Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 36A (Statistical Physics)

In the canonical ensemble, a one-particle velocity state has kinetic energy $E(\mathbf{v})=m|\mathbf{v}{|}^2/2$ and hence probability density proportional to $e^{-\beta E}$, where $\beta =(k_{B}T{)}^{-1}$. The three Cartesian Gaussian integrals normalize the density to the Maxwell-Boltzmann velocity distributionIntegrating over a spherical shell of radius $v$ and area $4\pi v^2$ gives the Maxwell speed distributionSinceits unique positive maximum is the Most probable speed in the Maxwell distribution

The $x$ component of the Maxwell-Boltzmann velocity distribution is a centered normal random variable, so $⟨v_x⟩=0$. Therefore

The same one-dimensional normal distribution hasBecause $\omega -{\omega }_0=({\omega }_0/c)v_x$, the frequency variance and standard deviation are

The Doppler relation is the affine change of variablesTransforming the one-dimensional Maxwell-Boltzmann velocity distribution therefore gives the normalized thermal Doppler broadening profileThusThe measured Gaussian line width satisfies ${\sigma }_{\omega }/{\omega }_0=\sqrt{k_{B}T/m}/c$, so the temperature follows from