The Landau distribution is a long-tailed unimodal distribution that resembles a normal/Gaussian distribution. This distribution’s fat tails decrease algebraically for large values of x. In this article, we will discuss the fundamentals of the Landau distribution and its importance in physics.
What is the Landau Distribution?
The Landau distribution describes energy loss fluctuations of a charged particle that passes through a medium. It is a continuous function, defined over the set of real numbers. The distribution, first proposed by Soviet physicist Lev Landau in 1944, has been pivotal in explaining the energy loss experienced by fast-moving particles in different areas of physics and chemistry.
The long upper tail of the Landau distribution is due to large fluctuations in high energy ionization. These fluctuations can cause the particle to lose a significant amount of energy, leading to the formation of a long tail. Typically, the particles will only lose a small amount of energy, leading to a peak in the distribution. However, in rare cases, some particles can lose a significant amount of energy, leading to a large fluctuation which forms the long tail.
The probability density function (PDF) can be written as 
The Landau distribution’s shape is determined by:
- The location parameter (μ), which tells you where on the horizontal axis a graph is centered, relative to the standard normal model.
- The scale parameter (σ), which stretches or squeezes the graph.
The calculation of the Landau distribution involves special functions, such as the complex error function, the Dawson function, and the Faddeeva function. These functions are not easy to work with and require specialized software packages, such as ROOT or Mathematica, to calculate values of the distribution. The Landau distribution can also be evaluated numerically using numerical integration techniques such as quadrature rules.
Where is it used?
The Landau distribution is commonly used in areas such as materials science and particle physics. It is used to explain the behavior of charged particles that travel through a medium, such as gases, liquids, and solids. This distribution is also used to describe the fluctuations in energy loss during the interactions of particles and photons with matter. In addition, the Landau distribution is used in semiconductor physics to explain the ionization potential of semiconductors.
 Wolfram Research (2010), LandauDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LandauDistribution.html (updated 2016).
 Glosser.ca, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons