Landau distribution

Landau distribution with mode at 2 and sigma of 1

In probability theory, the Landau distribution is a probability distribution named after Lev Landau.

Definition

The distribution is defined by the complex integral

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

where $c$ is any positive real number, and log refers to the logarithm base e, the natural logarithm.

For numerical purposes it is more convenient to use the following equivalent form of the integral,

$p(x) = \frac{1}{\pi} \int_0^\infty\! e^{-t \log t - x t} \sin(\pi t)\, dt.$

where log refers to the logarithm base e, the natural logarithm.

The Landau distribution is used in physics to describe the fluctuations in the energy loss of a charged particle passing through a thin layer of matter.

This distribution is a special case of the stable distribution with parameters α = 1, and β = 1.

The characteristic function may be expressed as:

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

where μ and c are real, which yields a Landau distribution shifted by μ and scaled by c.

Related distributions

• If $X \sim \textrm{Landau}(\mu,c)\,$ then $X + m \sim \textrm{Landau}(\mu + m ,c) \,$
• Landau distribution is a Stable distribution

References

• GSL manual, used under GFDL.

External links

• The Energy Loss Distribution of charged particles passing through matter (Use of the Landau Distribution in the high energy physics experiments")

Computing the Landau distribution

MATLAB script to compute the Landau distribution of a charged particle passing through a thin layer of matter.