Quantifying the transition from single file to Fickian diffusion

Single-file diffusion occurs when diffusing particles are confined to a narrow tube, which prohibits particles from passing one another. Observed in a variety of natural and engineered systems, single-file diffusion is marked by the subdiffusive mean-squared displacement of tracer particles. If the channel is barely wide enough to permit rare passing events, then the mean-squared displacement is subdiffusive at early times and diffusive at late times, and the transition between these two regimes is controlled by the average time it takes a particle to pass its neighbor. In this paper, we study how this so-called "hopping time" depends on confinement geometry, which is a problem previously studied in the chemical physics literature using a variety of theoretical methods, resulting in some conflicting predictions. Our approach leverages the theory of boundary homogenization to describe particle passing in terms of an effective "permeability," and we use the mathematical theory of strong localized perturbations to obtain explicit formulas for the permeability and hopping time. We confirm our analytical results by kinetic Monte Carlo simulations.