Evaporative cooling energy balance ($\alpha =(\eta +\kappa -3)/3$)

For a classical three-dimensional harmonic trap of fixed frequencies, $E=3N{k}_{B}T$. If each lost atom removes mean energy ${ϵ}_{\mathrm{ev}}=(\eta +\kappa )k_{B}T$, differential energy balance gives $d\ln T/d\ln N=(\eta +\kappa -3)/3$. The result assumes rapid elastic equilibration and excludes separate work from changing trap frequencies.