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

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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Figure 1: Mean FPT of a random walker in confinement.
Figure 2: Mean FPT of one-dimensional non-Markovian random walks.
Figure 3: Mean FPT of two- and three-dimensional non-Markovian random walks.

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Acknowledgements

This work was supported by ERC grant FPTOpt-277998.

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Correspondence to O. Bénichou or R. Voituriez.

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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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Guérin, T., Levernier, N., Bénichou, O. et al. Mean first-passage times of non-Markovian random walkers in confinement. Nature 534, 356–359 (2016). https://doi.org/10.1038/nature18272

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