Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 34D (Statistical Physics)

The ergodic hypothesis states that long-time averages of equilibrium observables along typical trajectories equal their ensemble averages on the accessible phase space with the same conserved quantities. It permits statistical mechanics to predict measured macroscopic quantities from equilibrium distributions rather than tracking a trajectory. In its microcanonical form the measure is on the constant-energy surface. It is an assumption about the relevant dynamics, not a consequence of particles being noninteracting; the calculations below use the equilibrium canonical ensemble at temperature $T$.

Take the floor at $z=0$, gas-particle mass $m$, $N$ atoms and box side $L$. The one-particle Hamiltonian is $p^2/(2m)+mgz$. With $\beta =1/(k_{B}T)$ and horizontal area $A=L^2$, the partition function isThis follows by independent Gaussian momentum integrals and the elementary height integral. Taking $E=-{\partial }_{\beta }\log Z_N$ givesFor an arbitrarily tall box compared with the gravitational scale height, $L\gg k_{B}T/(mg)$, the upper-wall correction is negligible andThe energy includes gravitational potential relative to the floor; the kinetic part alone is $3N{k}_{B}T/2$. Retaining the finite-height correction also gives the ordinary ideal-gas limit as $g\to 0$.

The equipartition theorem assigns $k_{B}T/2$ to each of the three quadratic momentum terms, giving kinetic energy $3N{k}_{B}T/2$. The gravitational term is linear in $z$, so it must not be counted as a quadratic degree of freedom. In the tall-box limit its distribution is proportional to $e^{-\beta mgz}$, with mean height $1/(\beta mg)$ and mean potential $k_{B}T$ per atom. Equivalently the generalized equipartition identity $⟨z\partial H/\partial z⟩=k_{B}T$ follows by integration by parts on $[0,\infty )$, where the boundary term vanishes. Three quadratic kinetic contributions plus one linear potential contribution give $5k_{B}T/2$ per atom. At finite $L$, the integration-by-parts boundary term is precisely the correction in part (i).

The Boltzmann distribution factors between momentum and position. Normalizing the height factor givesIn the tall-box limit this becomes $\boxed{\beta mg{e}^{-\beta mgz}dz}$ on $z\ge 0$, an exponential distribution with scale height $k_{B}T/(mg)$.

The Maxwell-Boltzmann velocity distribution is unaffected by the position-dependent potential, since the canonical density factors. Its speed density is proportional to $v^2{e}^{-m{v}^2/(2k_{B}T)}$ for $v\ge 0$. For $v>0$, differentiating its logarithm gives $2/v-mv/(k_{B}T)=0$, soThe density vanishes at zero and at infinity and has only this positive stationary point, so it is the maximum. It is the same at every height.