Landau distribution

Landau distribution
Probability density function ${\displaystyle \mu =0,\;c=\pi /2}$
Parameters	${\displaystyle c\in (0,\infty )}$ — scale parameter ${\displaystyle \mu \in (-\infty ,\infty )}$ — location parameter
Support	${\displaystyle \mathbb {R} }$
PDF	${\displaystyle {\frac {1}{\pi c}}\int _{0}^{\infty }e^{-t}\cos \left(\left(x-\mu \right){\frac {t}{c}}+{\frac {2t}{\pi }}\ln {\frac {t}{c}}\right)\,dt}$
Mean	Undefined
Variance	Undefined
MGF	Undefined
CF	${\displaystyle \exp \left(it\mu -{\frac {2ict}{\pi }}\log |t|-c|t|\right)}$

In probability theory, the Landau distribution is a probability distribution named after Lev Landau who used it in 1944.^[1]

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:

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

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 ${\displaystyle \ln }$ refers to the natural logarithm. In other words, it is the inverse Laplace transform of the function ${\displaystyle s^{s}}$.^[2]

The following real integral is equivalent to the above:

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

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters ${\displaystyle \alpha =1}$ and ${\displaystyle \beta =1}$,^[3] with characteristic function:^[4]

$${\displaystyle \varphi (t;\mu ,c)=\exp \left(it\mu -{\tfrac {2ict}{\pi }}\ln |t|-c|t|\right)}$$

where ${\displaystyle c\in (0,\infty )}$ and ${\displaystyle \mu \in (-\infty ,\infty )}$, which yields a density function:

$${\displaystyle p(x;\mu ,c)={\frac {1}{\pi c}}\int _{0}^{\infty }e^{-t}\cos \left(\left(x-\mu \right){\frac {t}{c}}+{\frac {2t}{\pi }}\ln {\frac {t}{c}}\right)\,dt,}$$

Taking ${\displaystyle \mu =0}$ and ${\displaystyle c={\frac {\pi }{2}}}$ we get the original form of ${\displaystyle p(x)}$ above.

Properties

The approximation function for ${\displaystyle \mu =0,\,c=1}$

• Translation: If ${\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,}$ then ${\displaystyle X+m\sim {\textrm {Landau}}(\mu +m,\,c)\,}$.
• Scaling: If ${\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,}$ then ${\displaystyle aX\sim {\textrm {Landau}}(a\mu -{\tfrac {2}{\pi }}ac\ln a,\,ac)}$.
• Sum: If ${\displaystyle X\sim {\textrm {Landau}}(\mu _{1},c_{1})}$ and ${\displaystyle Y\sim {\textrm {Landau}}(\mu _{2},c_{2})\,}$ then ${\displaystyle X+Y\sim {\textrm {Landau}}(\mu _{1}+\mu _{2},\,c_{1}+c_{2})}$.

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 ${\displaystyle \mu =0}$ and ${\displaystyle c=\pi /2}$, the pdf can be approximated^[5] using Lindhard theory which says:

$${\displaystyle p(x+\ln x-1+\gamma )\approx {\frac {\exp(-1/x)}{x(1+x)}},}$$

where ${\displaystyle \gamma }$ is Euler's constant.

A similar approximation ^[6][7] of ${\displaystyle p(x;\mu ,c)}$ for ${\displaystyle \mu =0}$ and ${\displaystyle c=1}$ is:

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

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.^[8]

Related distributions

• The Landau distribution is a stable distribution with stability parameter ${\displaystyle \alpha }$ and skewness parameter ${\displaystyle \beta }$ both equal to 1.

References

1. Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR). 8: 201.
2. Grupen, Claus; Shwartz, Boris (2008). Particle Detectors. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Vol. 26 (2nd ed.). Cambridge University Press. ISBN 9781009401494.^{[page needed]}
3. 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.
4. Zolotarev, V.M. (1986). One-dimensional stable distributions. Providence, R.I.: American Mathematical Society. ISBN 0-8218-4519-5.
5. "LandauDistribution—Wolfram Language Documentation".
6. Behrends, S. E.; Melissinos, A.C. Properties of argon-ethane/methane mixtures for use in proportional counters, Univ. of Rochester Preprint UR-776 (1981). doi:10.1016/0029-554X(81)90263-9.
7. Moyal, J.E. (1955). "Theory of ionization fluctuations". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 46 (374): 263–280. doi:10.1080/14786440308521076.
8. Bulyak E., Shul'ga N. (2022). "Landau distribution of ionization losses: history, importance, extensions". arXiv:2209.06387 [physics.plasm-ph].