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Mean first-passage times of non-Markovian random walkers in confinement

Nature volume 534, pages 356359 (16 June 2016) | Download Citation

Abstract

The first-passage time, defined as the time a random walker takes to reach a target point in a confining domain, is a key quantity in the theory of stochastic processes1. Its importance comes from its crucial role in quantifying the efficiency of processes as varied as diffusion-limited reactions2,3, target search processes4 or the spread of diseases5. Most methods of determining the properties of first-passage time in confined domains have been limited to Markovian (memoryless) processes3,6,7. However, as soon as the random walker interacts with its environment, memory effects cannot be neglected: that is, the future motion of the random walker does not depend only on its current position, but also on its past trajectory. Examples of non-Markovian dynamics include single-file diffusion in narrow channels8, or the motion of a tracer particle either attached to a polymeric chain9 or diffusing in simple10 or complex fluids such as nematics11, dense soft colloids12 or viscoelastic solutions13,14. Here we introduce an analytical approach to calculate, in the limit of a large confining volume, the mean first-passage time of a Gaussian non-Markovian random walker to a target. The non-Markovian features of the dynamics are encompassed by determining the statistical properties of the fictitious trajectory that the random walker would follow after the first-passage event takes place, which are shown to govern the first-passage time kinetics. This analysis is applicable to a broad range of stochastic processes, which may be correlated at long times. Our theoretical predictions are confirmed by numerical simulations for several examples of non-Markovian processes, including the case of fractional Brownian motion in one and higher dimensions. These results reveal, on the basis of Gaussian processes, the importance of memory effects in first-passage statistics of non-Markovian random walkers in confinement.

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Acknowledgements

This work was supported by ERC grant FPTOpt-277998.

Author information

Affiliations

  1. Laboratoire Ondes et Matière d’Aquitaine, University of Bordeaux, Unité Mixte de Recherche 5798, CNRS, F-33400 Talence, France

    • T. Guérin
  2. Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France

    • N. Levernier
    • , O. Bénichou
    •  & R. Voituriez
  3. Laboratoire Jean Perrin, CNRS/Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France

    • R. Voituriez

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All authors contributed equally to this work.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to O. Bénichou or R. Voituriez.

Reviewer Information Nature thanks K. Lindenberg and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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https://doi.org/10.1038/nature18272

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