Landau distribution

Short description: Probability distribution

Landau distribution

Probability density function

[[File:350px
μ=0,c=π/2|frameless]]

Parameters|

c∈(0,∞) — scale parameter

μ∈(−∞,∞) — location parameter
Support|  ℝ
PDF|  1πc∫0∞e−tcos⁡((x−μ)tc+2tπln⁡tc)dt
Mean|  Undefined
Variance|  Undefined
MGF|  Undefined
CF|  exp⁡(itμ−2ictπlog⁡|t|−c|t|)

In probability theory, the Landau distribution[1] 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.

## Contents

* 1 Definition
* 2 Properties
* 2.1 Approximations
* 3 Applications
* 4 Related distributions
* 5 References

## Definition

The probability density function, as written originally by Landau, is defined by the complex integral:

p(x)=12πi∫a−i∞a+i∞esln⁡s+xsds,

where a 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 ln refers to the natural logarithm. In other words it is the Laplace transform of the function ss.

The following real integral is equivalent to the above:

p(x)=1π∫0∞e−tln⁡t−xtsin⁡(πt)dt.

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters α=1 and β=1,[2] with characteristic function:[3]

φ(t;μ,c)=exp⁡(itμ−2ictπln⁡|t|−c|t|)

where c∈(0,∞) and μ∈(−∞,∞), which yields a density function:

p(x;μ,c)=1πc∫0∞e−tcos⁡((x−μ)tc+2tπln⁡tc)dt,

Taking μ=0 and c=π2 we get the original form of p(x) above.

## Properties

The approximation function for μ=0,c=1

* Translation: If X∼Landau(μ,c) then X+m∼Landau(μ+m,c).
* Scaling: If X∼Landau(μ,c) then aX∼Landau(aμ−2πacln⁡a,ac).
* Sum: If X∼Landau(μ1,c1) and Y∼Landau(μ2,c2) then X+Y∼Landau(μ1+μ2,c1+c2).

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 μ=0 and c=π/2, the pdf can be approximated[4] using Lindhard theory which says:

p(x+ln⁡x−1+γ)≈exp⁡(−1/x)x(1+x),

where γ is Euler's constant.

A similar approximation [5] of p(x;μ,c) for μ=0 and c=1 is:

p(x)≈12πexp⁡(−x+e−x2).

## 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[6].

## Related distributions

* The Landau distribution is a stable distribution with stability parameter α and skewness parameter β both equal to 1.

## References

1. ↑ Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR) 8: 201. http://e-heritage.ru/Book/10093344.
2. ↑ Gentle, James E. (2003). Random Number Generation and Monte Carlo Methods. Statistics and Computing (2nd ed.). New York, NY: Springer. p. 196. doi:10.1007/b97336. ISBN 978-0-387-00178-4.
3. ↑ Zolotarev, V.M. (1986). One-dimensional stable distributions. Providence, R.I.: American Mathematical Society. ISBN 0-8218-4519-5. https://archive.org/details/onedimensionalst00zolo_0.
4. ↑ "LandauDistribution—Wolfram Language Documentation". https://reference.wolfram.com/language/ref/LandauDistribution.html.
5. ↑ Behrens, S. E.; Melissinos, A.C.. Univ. of Rochester Preprint UR-776 (1981).
6. ↑ Bulyak E., Shul'ga N. (2022). "Landau distribution of ionization losses: history, importance, extensions". arXiv:2209.06387 [physics.plasm-ph].

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