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Analysis and Mathematical Physics

, Volume 5, Issue 4, pp 321–362 | Cite as

Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior

  • O. Bénichou
  • D. S. Grebenkov
  • L. Hillairet
  • L. Phun
  • R. Voituriez
  • M. ZinsmeisterEmail author
Article

Abstract

We consider a model of surface-mediated diffusion with alternating phases of bulk and surface diffusion for two geometries: the disk and rectangles. We develop a spectral approach to derive an exact formula for the mean exit time of a particle through a hole on the boundary. The spectral representation of the mean exit time through the eigenvalues of an appropriate self-adjoint operator is particularly well-suited to investigate the asymptotic behavior in the limit of large desorption rate \(\lambda \). For a point-like target, we show that the mean exit time diverges as \(\sqrt{\lambda }\). For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as \(\lambda \) tends to infinity. That implies that the pure bulk diffusion is never an optimal search strategy. We also investigate the influence of rectangle elongation onto the mean exit time, in particular, the dependence of the critical ratio of bulk and surface diffusion coefficients on the rectangle aspect ratio. We show that the intermittent search strategy can significantly outperform pure surface diffusion for elongated rectangles.

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • O. Bénichou
    • 1
  • D. S. Grebenkov
    • 2
  • L. Hillairet
    • 3
  • L. Phun
    • 3
  • R. Voituriez
    • 1
  • M. Zinsmeister
    • 3
    Email author
  1. 1.Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600)CNRS/UPMCParis CedexFrance
  2. 2.Laboratoire de Physique de la Matière Condensée (UMR 7643) CNRSEcole PolytechniquePalaiseauFrance
  3. 3.MAPMO (UMR 6628)Université d’OrléansOrléansFrance

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