First Passage Times for a Tracer Particle in Single File Diffusion and Fractional Brownian Motion

TL;DR

This paper analyzes the first passage time density of tracer particles in single-file diffusion and fractional Brownian motion, comparing simulations with analytical approximations to understand their universality and the effects of heterogeneity.

Contribution

It provides a detailed comparison of FPTDs in SFD and fBm, validating the universality class and highlighting limitations of analytical methods like WFA.

Findings

Willemski-Fixman approximation captures long-time power-law behavior for H >= 1/3.

Method of Images approximation does not accurately describe SFD FPTD.

Homogeneous SFD and fBm share the same scaled FPTD across all times.

Abstract

We investigate the full functional form of the first passage time density (FPTD) of a tracer particle in a single-file diffusion (SFD) system whose population is: (i) homogeneous, i.e., all particles having the same diffusion constant and (ii) heterogeneous, with diffusion constants drawn from a heavy-tailed power-law distribution. In parallel, the full FPTD for fractional Brownian motion [fBm - defined by the Hurst parameter, 0<H<1] is studied, of interest here as fBm and SFD systems belong to the same universality class. Extensive stochastic (non-Markovian) SFD and fBm simulations are performed and compared to two analytical Markovian techniques: the Method of Images approximation (MIA) and the Willemski-Fixman approximation (WFA). We find that the MIA cannot approximate well any temporal scale of the SFD FPTD. Our exact inversion of the Willemski-Fixman integral equation captures the long-time power-law exponent, when H >= 1/3, as predicted by Molchan [1999] for fBm. When H<1/3, which includes homogeneous SFD (H=1/4), and heterogeneous SFD (H<1/4), the WFA fails to agree with any temporal scale of the simulations and Molchan's long-time result. SFD systems are compared to their fBm counter parts; and in the homogeneous system both scaled FPTDs agree on all temporal scales including also, the result by Molchan, thus affirming that SFD and fBm dynamics belong to the same universality class. In the heterogeneous case SFD and fBm results for heterogeneity-averaged FPTDs agree in the asymptotic time limit. The non-averaged heterogeneous SFD systems display a lack of self-averaging. An exponential with a power-law argument, multiplied by a power-law pre-factor is shown to describe well the FPTD for all times for homogeneous SFD and sub-diffusive fBm systems.