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 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.

💡 Research Summary

The paper extends the classic single‑file diffusion model, in which hard‑sphere particles move stochastically along a one‑dimensional channel without overtaking each other, to incorporate three layers of complexity: (i) heterogeneity of diffusion coefficients, (ii) non‑uniform initial particle density, and (iii) a renewal anomalous waiting‑time distribution. In the standard Brownian file, each particle’s waiting time between successive jumps is exponentially distributed, leading to normal diffusion of the tagged particle with a mean‑square displacement (MSD) that grows as a sub‑linear power of time due to the single‑file constraint. Here the authors retain the exponential jump‑attempt synchrony (renewal) but replace the exponential tail with a heavy‑tailed power‑law ψα(τ)∝τ−(1+α) where 0<α<1. All particles attempt to jump simultaneously at each renewal epoch, which distinguishes the model from non‑renewal anomalous files where jump attempts are asynchronous.

The central analytical result is an exact transformation linking the probability density function (PDF) of the ordinary Brownian file, P(x,t), to that of the renewal‑anomalous file, Q(x,t). In Laplace space the relation reads Q(s)=s^{α−1}P(s^{α}), which corresponds to a time‑rescaling t→t^{α}. By applying this transformation to the known MSD of the heterogeneous Brownian file, ⟨r²(t)⟩_norm, the authors obtain a compact scaling law for the anomalous case: ⟨r²(t)⟩_anom =