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

TL;DR

This paper derives distance-dependent correlation functions for a quasi-one-dimensional system of hard disks by exploiting the canonical NL T partition function for a finite and thermodynamic limit in a pore. It provides analytical expressions for the translational PDF $g(R)$ and the next-neighbor PDF $g_1(R)$, obtained by decomposing the partition function around fixed longitudinal separations and performing a saddle-point analysis. The results show exponentially decaying correlations with a nonmonotonic correlation length that peaks at density $ ho=1$ and recovers the 1D Tonks gas limit as the pore width vanishes; comparisons with molecular dynamics simulations reveal good agreement at extreme densities, with nuanced shifts at intermediate densities likely due to boundary conditions and finite-size effects. The study offers a rigorous analytical route to understand ordering and entropy-driven defects in q1D pores, with potential implications for transport in confined geometries and related cold-atom analogies.

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.

Figures (6)

• Figure 1: Definition of the pair distribution function $\,g(R)\,$ and the three PFs: $\,Z_{n, R}\,$ is for $\,n-1\,$ free moving disks in the space $\,R\,$ in which there are $\,n\,$ neighbors of disk $\,0\,$; $\,Z_{N-n, L-R}\,$ is for $\,N-n\,$ free disks in the space $\,L-R\,$; and $\,Z_{N}\,$ is for $\,N\,$ free disks in the space $\,L\,$.
• Figure 2: Part a): Theoretical results (solid lines) and MD simulation data (symbols) for pair distribution function $\,g_{1}(R)\,$ for pore width $\,\Delta =0.5\,$ and four densities: 1 -- $\,\rho =0.8\,$; 2 -- 1.01; 3 -- 1.053; 4 -- 1.111 . Part b): Theoretical results for shifted densities $\,\rho =1.034\,$ and $\,\rho =1.065\,$ (thick solid lines) which are practically indistinguishable of the MD simulation data (symbols) for densities $\,\rho=1.01\,$ (right peak) and $\,\rho =1.053\,$ (leftt peak), respectively. For comparison, the thin solid lines 2 and 3 [the same as in part a)] show theoretical curves for actual (non shifted) densities $\,\rho =1.01\,$ and 1.053 , respectively .
• Figure 3: Theoretical pair distribution function $\,g_{1}(R)\,$ for pore width $\,\Delta =0.866\,$ and four different densities: 1 -- $\,\rho=0.6\,$; 2 -- 1; 3 -- 1.4 and 4 -- 1.6 .
• Figure 4: Theoretical PDF $\,g(R)\,$ (the dashed line) superimposed on MD simulation data (the solid line and symbols) for the case of pore width $\,\Delta =0.5\,$ and density $\,\rho =1\,$. The MD data are shown for two distinct sizes of the simulated system, i.e., $\,N=400\,$ (the green color) and $\,N=2000\,$ (the red color). Part a) for distances $\,R<5\,$ and part b) for distances $\,R<200\,$. In part b) the theoretical curve is cropped in the range $\,15<R/d<50\,$ for better visualization of the simulation data.
• Figure 5: Dependence of the correlation length $\,\xi\,$ on density $\,\rho\,$ for the case of pore width $\,\Delta =0.5\,$.