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

- 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(omega) distribution of the random variable omega = tau(1)/(tau(1) + tau(2)), which is a measure for how similar the first passage times tau(1) and tau(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(omega) 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(omega), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that largeWe 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(omega) distribution of the random variable omega = tau(1)/(tau(1) + tau(2)), which is a measure for how similar the first passage times tau(1) and tau(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(omega) 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(omega), 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.…

Author details: | Thiago G. Mattos, Carlos Mejia-Monasterio, Ralf MetzlerORCiDGND, Gleb Oshanin |
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DOI: | https://doi.org/10.1103/PhysRevE.86.031143 |

ISSN: | 1539-3755 |

Parent title (English): | Physical review : E, Statistical, nonlinear and soft matter physics |

Publisher: | American Physical Society |

Place of publication: | College Park |

Publication type: | Article |

Language: | English |

Year of first publication: | 2012 |

Year of completion: | 2012 |

Release date: | 2017/03/26 |

Volume: | 86 |

Issue: | 3 |

Page number: | 8 |

Funder: | European Science Foundation; Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme [PIRSES-GA-2010-269139]; Academy of Finland within the FiDiPro program; ESF Research Network "Exploring the Physics of Small Devices" |

Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |

Peer review: | Referiert |