Equilibrium height distribution in a uniform gravitational field
= Equilibrium height distribution in a uniform gravitational field
For a dilute gas with a floor at $z=0$, constant $g$, uniform cross-section and no relevant ceiling, $P(z)=\beta mg\,e^{-\beta mgz}$. Its mean gravitational energy is $k_BT$, and $P(z>h)=e^{-\beta mgh}$. A linear potential contributes $k_BT$, not the $k_BT/2$ of a quadratic degree of freedom.