrt{\frac{2RT}{M}}\]
2. 平均速率 \(\left\langle v \right\rangle\): \[\left\langle v \right\rangle = \int_0^{\infty} v F(v)\mathrm{d}v = \sqrt{\frac{8kT}{\pi m}}\]
3. 方均根速率 \(v_{\text{rms}}\): \[v_{\text{rms}} = \left( \int_0^{\infty} v^2 F(v)\mathrm{d}v \right)^{1/2} = \sqrt{\frac{3kT}{m}}\]

不考虑分子动能: \[f(r) = C \mathrm{e}^{- \frac{E_p}{kT}}\] E.g. 重力势能下分子分布: \[n = n(0)\mathrm{e}^{- \frac{Mgh}{kT}}\]

\[f(r, v) = C \mathrm{e}^{- \frac{E}{k T}}, \ \mathrm{d}N = N f(r, v)\mathrm{d}r \mathrm{d}v\] \[\Rightarrow n_1 = n_2 \exp \left( - \frac{\epsilon_1 - \epsilon_2}{kT} \right)\] \[\Rightarrow T = \frac{\epsilon_1 - \epsilon_2}{k\ln (n_2/n_1)}\]

每一个分子的每一个自由度平均动能都是 \(\frac{kT}{2}\). \[\Rightarrow\bar{\epsilon} = \frac{1}{2}i k T, \ i = t + r + 2v.\] \(2v\) 的原因是振动自由度还有势能, 其大小与振动平均动能相同.