TY - JOUR

T1 - Narrow escape, part I

AU - Singer, A.

AU - Schuss, Z.

AU - Holcman, D.

AU - Eisenberg, R. S.

N1 - Funding Information:
This research was partially supported by research grants from the Israel Science Foundation, US-Israel Binational Science Foundation, and the NIH Grant No. UPSHS 5 RO1 GM 067241. D. H. is incumbent to the Madeleine Haas Russell Career Development Chair, his research is partially supported by the program “Chaire d’Excellence”.

PY - 2006/2

Y1 - 2006/2

N2 - A Brownian particle with diffusion coefficient D is confined to a bounded domain Ω by a reflecting boundary, except for a small absorbing window ∂Ωa. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. In the three-dimensional case, we construct an asymptotic approximation when the window is an ellipse, assuming the large semi axis a is much smaller than |Ω|1/3 (|Ω| is the volume), and show that the mean escape time is Eτ ∼ |Ω|/2π Da K(e), where e is the eccentricity and K(·) is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula Eτ ∼ |Ω|/4aD, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion Eτ = |Ω|/4aD[1 + a/R log R/a + O(a/R)]. This result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. If Ω is a two-dimensional bounded Riemannian manifold with metric g and ε = |∂Ωa|g ≪ 1, we show that Eτ = |Ω|g/Dπ[log 1/ε + O(1)]. This result is applicable to diffusion in membrane surfaces.

AB - A Brownian particle with diffusion coefficient D is confined to a bounded domain Ω by a reflecting boundary, except for a small absorbing window ∂Ωa. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. In the three-dimensional case, we construct an asymptotic approximation when the window is an ellipse, assuming the large semi axis a is much smaller than |Ω|1/3 (|Ω| is the volume), and show that the mean escape time is Eτ ∼ |Ω|/2π Da K(e), where e is the eccentricity and K(·) is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula Eτ ∼ |Ω|/4aD, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion Eτ = |Ω|/4aD[1 + a/R log R/a + O(a/R)]. This result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. If Ω is a two-dimensional bounded Riemannian manifold with metric g and ε = |∂Ωa|g ≪ 1, we show that Eτ = |Ω|g/Dπ[log 1/ε + O(1)]. This result is applicable to diffusion in membrane surfaces.

KW - Brownian motion

KW - Exit problem

KW - Singular perturbations

UR - http://www.scopus.com/inward/record.url?scp=33644500473&partnerID=8YFLogxK

U2 - 10.1007/s10955-005-8026-6

DO - 10.1007/s10955-005-8026-6

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AN - SCOPUS:33644500473

SN - 0022-4715

VL - 122

SP - 437

EP - 463

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3

ER -