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 straight phi(c) is 1. However, if we relax this condition slightly such that straight phi(c)=1-delta, we find a series of critical temperatures Tc(i)=mgD(N+1-i)/mu0 with mu0=1/delta-1, at which the ith particle undergoes the liquid-solid transition. The functional form of the onset temperature, Tc(1)=mgDN/mu0, 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.