# 麦克斯韦-玻尔兹曼分布

$Z = \sqrt{X_1^2+X_2^2+X_3^2}$

## 推导

$\frac{N_i}{N} = \frac{g_i \exp\left(-E_i/kT \right) } { \sum_{j}^{} g_j \,{\exp\left(-E_j/kT\right)} } \qquad\qquad (1)$

### 动量向量的分布

$E=\frac{p^2}{2m} \qquad\qquad (2)$

$\frac{N_i}{N} = \frac{1}{Z} \exp \left[ -\frac{p_x^2 + p_y^2 + p_z^2}{2mkT} \right] \qquad\qquad (3)$

$f_\mathbf{p} (p_x, p_y, p_z) = \frac{c}{Z} \exp \left[ -\frac{p_x^2 + p_y^2 + p_z^2}{2mkT} \right]. \qquad\qquad (4)$

$c = \frac{Z}{(2 \pi mkT)^{3/2}}. \qquad\qquad (5)$

$f_\mathbf{p} (p_x, p_y, p_z) = \left( \frac{1}{2 \pi mkT} \right)^{3/2} \exp \left[ -\frac{p_x^2 + p_y^2 + p_z^2}{2mkT} \right]. \qquad\qquad (6)$

### 能量的分布

$f_E\,dE=f_p\left(\frac{dp}{dE}\right)\,dE =2\sqrt{\frac{E}{\pi(kT)^3}}~\exp\left[\frac{-E}{kT}\right]\,dE. \qquad \qquad(7)$

$f_E(E)\,dE=\chi^2(x;3)\,dx$

$x=\frac{2E}{kT}.\,$

### 速度向量的分布

$f_\mathbf{v} d^3v = f_\mathbf{p} \left(\frac{dp}{dv}\right)^3 d^3v$

$f_\mathbf{v} (v_x, v_y, v_z) = \left(\frac{m}{2 \pi kT} \right)^{3/2} \exp \left[- \frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT} \right], \qquad\qquad$

$f_\mathbf{v} \left(v_x, v_y, v_z\right)\, dv_x\, dv_y\, dv_z.$

$f_v \left(v_x, v_y, v_z\right) = f_v (v_x)f_v (v_y)f_v (v_z)$

$f_v (v_i) = \sqrt{\frac{m}{2 \pi kT}} \exp \left[ \frac{-mv_i^2}{2kT} \right]. \qquad\qquad$

### 速率的分布

$f(v) = \sqrt{\frac{2}{\pi}\left(\frac{m}{kT}\right)^3}\, v^2 \exp \left(\frac{-mv^2}{2kT}\right)$

$v = \sqrt{v_x^2 + v_y^2 + v_z^2}$

### 典型的速率

#### 最概然速率

$\frac{df(v)}{dv} = 0$

$v_p = \sqrt { \frac{2kT}{m} } = \sqrt { \frac{2RT}{M} }$

#### 平均速率

$\langle v \rangle = \int_0^{\infin} v \, f(v) \, dv= \sqrt { \frac{8kT}{\pi m}}= \sqrt { \frac{8RT}{\pi M}}$

#### 均方根速率

$v_\mathrm{rms} = \left(\int_0^{\infin} v^2 \, f(v) \, dv \right)^{1/2}= \sqrt { \frac{3kT}{m}}= \sqrt { \frac{3RT}{M} }$

#### 三种典型速率的关系

$v_p : v_{average} : v_\mathrm{rms} \approx 1:1.128:1.224.$[1]

### 相对论气体的速率分布

$f(\gamma) = \frac {\gamma^2 \beta }{\theta K_2(1/\theta)} \mathrm{exp} \left( - \frac {\gamma}{\theta} \right) \qquad (11)$

## 参考文献

1. ^ 秦允豪. 热学. 高等教育出版社. : 65页. ISBN 978-7-04-013790-3.
2. ^ Synge, J.L., The relativistic gas, Noord-Holland, 1957