Small system size effects in single-file diffusion

Single-file diffusion (SFD) is the diffusive motion of particles in one-dimension with the constraint that particles can not pass each other. We model SFD in a finite system using $n$ random walkers on a periodic lattice with an average of $m$ empty sites between walkers. The time-dependence of the mean-squared displacement (MSD) is found using a Monte Carlo simulation as a function of $n$ for small systems with $n \le 500$ walkers. For short times, the increase in the MSD is approximately proportional to the time $t$ for all systems. For longer times and large $n$, the MSD rolls over and approaches the expected asymptotic behavior, MSD $\propto t^{1/2}$. However, for small $n$ the MSD approaches a constant value because the finite system size limits the maximum spread of the walkers.