Static and dynamical properties of a hard-disk fluid confined to a narrow channel

Abstract

The thermodynamic properties of disks moving in a channel sufficiently narrow that they can collide only with their nearest neighbors can be solved exactly by determining the eigenvalues and eigenfunctions of an integral equation. Using it we have determined the correlation length $\xi$ of this system. We have developed an approximate solution which becomes exact in the high density limit. It describes the system in terms of defects in the regular zigzag arrangement of disks found in the high-density limit. The correlation length is then effectively the spacing between the defects. The time scales for defect creation and annihilation are determined with the help of transition-state theory, as is the diffusion coefficient of the defects, and these results are found to be in good agreement with molecular dynamics simulations. On compressing the system with the Lubachevsky--Stillinger procedure, jammed states are obtained whose packing fractions $\phi_J$ are a function of the compression rate $\gamma$. We find a quantitative explanation of this dependence by making use of the Kibble--Zurek hypothesis. We have also determined the point-to-set length scale $\xi_{PS}$ for this system. At a packing fraction $\phi$ close to its largest value $\phi_{\text{max}}$, $\xi_{PS}$ has a simple power law divergence, $\xi_{PS} \sim 1/(1-\phi/\phi_{\text{max}})$, while $\xi$ diverges much faster, $\ln(\xi) \sim 1/(1-\phi/\phi_{\text{max}})$.