Land distribution

The Landau distribution is a continuous probability distribution named after Lev Landau . Because of their long tail , the moments of distribution (such as the expected value and the variance ) are not defined. The land distribution is a special case of the Lévy stable distributions .

definition

Density p (x) of a Landau distribution

The probability density of the standard Landau distribution is given by the complex integral

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

where is the natural logarithm and is any positive real number that does not affect the result. For numerical purposes, the following equivalent form is more suitable ${\displaystyle \log }$ ${\displaystyle c}$

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

The distribution can be approximated by the following closed expression

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

The Landau distribution is a special case of the Lévy-stable distributions with the parameters and . ${\displaystyle \alpha =1}$ ${\displaystyle \beta =1}$

The characteristic function is

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

with real numbers , and the sign function . The function generates a land distribution that is shifted by and scaled by. ${\displaystyle \mu }$ ${\displaystyle c}$ ${\displaystyle \operatorname {sgn} }$ ${\displaystyle \mu }$ ${\displaystyle c}$

properties

• If so , then applies . ${\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,}$ ${\displaystyle X+m\sim {\textrm {Landau}}(\mu +m,c)\,}$

application

The Landau distribution describes the fluctuations in the escape of energy from a thin layer due to impact ionization .

Web links

• Land distribution in the Data Analysis BriefBook of the COZY-11 experiment at Forschungszentrum Jülich, Rudolf K. Bock, April 7, 1998

Individual evidence

1. ↑ L. Landau: On the energy loss of fast particles by ionization . In: J. Phys. (USSR) . 8, 1944, p. 201.
2. ^ SE Behrens, AC Melissinos: Univ. of Rochester Preprint UR-776 (1981) .
3. ^ Interaction of Charged Particles . Archived from the original on June 30, 2012. Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Retrieved April 14, 2014. 
4. ↑ James E. Gentle: Random Number Generation and Monte Carlo Methods  (= Statistics and Computing), 2nd. Edition, Springer, New York, NY 2003, ISBN 978-0-387-00178-4 , p. 196, doi : 10.1007 / b97336 .
5. ↑ S. Meroli: Energy loss measurement for charged particles in very thin silicon layers . In: JINST . 6, 2011, p. 6013. doi : 10.1088 / 1748-0221 / 6/06 / P06013 .