Condensation of hard rods under gravity:  Exact results in one dimension

We present exact results for the density profile of a one dimensional array of N hard rods of diameter D and mass m under gravity g. For a strictly one dimensional system, the liquid-solid transition occurs at zero temperature, because the close-packed density ${\phi }_c$ is 1. However, if we relax this condition slightly such that ${\phi }_c=1\mathrm{}-\delta ,$ we find a series of critical temperatures $T_c^{\mathrm{}\left(i\right)\mathrm{}}=mgD(N+1\mathrm{}-i)/{\mu }_0$ with ${\mu }_0=1/\delta -1,$ at which the ith particle undergoes the liquid-solid transition. The functional form of the onset temperature, $T_c^{\mathrm{}\left(1\right)\mathrm{}}=mgDN/{\mu }_0,$ is consistent with the previous result [Physica A 271, 192 (1999)] obtained by the Enskog equation. We also show that the increase in the center of mass is linear in T before the transition, but it becomes quadratic in T after the transition because of the formation of solid near the bottom.