TY - JOUR

T1 - Narrow escape, part II

T2 - The circular disk

AU - Singer, A.

AU - Schuss, Z.

AU - Holcman, D.

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.” The authors thank R. S. Eisenberg for critical review of the manuscript.

PY - 2006/2

Y1 - 2006/2

N2 - We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ωa, which is absorbing. As ε = |∂Ωa|/|∂Ω decreases to zero the mean time to absorption in ∂Ωa, denoted Eτ, becomes infinite. The narrow escape problem is to find an asymptotic expansion of Eτ for ε ≪ 1. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general domain, Eτ = |Ω|/Dπ[log 1/ε + O(1)], (D is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is E[τ|x(0) = 0] = R2/D[log 1/ε + log 2 + 1/4 + Oε]. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because log 1/ε is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into ∂Ωa at the endpoints of ∂Ωa, and find the value of the flux near the center of the window.

AB - We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ωa, which is absorbing. As ε = |∂Ωa|/|∂Ω decreases to zero the mean time to absorption in ∂Ωa, denoted Eτ, becomes infinite. The narrow escape problem is to find an asymptotic expansion of Eτ for ε ≪ 1. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general domain, Eτ = |Ω|/Dπ[log 1/ε + O(1)], (D is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is E[τ|x(0) = 0] = R2/D[log 1/ε + log 2 + 1/4 + Oε]. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because log 1/ε is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into ∂Ωa at the endpoints of ∂Ωa, and find the value of the flux near the center of the window.

KW - Exit problem

KW - Planar Brownian motion

KW - Singular perturbations

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U2 - 10.1007/s10955-005-8027-5

DO - 10.1007/s10955-005-8027-5

M3 - Article

AN - SCOPUS:33644532542

SN - 0022-4715

VL - 122

SP - 465

EP - 489

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3

ER -