Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 14E / a / Solution

The Bose--Einstein and Fermi--Dirac occupation factors areIn the stated nonrelativistic dilute regime, $E-\mu$ is dominated by $m{c}^2$, so the exponential is large and either denominator is asymptotic to $\exp ((E-\mu )/(k_{B}T))$. Integrating the resulting Maxwell--Boltzmann occupation over momentum states gives

For $E=m{c}^2+p^2/(2m)$, integration by parts or differentiation of the supplied Gaussian integral givesConsequently