First passages in bounded domains: When is the mean first passage time meaningful?

Thiago G. Mattos, Carlos Mejía-Monasterio, Ralf Metzler, and Gleb Oshanin

Phys. Rev. E 86, 031143 – Published 27 September 2012

Abstract

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability $P (ω)$ distribution of the random variable $ω = τ_{1} / (τ_{1} + τ_{2})$ , which is a measure for how similar the first passage times $τ_{1}$ and $τ_{2}$ are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether $P (ω)$ represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of $P (ω)$ , characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

Received 7 August 2012

DOI:https://doi.org/10.1103/PhysRevE.86.031143

Authors & Affiliations

Thiago G. Mattos^*

Max Planck Institute for Intelligent Systems, Heisenbergstraße 3, 70569 Stuttgart, Germany

Carlos Mejía-Monasterio

Laboratory of Physical Properties, Technical University of Madrid, Avenida Complutense s/n, 28040 Madrid, Spain and Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 Helsinki, Finland

Ralf Metzler

Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany and Physics Department, Tampere University of Technology, Korkeakoulunkatu 3, FIN-33101 Tampere, Finland

Gleb Oshanin

Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France

^*Corresponding author; tgmattos@is.mpg.de

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Issue

Vol. 86, Iss. 3 — September 2012

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Images

Figure 1
Trajectories of two Brownian walkers starting at the same initial position $(r_{0}, θ_{0})$ inside a bounded pie-wedge domain with opening angle $Θ$ as well as absorbing radial boundaries (dashed lines) and reflecting boundary (solid line) at $r = 1$ . The values of the first passage times to the adsorbing boundaries are used to construct the random variable $ω$ .Reuse & Permissions
Figure 2
(a) First passage time distribution $Ψ_{(ρ_{0}, θ_{0})} (τ)$ for a Brownian walk in a pie-wedge domain with opening angle $Θ = π / 2$ . Different colors and symbols correspond to different starting points: $(ρ_{0} = 0.76, θ_{0} = - 0.38 Θ$ ) (dark blue/triangles), $(0.76, - 0.23 Θ)$ (beige/squares), and $(0.76, 0)$ (light blue/circles). The dashed straight line indicates the intermediate power-law decay $Ψ (τ) \sim 1 / τ^{2}$ , Eq. (15). (b) The corresponding distribution $P (ω)$ with the same color and symbol coding. In grey scale, dark blue corresponds to the darkest shade of grey, light blue to dark grey, and beige to light grey.Reuse & Permissions
Figure 3
Dependence of the parameter $χ$ on the starting angle $θ_{0}$ for fixed $ρ_{0} = 0.76$ in a pie-wedge domain with opening angle $Θ = π / 2$ . The three insets show the shape of the uniformity distribution $P (ω)$ for the $θ_{0}$ values indicated by the arrows.Reuse & Permissions
Figure 4
Phase chart for the shape of the uniformity distribution $P (ω)$ in three different pie-wedge domains: (a) $Θ = 3 π / 4$ , (b) $Θ = π / 2$ , (c) $Θ = π / 3$ . The color of the symbols is light blue if $χ < - χ_{☆}$ , dark blue if $χ > χ_{☆}$ , and beige if $| χ | < χ_{☆}$ , where we chose $χ_{☆} = 0.25$ . For each initial location, $P (ω)$ was computed from a sample of $N = 10^{5}$ random trajectories.Reuse & Permissions
Figure 5
Phase chart for the shape of the uniformity distribution $P (ω)$ for a Brownian walker in the unit circle with reflective BCs (solid lines) and an aperture of size $Θ$ (absorbing BC, dashed lines). (a) $Θ = π / 18$ , (b) $Θ = π / 2$ , (c) $Θ = π$ , and (d) $Θ = 2 π$ . Starting locations are colored light blue if $χ < - χ_{☆}$ , dark blue if $χ > χ_{☆}$ , and beige if $| χ | < χ_{☆}$ , where $χ_{☆} = 1$ . In the lower panel, the relative error of the FPT $ɛ$ is shown as a function of the initial radial position $r$ along the horizontal diameter of the unit circle with $Θ = π / 2$ , panel (b): $r = 0$ is associated with the center of the circle and the point $r = 1$ lies over the absorbing boundary.Reuse & Permissions
Figure 6
Phase chart of the shape of the uniformity distribution $P (ω)$ for symmetric triangles with central angle $π / 2$ (a) and $2 π / 3$ (c), and for the asymmetric triangle with angles $2 π / 3, π / 4, π / 12$ (b). The color of the symbols is light blue if $χ < - χ_{☆}$ , dark blue if $χ > χ_{☆}$ , and beige if $| χ | < χ_{☆}$ , with $χ_{☆} = 0.25$ .Reuse & Permissions

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