Planar effusion flux ($\Gamma =w{n}_2⟨v⟩/\pi$)

For an isotropic two-dimensional velocity distribution, a straight opening of width $w$ in a region of number per area $n_2$ loses particles at the displayed rate. The angular integral is ${\int }_{-\pi /2}^{\pi /2}\cos \theta d\theta /(2\pi )=1/\pi$. At fixed equilibrium kinetic energy the speed distribution scales as $m^{-1/2}$, so the rate has the same scaling. A fixed second moment alone does not determine the first moment without a distribution assumption.