Abstract

We sum the canonical partition function for a system of hard rods in a box of finite length in the presence of a linear external potential (gravity). From the canonical punition function closed expressions for the pressure at the top and bottom walls, and the chemical potential follow. The canonical number density and higher distribution functions are also determined. In particular it is shown that the number densities at the extremes of the box are proportional to the associated pressures at those points even though this is not generally true in the bulk of the system. It also is shown that the system is naturally divided in two wall zones and, if the density is low enough, a central zone as it is the case for the free field system. An expression for the local pressure is also derived and it is found that, in the thermodynamic limit and in a sufficiently weak external potential, an exact local relation between the number density and the pressure profile (equation of state) exists in the canonical ensemble within the central region. We also compute the grand canonical partition function for the system and generalize some results from other authors. © 1997 American Institute of Physics.