Effusive cooling ($T\propto N^{1/3}$)

If a classical monatomic gas re-equilibrates by collisions while losing atoms by effusion, the escaping atoms carry mean kinetic energy $2k_{B}T$. Without wall heating, conservation of energy gives $d(3N{k}_{B}T/2)=2k_{B}TdN$ and therefore $T\propto N^{1/3}$. This assumes a fixed container volume and a small aperture in the molecular-flow regime.