Condensation of hard rods under gravity: Exact results in one dimension - art. no. 051506

We present exact results for the density profile of a one dimensional array of N hard rods of diameter D and mass til under gravity g. For a strictly one dimensional system, the liquid-solid transition occurs at zero temperat ure, because the close-packed density phi (c) is 1. However, if we relax th is condition slightly such that phi (c) = 1 - delta, we find a series of cr itical temperatures T-c((i)) = mgD(N+1 - i)/mu (o) with mu (o) = 1/delta - 1, at which the ith particle undergoes the liquid-solid transition. The fun ctional form of the onset temperature, T-c((1)) =mgDN/mu (o) 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 i n T before the transition, but it becomes quadratic in T after the transiti on because of the formation of solid near the bottom.