Maxwell-Boltzmann velocity distribution ($f(\mathbf{v})$)

For particles of mass $m$ in a classical gas at temperature $T$, the normalized velocity density isIts Cartesian components are independent centered normal random variables of variance $k_{B}T/m$.

Integrating the Maxwell-Boltzmann velocity distribution over directions gives

The maximum of the Maxwell speed distribution occurs at

Doppler broadening is the spread of an observed spectral line caused by the distribution of emitter velocities along the line of sight.

If nonrelativistic emitters of mass $m$ have rest-frame angular frequency ${\omega }_0$, then the one-dimensional Maxwell-Boltzmann velocity distribution produces a Gaussian line with standard deviationMeasuring the line width therefore determines the gas temperature when nonthermal broadening is negligible or separately modeled.