# 麦克斯韦-玻尔兹曼分布

（重定向自麦克斯韦速度分布律

${\displaystyle Z={\sqrt {X_{1}^{2}+X_{2}^{2}+X_{3}^{2}}}}$

## 推导

${\displaystyle {\frac {N_{i}}{N}}={\frac {g_{i}\exp \left(-E_{i}/kT\right)}{\sum _{j}^{}g_{j}\,{\exp \left(-E_{j}/kT\right)}}}\qquad \qquad (1)}$

### 动量向量的分布

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

${\displaystyle {\frac {N_{i}}{N}}={\frac {1}{Z}}\exp \left[-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right]\qquad \qquad (3)}$

${\displaystyle f_{\mathbf {p} }(p_{x},p_{y},p_{z})={\frac {c}{Z}}\exp \left[-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right].\qquad \qquad (4)}$

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

${\displaystyle f_{\mathbf {p} }(p_{x},p_{y},p_{z})=\left({\frac {1}{2\pi mkT}}\right)^{3/2}\exp \left[-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right].\qquad \qquad (6)}$

### 能量的分布

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

${\displaystyle f_{E}(E)\,dE=\chi ^{2}(x;3)\,dx}$

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

### 速度向量的分布

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

${\displaystyle f_{\mathbf {v} }(v_{x},v_{y},v_{z})=\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left[-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right],\qquad \qquad }$

${\displaystyle f_{\mathbf {v} }\left(v_{x},v_{y},v_{z}\right)\,dv_{x}\,dv_{y}\,dv_{z}.}$

${\displaystyle f_{v}\left(v_{x},v_{y},v_{z}\right)=f_{v}(v_{x})f_{v}(v_{y})f_{v}(v_{z})}$

${\displaystyle f_{v}(v_{i})={\sqrt {\frac {m}{2\pi kT}}}\exp \left[{\frac {-mv_{i}^{2}}{2kT}}\right].\qquad \qquad }$

### 速率的分布

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

${\displaystyle v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}$

### 典型的速率

#### 最概然速率(最大可能速率)

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

${\displaystyle v_{p}={\sqrt {\frac {2kT}{m}}}={\sqrt {\frac {2RT}{M}}}}$

#### 平均速率

${\displaystyle \langle v\rangle =\int _{0}^{\infty }v\,f(v)\,dv={\sqrt {\frac {8kT}{\pi m}}}={\sqrt {\frac {8RT}{\pi M}}}}$

#### 均方根速率

${\displaystyle v_{\mathrm {rms} }=\left(\int _{0}^{\infty }v^{2}\,f(v)\,dv\right)^{1/2}={\sqrt {\frac {3kT}{m}}}={\sqrt {\frac {3RT}{M}}}}$

#### 三种典型速率的关系

${\displaystyle v_{p}:v_{average}:v_{\mathrm {rms} }\approx 1:1.128:1.224}$ [1]

### 相对论气体的速率分布

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

## 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