Landau distribution

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 Laplace transform of the function ${\displaystyle s^{s}}$.

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 ${\displaystyle \beta =1}$,^[2] with characteristic function:^[3]

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

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.

• 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.

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]

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