[Paper Review] Hard disks confined within a narrow channel

J. M. Brader, E. Di Bernardo+1 more|arXiv (Cornell University)|Jan 23, 2026

Material Dynamics and Properties0 citations

TL;DR

The paper uses inhomogeneous Percus-Yevick theory to study hard disks confined in a narrow channel, showing accurate dimensional crossover from 2D to 1D and predicting a zigzag ordering at high packing.

ABSTRACT

We employ inhomogeneous integral equation theory to investigate the equilibrium properties of hard disks confined to a channel of width $L$ by hard parallel walls. If the channel width is narrowed below two disk diameters, then the system enters a quasi one-dimensional regime for which the particles cannot move past each other. In the limit when $L$ is equal to one particle diameter the system reduces to the one-dimensional bulk along the center of the channel. We study first the dimensional crossover properties of the inhomogeneous Percus-Yevick (PY) integral equation as $L$ is reduced and then investigate the behaviour of a quasi one-dimensional system as the packing of the particles is increased for a fixed value of $L$. We find that the inhomogeneous PY equation is highly accurate for situations of quasi one-dimensional confinement and that it predicts the onset of a structural transition to a zigzag state at higher packing. The excellent performance of this integral equation method and the ease with which it handles confinement-induced dimensional crossover is a consequence of the improved resolution which comes from treating explicitly the inhomogeneous two-body correlation functions.

Motivation & Objective

• Investigate equilibrium properties of hard disks confined between parallel hard walls with channel width L.
• Examine how confinement induces dimensional crossover from 2D to 1D and how the theory handles it.
• Assess how packing affects density profiles and two-body correlations in quasi-1D confinement.

Proposed method

• Formulate the inhomogeneous Ornstein-Zernike (OZ) equation for systems with external confinement.
• Apply the inhomogeneous Percus-Yevick closure to relate h and c under confinement (c = (e^{-βφ}−1)(h−c+1)).
• Use the Lovett–Mou–Buff–Wertheim (LMBW) sum-rule to connect the one-body density to two-body correlations.
• Leverage the exact 1D Percus–Yevick solution for hard rods as a benchmark and to demonstrate dimensional crossover.
• Employ the Percus functional framework and its functional derivatives to obtain c^(1) and thus h and g in inhomogeneous settings.
• Analyze dimensional crossover by progressively narrowing the channel from 2D to 1D and show natural reduction to 1D theory.

Figure 1: Dimensional crossover imposed by the sequential application of confining potentials. For example, trapping a 3D system of hard spheres between parallel hard walls with a separation $L\!=\!d$ recovers a 2D bulk system of hard disks. Introducing a further pair of confining walls then reduces

Experimental results

Research questions

• RQ1 How does the inhomogeneous PY theory describe the dimensional crossover from 2D hard disks to 1D hard rods as channel width L decreases?
• RQ2 How accurate is the inhomogeneous PY closure for predicting density profiles and two-body correlations in quasi-1D confinement?
• RQ3 Can the theory capture the onset of long-range longitudinal order (zigzag state) at high packing?
• RQ4 Does the approach reproduce exact results in the 1D limit and agree with known solutions for quasi-1D systems?

Key findings

• The inhomogeneous PY equation accurately describes quasi-1D confinement and naturally reduces to the exact 1D solution as L → 1.
• Density profiles become highly peaked at the center while walls induce layering as confinement tightens.
• The inhomogeneous two-body correlation functions reveal onset of long-range longitudinal order (zigzag state) at higher packing.
• The method agrees well with exact quasi-1D solutions for packing up to near close-packing in the zigzag regime.
• Dimensional crossover is handled without fine-tuning, outperforming some FMT approaches in maintaining correct 0D/1D limits.

Figure 3: Dimensional crossover from hard disks to hard rods. Panel A shows, in shades of green, the one-body density profiles for a system of hard-disk particles between two hard walls, as the slit width $L$ is reduced from $5$ to $1.2$ (for particle diameter $d$ set to unity). For all profiles the