Tracer Diffusion beyond Gaussian Behavior: Explicit Results for General Single-File Systems

Single-file systems, in which particles diffuse in narrow channels while not overtaking each other, is a fundamental model for the tracer subdiffusion observed in confined geometries, such as in zeolites or carbon nanotubes. Twenty years ago, the mean squared displacement of a tracer was determined at large times, for any diffusive single-file system. Since then, for a general single-file system, even the determination of the fourth cumulant, which probes the deviation from Gaussianity, has remained an open question. Here, we fill this gap and provide an explicit formula for the fourth cumulant of an arbitrary single-file system. Our approach also allows us to quantify the perturbation induced by the tracer on its environment, encoded in the correlation profiles. These explicit results constitute a first step towards obtaining a closed equation for the correlation profiles for arbitrary single-file systems.