Maxwell–Boltzmann distribution

Remarks

The function $M/n$ is the density of the normal distribution on $\R^3$ with mean $u$ and covariance $(kT/m) I$. In the kinetic description of a gas, $M$ has number density $n$, bulk velocity $u$ and temperature $T$. For $u = 0$, the speed $s = \abs{v}$ has the density $4\pi s^2 (m/(2\pi kT))^{3/2} e^{-ms^2/(2kT)}$ on $(0, \infty)$, after normalising by $n$. Its most probable value is $\sqrt{2kT/m}$, its mean is $\sqrt{8kT/(\pi m)}$, and its root mean square is $\sqrt{3kT/m}$.

Example: check of the temperature for $u = 0$. Each component $v_i$ has variance $kT/m$ under $M/n$, so $\int \tfrac{m}{2} \abs{v}^2 M \, dv = n \cdot \tfrac{m}{2} \cdot 3 \cdot \tfrac{kT}{m} = \tfrac32 n k T$, as required.

Non-example: the uniform distribution of velocities on a ball $\set{\abs{v} \le R}$ is not a Maxwellian, since it vanishes outside the ball while every Maxwellian is positive everywhere.