Landau distribution


In probability theory, the Landau distribution is a probability distribution named after Lev Landau.
Because of the distribution's "fat" tail, the moments of the distribution, such as mean or variance, are undefined. The distribution is a particular case of stable distribution.

Definition

The probability density function, as written originally by Landau, is defined by the complex integral:
where is an arbitrary positive real number, meaning that the integration path can be any parallel to the imaginary axis, intersecting the real positive semi-axis, and refers to the natural logarithm.
In other words it is the Laplace transform of the function.
The following real integral is equivalent to the above:
The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters and, with characteristic function:
where and, which yields a density function:
Taking and we get the original form of above.

Properties

[Image:Landau_pdf.svg|300px|thumb|right|The approximation function for ]
  • Translation: If then
  • Scaling: If then
  • Sum: If and then
These properties can all be derived from the characteristic function.
Together they imply that the Landau distributions are closed under affine transformations.

Approximations

In the "standard" case and, the pdf can be approximated using Lindhard theory which says:
where is Euler's constant.
A similar approximation of for and is:

Applications

In nuclear and particle physics, the Landau distribution appears as a probability that a fast particle with a given initial energy will lose a given energy after passing the layer of matter with given thickness.

Related distributions