^{1,a)}and Tobias Ambjörnsson

^{1}

### 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, *H* ∈ (0, 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 [Commun. Math. Phys.205, 97 (1999)10.1007/s002200050669] 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.

We would like to acknowledge the following persons for their insightful discussions on various aspects of this investigation: Michael Lomholt, Ludvig Lizana, Michaela Schad, and Sigurður Æ. Jónsson. Computer time was provided by LUNARC at Lund University.

I. INTRODUCTION

II. APPROXIMATING THE FPTD

A. The Willemski-Fixman approximation

B. The method of images approximation

C. Conjecture for the FPTD

III. SIMULATIONS

A. Simulating SFD

B. Simulating fBm

C. Collapsing data

IV. RESULTS

A. SFD simulations

B. Long-time SFD asymptotics

C. SFD versus fBm

D. SFD data versus approximations

E. fBm and WFA

F. Conjecture results

V. SUMMARY AND DISCUSSION

## Figures

Schematic representation of the lattice simulations of a SFD system. All particles (including the tracer—here in black) move under Brownian motion and are hard-core (the particles cannot occupy the same lattice site), meaning that they cannot pass each other, keeping their order for all times. Hence the tracer is in the center of all other particles for all times. The top panel shows the start of the simulation in thermal equilibrium, whereas the bottom panel shows that after some time, the tracer has achieved a first passage event.

Schematic representation of the lattice simulations of a SFD system. All particles (including the tracer—here in black) move under Brownian motion and are hard-core (the particles cannot occupy the same lattice site), meaning that they cannot pass each other, keeping their order for all times. Hence the tracer is in the center of all other particles for all times. The top panel shows the start of the simulation in thermal equilibrium, whereas the bottom panel shows that after some time, the tracer has achieved a first passage event.

Collapsed (log-log) plot of FPTD for a *homogeneous* SFD system with different absorption points and fBm FPTD (for simulation parameters see Tables I and III, respectively). Immediately, the collapsed plot shows that homogeneous SFD and fBm have the same FPTD dynamics over all time frames in the correct scaling, Sec. III. The MIA, Eq. (12), and the WFA, Eq. (5), are shown for comparison, both of which show poor agreement. The averaged proposed functional form, , Eq. (13), is constructed by collapsing all simulated data and fitting (see Appendix C), keeping the power-law exponent in the pre-factor fixed to *H* − 2; with *H* = 1/4, and setting *C* according to Eq. (14). The parameters (Table II) were then averaged and a single mean curve plotted. This conjecture shows excellent agreement with both anomalous diffusive systems on all time scales.^{63} The Molchan long-time prediction^{52} is given to guide the eye. The fBm FPTD consists of two different absorption points (Δ*x* = 50, Δ*x* = 100, 6 × 10^{4} simulations)^{64} and displays good agreement with SFD results. (Inset) The short-time regime (linear axes) agreement between homogeneous SFD, fBm, and . Remaining simulation details are presented in Tables I, III, and II. The subscript *s* on each axis variable denotes “scaled,” namely *t* _{ s } = ϖ*t* and *f* _{ s } = ϖ^{−1} *f*(*t*), where ϖ = *C* ^{1/2H }(Δ*x*)^{−1/H }.

Collapsed (log-log) plot of FPTD for a *homogeneous* SFD system with different absorption points and fBm FPTD (for simulation parameters see Tables I and III, respectively). Immediately, the collapsed plot shows that homogeneous SFD and fBm have the same FPTD dynamics over all time frames in the correct scaling, Sec. III. The MIA, Eq. (12), and the WFA, Eq. (5), are shown for comparison, both of which show poor agreement. The averaged proposed functional form, , Eq. (13), is constructed by collapsing all simulated data and fitting (see Appendix C), keeping the power-law exponent in the pre-factor fixed to *H* − 2; with *H* = 1/4, and setting *C* according to Eq. (14). The parameters (Table II) were then averaged and a single mean curve plotted. This conjecture shows excellent agreement with both anomalous diffusive systems on all time scales.^{63} The Molchan long-time prediction^{52} is given to guide the eye. The fBm FPTD consists of two different absorption points (Δ*x* = 50, Δ*x* = 100, 6 × 10^{4} simulations)^{64} and displays good agreement with SFD results. (Inset) The short-time regime (linear axes) agreement between homogeneous SFD, fBm, and . Remaining simulation details are presented in Tables I, III, and II. The subscript *s* on each axis variable denotes “scaled,” namely *t* _{ s } = ϖ*t* and *f* _{ s } = ϖ^{−1} *f*(*t*), where ϖ = *C* ^{1/2H }(Δ*x*)^{−1/H }.

Collapsed plot of FPTD for a *heterogeneous* SFD system (Table I) with different absorption points. Also within, fBm FPTD is plotted for *H* = 1/6, see Table III. The long-time dynamics for both systems agree very well with each other and also with Molchan's equation, Eq. (1). In the very short-time the systems part, most likely because of the complex nature of heterogeneity-averaged SFD systems (see inset). The MIA, Eq. (12), and the WFA, Eq. (5), are shown for comparison. Both approximations show ill agreement with the simulated data for both systems and the theory. , Eq. (13), was fitted to the fBm data (displaying excellent agreement), Table III, as opposed to the SFD data, due to its poor fit (discussion in inset caption here, and Sec. IV).^{65} The data are scaled (signified by subscript *s*) using Eq. (15), with *H* = 1/6 (since α = 1/2)—see Fig. 2 caption for further explanation of the scaling. All remaining simulation details are presented in Tables I and III. (Inset) Collapsed plot of the FPTD for 3 different sets of heterogeneous friction constants, kept constant for each simulation. For this non-averaged case we use *x* _{ c } = 50 for all simulations, with all other system parameters displayed in Table I. The inset illustrates the fact that no self-averaging takes place in this heterogeneity-averaged system (see Sec. IV for further discussion).

Collapsed plot of FPTD for a *heterogeneous* SFD system (Table I) with different absorption points. Also within, fBm FPTD is plotted for *H* = 1/6, see Table III. The long-time dynamics for both systems agree very well with each other and also with Molchan's equation, Eq. (1). In the very short-time the systems part, most likely because of the complex nature of heterogeneity-averaged SFD systems (see inset). The MIA, Eq. (12), and the WFA, Eq. (5), are shown for comparison. Both approximations show ill agreement with the simulated data for both systems and the theory. , Eq. (13), was fitted to the fBm data (displaying excellent agreement), Table III, as opposed to the SFD data, due to its poor fit (discussion in inset caption here, and Sec. IV).^{65} The data are scaled (signified by subscript *s*) using Eq. (15), with *H* = 1/6 (since α = 1/2)—see Fig. 2 caption for further explanation of the scaling. All remaining simulation details are presented in Tables I and III. (Inset) Collapsed plot of the FPTD for 3 different sets of heterogeneous friction constants, kept constant for each simulation. For this non-averaged case we use *x* _{ c } = 50 for all simulations, with all other system parameters displayed in Table I. The inset illustrates the fact that no self-averaging takes place in this heterogeneity-averaged system (see Sec. IV for further discussion).

Collapsed plot of FPTD of sub-diffusive fBm with *H* = 0.35 (top), *H* = 0.45 (bottom) (simulation parameters: Table III). Both panels show that the WFA has the correct heavy-tailed gradient (compare to Molchan's prediction). The fBm also agrees with Eq. (1), as the tail is fixed (*H* − 2) and the data are modeled well by our conjecture, Eq. (13). Our conjecture models all time scales well, see Table III for quantitative details.

Collapsed plot of FPTD of sub-diffusive fBm with *H* = 0.35 (top), *H* = 0.45 (bottom) (simulation parameters: Table III). Both panels show that the WFA has the correct heavy-tailed gradient (compare to Molchan's prediction). The fBm also agrees with Eq. (1), as the tail is fixed (*H* − 2) and the data are modeled well by our conjecture, Eq. (13). Our conjecture models all time scales well, see Table III for quantitative details.

Collapsed plot of FPTD for super-diffusive fBm, *H* = 3/4, see Table III. Both the WFA and the fBm data only agree with Eq. (1) in the long time, as expected. Using the conjecture, Eq. (13), keeping *H* fixed, the 2 remaining degrees of freedom (γ, β, see Table III) cannot account for the discrepancy seen between the our conjectured FPTD and the fBm data. It is apparent that the MIA fails on all time scales. (Inset) Collapsed plots on linear axes illustrate short-time dynamics (fBm data as crosses).

Collapsed plot of FPTD for super-diffusive fBm, *H* = 3/4, see Table III. Both the WFA and the fBm data only agree with Eq. (1) in the long time, as expected. Using the conjecture, Eq. (13), keeping *H* fixed, the 2 remaining degrees of freedom (γ, β, see Table III) cannot account for the discrepancy seen between the our conjectured FPTD and the fBm data. It is apparent that the MIA fails on all time scales. (Inset) Collapsed plots on linear axes illustrate short-time dynamics (fBm data as crosses).

## Tables

SFD simulation parameters. Not applicable is abbreviated to N.A.

SFD simulation parameters. Not applicable is abbreviated to N.A.

Homogeneous fit parameters. is the normalized chi-squared parameter. Raw data placed into 30 natural log-bins (see Appendix C).

Homogeneous fit parameters. is the normalized chi-squared parameter. Raw data placed into 30 natural log-bins (see Appendix C).

fBm simulation and fit parameters. Each simulation has: *C* = 5; ; *N* = 6 × 10^{4}. Raw data placed in 50 natural log-bins, fitted to parameters (γ, β) (see Appendix C) to test the applicability of our simple conjecture, Eq. (13). is the normalized chi-squared parameter.

fBm simulation and fit parameters. Each simulation has: *C* = 5; ; *N* = 6 × 10^{4}. Raw data placed in 50 natural log-bins, fitted to parameters (γ, β) (see Appendix C) to test the applicability of our simple conjecture, Eq. (13). is the normalized chi-squared parameter.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content