Canonical partition function and distance dependent correlation functions of a quasi-one-dimensional system of hard disks

TL;DR

This paper derives analytical formulas for correlation functions in a quasi-one-dimensional system of hard disks, revealing short-range order and non-monotonic correlation length behavior depending on density and pore width.

Contribution

It provides new analytical expressions for distance-dependent correlation functions in a q1D hard disk system, linking thermodynamics and ordering to system parameters.

Findings

Correlation functions decay exponentially with distance.

Correlation length peaks at a specific density (Nd/L=1).

Theoretical results agree with molecular dynamics simulations.

Abstract

The canonical NLT partition function of a quasi-one dimensional (q1D) one-file system of equal hard disks [J. Chem Phys. 153, 144111 (2020)] provides an analytical description of the thermodynamics and ordering in this system (a pore) as a function of linear density Nd/L where d is the disk diameter. We derive the analytical formulae for the distance dependence of the translational pair distribution function and the distribution function of distances between next neighbor disks, and then demonstrate their use by calculating the translational order in the pore. In all cases, the order is found to be of a short range and to exponentially decay with the disks' separation. The correlation length presented for different pore widths and densities shows a non-monotonic dependence with a maximum at Nd/L=1 and tends to the 1D value for a vanishing pore width. The results indicate a special role of this density when the pore length L is equal exactly to N disk diameters. A comparison between the theoretical results for an infinite system and the results of a molecular dynamics simulation for a finite system with periodic boundary conditions is presented and discussed.