Wall induced density profiles and density correlations in confined Takahashi lattice gases

TL;DR

This paper develops a formalism to analyze the static properties, density profiles, and correlations of particles on a one-dimensional lattice confined by walls, with applications to the Takahashi and hard rod lattice gases.

Contribution

It introduces a general recursive approach to compute thermodynamic quantities, density distributions, and correlations for confined lattice gases, including explicit solutions for the hard rod case.

Findings

Derived linear recursion relations for partition functions.

Explicit density and correlation profiles near walls.

Identified constant occupation regions in the canonical ensemble.

Abstract

We propose a general formalism to study the static properties of a system composed of particles with nearest neighbor interactions that are located on the sites of a one-dimensional lattice confined by walls (``confined Takahashi lattice gas''). Linear recursion relations for generalized partition functions are derived, from which thermodynamic quantities, as well as density distributions and correlation functions of arbitrary order can be determined in the presence of an external potential. Explicit results for density profiles and pair correlations near a wall are presented for various situations. As a special case of the Takahashi model we consider in particular the hard rod lattice gas, for which a system of nonlinear coupled difference equations for the occupation probabilities has been presented previously by Robledo and Varea. A solution of these equations is given in terms of the solution of a system of independent linear equations. Moreover, for zero external potential in the hard rod system we specify various central regions between the confining walls, where the occupation probabilities are constant and the correlation functions are translationally invariant in the canonical ensemble. In the grand canonical ensemble such regions do not exist.