Imperfect narrow escape problem

T. Guérin, M. Dolgushev, O. Bénichou, and R. Voituriez
Phys. Rev. E 107, 034134 – Published 23 March 2023

Abstract

We consider the kinetics of the imperfect narrow escape problem, i.e., the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity κ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semianalytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large-reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the “constant flux approximation”; we show that this approximation turns out to give exactly the next-to-leading-order term of the small-reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above-mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

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  • Received 27 September 2022
  • Accepted 2 March 2023

DOI:https://doi.org/10.1103/PhysRevE.107.034134

©2023 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

T. Guérin1, M. Dolgushev2, O. Bénichou2, and R. Voituriez2,3

  • 1Laboratoire Ondes et Matière d'Aquitaine, CNRS, UMR 5798, Université de Bordeaux, F-33400 Talence, France
  • 2Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), F-75005 Paris, France
  • 3Sorbonne Université, CNRS, Laboratoire Jean Perrin (LJP), F-75005 Paris, France

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Issue

Vol. 107, Iss. 3 — March 2023

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