Condensation of hard rods under gravity: Exact results in one dimension
Abstract
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 φc is 1. However, if we relax this condition slightly such that φc=1-δ, we find a series of critical temperatures T(i)c=mgD(N+1-i)/μ0 with μ0=1/δ-1, at which the ith particle undergoes the liquid-solid transition. The functional form of the onset temperature, T(1)c=mgDN/μ0, 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.
- Publication:
-
Physical Review E
- Pub Date:
- May 2001
- DOI:
- 10.1103/PhysRevE.63.051506
- arXiv:
- arXiv:cond-mat/0007494
- Bibcode:
- 2001PhRvE..63e1506H
- Keywords:
-
- 64.70.Dv;
- 51.30.+i;
- 45.70.Vn;
- Solid-liquid transitions;
- Thermodynamic properties equations of state;
- Granular models of complex systems;
- traffic flow;
- Condensed Matter - Statistical Mechanics;
- Condensed Matter - Materials Science
- E-Print:
- doi:10.1103/PhysRevE.63.051506