Note on One-Dimensional Lattice Gas in External Fields

Abstract

Abstract The fundamental equations of the scaled particle theory are derived for a one-dimensional lattice gas in an external potential. These equations relate the work required to add a particle, at a fixed point, to a N – 1 particle system to the activity and to a series of coordinate distribution functions. The equations hold in any dimension, and replacing sums by integrals, describe continuum fluids. The rigid lattice gas is solved by these means. When nearest neighbors interact, in a positive, increasing external potential, a formal solution is obtained by a matrix method. The grand partition function, in the infinite length limit, depends only on a single eigenvalue of an infinite product of matrices. The one-particle distribution, in this limit, is reduced to a series of terminating continued fractions, which is readily approximated in the high coordinate or low activity limit. Lastly, it is shown that the zeros of the grand partition function lie on the negative real axis of the complex activity ...