First Passage Distributions in a Collective Model of Anomalous Diffusion with Tunable Exponent

TL;DR

This paper models anomalous diffusion using a superposition of linear modes with broad relaxation times, analyzing first passage distributions and absorption properties for different subdiffusion exponents.

Contribution

It introduces a model linking mode superposition to anomalous diffusion and characterizes first passage distributions with variable exponents through numerical simulations.

Findings

Absorption probability densities vary with the anomalous diffusion exponent.

Distributions exhibit features characterized by exponents dependent on alpha.

Quantitative differences are observed between polymer translocation and tagged monomer diffusion.

Abstract

We consider a model system in which anomalous diffusion is generated by superposition of underlying linear modes with a broad range of relaxation times. In the language of Gaussian polymers, our model corresponds to Rouse (Fourier) modes whose friction coefficients scale as wavenumber to the power $2-z$. A single (tagged) monomer then executes subdiffusion over a broad range of time scales, and its mean square displacement increases as $t^\alpha$ with $\alpha=1/z$. To demonstrate non-trivial aspects of the model, we numerically study the absorption of the tagged particle in one dimension near an absorbing boundary or in the interval between two such boundaries. We obtain absorption probability densities as a function of time, as well as the position-dependent distribution for unabsorbed particles, at several values of $\alpha$. Each of these properties has features characterized by exponents that depend on $\alpha$. Characteristic distributions found for different values of $\alpha$ have similar qualitative features, but are not simply related quantitatively. Comparison of the motion of translocation coordinate of a polymer moving through a pore in a membrane with the diffusing tagged monomer with identical $\alpha$ also reveals quantitative differences.