Renewal-anomalous-heterogeneous files

Renewal-anomalous-heterogeneous files are solved. A simple file is made of Brownian hard spheres that diffuse stochastically in an effective 1D channel. Generally, Brownian files are heterogeneous: the spheres' diffusion coefficients are distributed and the initial spheres' density is non-uniform. In renewal-anomalous files, the distribution of waiting times for individual jumps is not exponential as in Brownian files, yet obeys: ψα(t)∼t−1−α, 0<α<1. The file is renewal as all the particles attempt jumping at the same time. It is shown that the mean square displacement (MSD) in a renewal-anomalous-heterogeneous file, 〈r2〉, obeys, 〈r2〉∼〈r2〉nrmlα, where 〈r2〉nrml is the MSD in the corresponding Brownian file. This scaling is an outcome of an exact relation (derived here) connecting probability density functions of Brownian files and renewal-anomalous files. It is also shown that non-renewal-anomalous files are slower than the corresponding renewal ones.