Rotational equipartition in a rough-ball gas

Rough elastic collisions can exchange rotational and translational kinetic energy. If all three translational and all three rotational degrees of freedom of a rigid ball reach classical thermal equilibrium, their six quadratic terms contributeper ball, by the equipartition theorem.

In a constant downward gravitational field above a floor at $z=0$, with no upper boundary, the equilibrium height distribution is exponential:Its mean gravitational potential energy is $⟨mgz⟩=k_{B}T$, so the total mechanical mean energy is $4k_{B}T$. The field is uniform; the height distribution is not.

This result assumes the rough collisions actually thermalize rotation, sufficient room for the exponential height tail and no energy loss. Smooth elastic spheres need not exchange rotational energy with translation. A finite ceiling changes the height distribution and its mean potential energy.

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.