= Solution
The <Boltzmann distribution> assigns probability $P_i=e^{-E_i/(k_BT)}/Z$ to each microstate. For the <equilibrium height distribution in a uniform gravitational field>, take a dilute gas above a floor in a constant gravitational field. Uniform cross-section and no relevant upper boundary give the normalized height density
$$
P(z)=\frac{mg}{k_BT}e^{-mgz/(k_BT)},\qquad z\geq0.
$$
Thus
$$
\boxed{\langle U_g\rangle=\int_0^\infty mgz\,P(z)\,dz=k_BT.}
$$
The gravitational potential is linear in height, so its mean is not a quadratic-mode contribution of $k_BT/2$.
The <equipartition> states that each independent quadratic term in a classical equilibrium Hamiltonian contributes mean energy $k_BT/2$, provided the relevant degrees of freedom are thermally accessible. Equivalently, $\langle x_i\partial H/\partial x_i\rangle=k_BT$ when the phase-space integration by parts has vanishing boundary terms. Three translational velocity components therefore contribute $3k_BT/2$.
For <rotational equipartition in a rough-ball gas>, collisions exchange both translational and rotational energy. In the intended three-dimensional classical equilibrium model there are three translational and three rotational quadratic terms, giving $3k_BT$ in total <kinetic energy> per ball. Adding $\langle U_g\rangle=k_BT$, and conserving the initial energy $mgh$ per ball, gives
$$
\boxed{mgh=\left(\tfrac32+\tfrac32+1\right)k_BT=4k_BT,\qquad T=\frac{mgh}{4k_B}.}
$$
Roughness is relevant because it allows rotation to thermalize. The <effective mechanical temperature of released rough balls> describes the balls' mechanical degrees of freedom, not their internal material <temperature>; finite ball size and excluded-volume effects are neglected.<Classical Gases by David Tong>[Sections 2.2.1 and 2.3: equipartition and classical thermal distributions]{ref}
The height tail is
$$
\boxed{P(z>h)=\int_h^\infty P(z)\,dz=e^{-mgh/(k_BT)}=e^{-4}=0.0183\simeq1.83\%.}
$$
This can be nonzero because collisions redistribute energy between balls; an individual ball is not restricted to its original energy. The exponential distribution is the large-number statistical approximation under the stated effective-equilibrium assumption.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2015/paper-2-gravitational-height.png]
{title=The thermal height distribution puts about 1.83 percent of the rough balls above their release height}
{source=paper-2-gravitational-height.py}
Back to article page
