Rotational equipartition in a rough-ball gas

Rough elastic collisions can exchange rotational and translational kinetic energy. For three-dimensional rigid balls with all three rotations thermalized, these six quadratic terms contribute $3k_{B}T$ per particle. A uniform gravitational height distribution adds $k_{B}T$, giving total mechanical mean energy $4k_{B}T$.

If each initially stationary ball is released at height $h$ and the subsequent isolated dilute assembly reaches effective classical equilibrium, energy conservation gives $4k_{B}T=mgh$. The height-tail probability above the release height is then $e^{-4}$. The effective temperature refers to mechanical motion, not the material's internal temperature.