Condensation of Hard Spheres Under Gravity: Exact Results in One Dimension

We present exact results for the density profile of the one dimensional array of N hard spheres 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-pakced density, $\phi_c$, is one. However, if we relax this condition slightly such that $phi_c=1-\delta$, we find a series of critical temperatures T_c^i=mgD(N+1-i)/\mu_o with \mu_o=const, at which the i-th particle undergoes the liquid-solid transition. The functional 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 in T before the transition, but it becomes quadratic in T after the transition because of the formation of solid near the bottom.