= 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 contribute
$$
\langle E_{\rm kin}\rangle=3k_BT
$$
per 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:
$$
p(z)=\frac{mg}{k_BT}e^{-mgz/(k_BT)},\qquad z\ge0.
$$
Its mean <gravitational potential energy> is $\langle mgz\rangle=k_BT$, so the total mechanical mean <energy> is $4k_BT$. 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>.
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