Landau distribution

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.

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

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

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:

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

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]

$${\displaystyle \phi (t;\mu ,c)=\exp (it\mu -\frac{2ict}{\pi }\ln |t|-c|t|)}$$

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

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

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

• Translation: If ${\displaystyle X\sim \text{<mrow data-mjx-texclass="ORD">Landau</mrow>}(\mu ,c)}$ then ${\displaystyle X+m\sim \text{<mrow data-mjx-texclass="ORD">Landau</mrow>}(\mu +m,c)}$.
• Scaling: If ${\displaystyle X\sim \text{<mrow data-mjx-texclass="ORD">Landau</mrow>}(\mu ,c)}$ then ${\displaystyle aX\sim \text{<mrow data-mjx-texclass="ORD">Landau</mrow>}(a\mu -\frac{2}{\pi }ac\ln a,ac)}$.
• Sum: If ${\displaystyle X\sim \text{<mrow data-mjx-texclass="ORD">Landau</mrow>}({\mu }_1,c_1)}$ and ${\displaystyle Y\sim \text{<mrow data-mjx-texclass="ORD">Landau</mrow>}({\mu }_2,c_2)}$ then ${\displaystyle X+Y\sim \text{<mrow data-mjx-texclass="ORD">Landau</mrow>}({\mu }_1+{\mu }_2,c_1+c_2)}$.

These properties can all be derived from the characteristic function. Together they imply that the Landau distributions are closed under affine transformations.

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:

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

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]

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

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Landau distribution. Read more