From anomalous diffusion in polygons to a transport locking relation

We study particle transport in a class of open channels of finite length, made of identical cells of connected open polygonal billiards with parallel boundaries. In these systems, the mean square displacement (MSD) grows in time faster than linearly. We show that, irrespective of the geometry of these channels, the distribution of the first return times decays algebraically with two different exponents, separated by a crossover region determined by the MSD. We find that the distribution of first return times satisfies a simple scaling form. In turn, the transmission coefficient-defined as the fraction of trajectories that, starting at the cell at the origin, escape the channel through the other boundary-decays algebraically with the size of the system and, as a signature of nonrecurrent transport, sometimes slower. From these two processes, we derive a locking relation among the scaling exponents for the asymptotic behavior of the MSD, the times of first return to the origin, and the way transmission decays with the system size, showing that these three processes are interdependent. The locking relation holds for diffusive processes as well as for fractional Brownian motion with an arbitrary Hurst exponent. We argue that the locking relation may be valid for many other transport processes, Markovian or not, with finite MSD.