# 朗道分布

Landau distribution with mode at 2

## 定义

${\displaystyle p(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\!e^{s\log s+xs}\,ds,}$

${\displaystyle p(x)={\frac {1}{\pi }}\int _{0}^{\infty }\!e^{-t\log t-xt}\sin(\pi t)\,dt,}$

${\displaystyle p(x)={\frac {1}{\sqrt {2\pi }}}\exp \left\{-{\frac {1}{2}}(x+e^{-x})\right\},}$

${\displaystyle \varphi (t;\mu ,c)=\exp \!{\Big [}\;it\mu -|c\,t|(1+{\tfrac {2i}{\pi }}\log(|t|)){\Big ]},}$

## 相关性质

• ${\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,}$${\displaystyle X+m\sim {\textrm {Landau}}(\mu +m,c)\,}$
• 朗道分布是一种稳定分布

## 参考文献

1. ^ Landau, L. On the energy loss of fast particles by ionization. J. Phys. (USSR). 1944, 8: 201.
2. ^ Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).
3. ^ Interaction of Charged Particles. [14 April 2014]. （原始内容存档于2012年6月30日）.
4. ^ Gentle, James E. Random Number Generation and Monte Carlo Methods. Statistics and Computing 2nd. New York, NY: Springer. 2003: 196. ISBN 978-0-387-00178-4. doi:10.1007/b97336.
5. ^ Meroli, S. Energy loss measurement for charged particles in very thin silicon layers. JINST. 2011, 6: 6013.