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:

p ( x ) = 1 2 π i ∫ a − i ∞ a + i ∞ e s ln ⁡ s + x s d s , {\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 ln {\displaystyle \ln } refers to the natural logarithm. In other words, it is the Laplace transform of the function s s {\displaystyle s^{s}}.

The following real integral is equivalent to the above:

p ( x ) = 1 π ∫ 0 ∞ e − t ln ⁡ t − x t sin ⁡ ( π t ) d t . {\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 α = 1 {\displaystyle \alpha =1} and β = 1 {\displaystyle \beta =1}, with characteristic function:

φ ( t ; μ , c ) = exp ⁡ ( i t μ − 2 i c t π ln ⁡ | t | − c | t | ) {\displaystyle \varphi (t;\mu ,c)=\exp \left(it\mu -{\tfrac {2ict}{\pi }}\ln |t|-c|t|\right)}

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

p ( x ; μ , c ) = 1 π c ∫ 0 ∞ e − t cos ⁡ ( ( x − μ ) t c + 2 t π ln ⁡ t c ) d t , {\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 μ = 0 {\displaystyle \mu =0} and c = π 2 {\displaystyle c={\frac {\pi }{2}}} we get the original form of p ( x ) {\displaystyle p(x)} above.

Properties

The approximation function for μ = 0 , c = 1 {\displaystyle \mu =0,\,c=1}
  • Translation: If X ∼ Landau ( μ , c ) {\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,} then X + m ∼ Landau ( μ + m , c ) {\displaystyle X+m\sim {\textrm {Landau}}(\mu +m,\,c)\,}.
  • Scaling: If X ∼ Landau ( μ , c ) {\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,} then a X ∼ Landau ( a μ − 2 π a c ln ⁡ a , a c ) {\displaystyle aX\sim {\textrm {Landau}}(a\mu -{\tfrac {2}{\pi }}ac\ln a,\,ac)}.
  • Sum: If X ∼ Landau ( μ 1 , c 1 ) {\displaystyle X\sim {\textrm {Landau}}(\mu _{1},c_{1})} and Y ∼ Landau ( μ 2 , c 2 ) {\displaystyle Y\sim {\textrm {Landau}}(\mu _{2},c_{2})\,} then X + Y ∼ Landau ( μ 1 + μ 2 , c 1 + c 2 ) {\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 μ = 0 {\displaystyle \mu =0} and c = π / 2 {\displaystyle c=\pi /2}, the pdf can be approximated using Lindhard theory which says:

p ( x + ln ⁡ x − 1 + γ ) ≈ exp ⁡ ( − 1 / x ) x ( 1 + x ) , {\displaystyle p(x+\ln x-1+\gamma )\approx {\frac {\exp(-1/x)}{x(1+x)}},}

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

A similar approximation of p ( x ; μ , c ) {\displaystyle p(x;\mu ,c)} for μ = 0 {\displaystyle \mu =0} and c = 1 {\displaystyle c=1} is:

p ( x ) ≈ 1 2 π exp ⁡ ( − x + e − x 2 ) . {\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.

Related distributions

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