Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution (not to be confused with the other Maxwell-Boltzmann distribution) is a real, continuous Probability distribution. For a Random variable XX, the Probability density function is

f(x)=2πx2a3e−x2/2a2f(x)=\sqrt{ \frac{2}{\pi} } \frac{x^{2}}{a^{3}} e^{-x^{2}/2a^{2}}

where a>0∈Ra>0\in \mathbb{R} is a parameter. It is commonly employed in physics to model the velocity distribution of particles in an ideal gas.

3D monoatomic ideal gas

In a 3D monoatomic ideal gas, the distribution measures the speed of the particles v2=vx2+vy2+vz2v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}. In this case, a=kBT/ma=\sqrt{ k_{B}T/m } and the PDF is

f(v)=2π(mkBT)3/2v2e−mv2/2kBTf(v)=\sqrt{ \frac{2}{\pi} } \left( \frac{m}{k_{B}T} \right)^{3/2} v^{2}e^{- mv^{2}/2k_{B}T}

where kBk_{B} is the Boltzmann constant, TT is the temperature and mm is the mass of the particles.

2D monoatomic ideal gas

A similar distribution can be derived in a 2D gas as well:

f(v)=mkBTve−mv2/2kBTf(v)=\frac{m}{k_{B}T} ve^{-mv^{2}/2k_{B}T}

NN-dimensional generalization

The distribution can be generalized to an arbitrary number of dimensions as

f(v)=A(m2πkBT)N/2vN−1e−mv2/2kBTf(v)=A\left( \frac{m}{2\pi k_{B}T} \right)^{N/2}v^{N-1}e^{-mv ^{2}/2k_{B}T}

where AA is a normalization constant.

Relation to other distributions

It is a specific case of the Chi-square distribution, with 3 degrees of freedom (representing the components of the velocity vector).