Renewal-anomalous-heterogeneous files

TL;DR

This paper investigates renewal-anomalous heterogeneous files composed of diffusing particles with non-uniform properties, deriving a scaling relation for their mean square displacement and comparing their dynamics to non-renewal files.

Contribution

It introduces a new class of renewal-anomalous heterogeneous files and derives an exact relation linking their MSD to that of Brownian files, revealing their slower dynamics.

Findings

MSD scales as the power of the MSD in Brownian files with exponent alpha.

Renewal-anomalous files are slower than non-renewal-anomalous files.

An exact relation connecting probability density functions of the two file types was derived.

Abstract

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 exponential as in Brownian files, yet obeys: {\psi}_{\alpha} (t)~t^(-1-{\alpha}), 0<{\alpha}<1. The file is renewal as all the particles attempt to jump at the same time. It is shown that the mean square displacement (MSD) in a renewal-anomalous-heterogeneous file, <r^2>, obeys, <r^2>~[<r^2>_{nrml}]^{\alpha}, where <r^2 >_{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.