Narrow Escape, Part II: The Circular Disk
By A. Singer, Zeev Schuss, David Holcman. 2004
139 citations recorded by the source corpus.
We consider Brownian motion in a circular disk Ω, whose boundary $$\partial\Omega$$ is reflecting, except for a small arc, $$\partial\Omega_a$$, which is absorbing. As $$\varepsilon=|\partial\Omega_a|/|\partial \Omega|$$ decreases to zero the mean time to absorption in $$\partial\Omega_a$$, denoted $$E\tau$$, becomes infinite. The narrow escape problem is to find an asymptotic expansion of $$E\tau$$ for $$\varepsilon\ll1$$. 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, $$E\tau = {\frac{|\Omega|}{D\pi}}\big[\log{\frac{1}{\varepsilon}}+O(1)\big],$$ ($$D$$ 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 $$E[\tau\,|\,{\boldmath x}(0)={\bf 0}]={\frac{R^2}{D}}\big[\log{\frac{1}{\varepsilon}}+ \log 2 +{ \frac{1}{4}} + O(\varepsilon)\big]$$. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because $$\log{\frac{1}{\varepsilon}}$$ is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into $$\partial\Omega_a$$ at the endpoints of $$\partial\Omega_a$$, and find the value of the flux near the center of the window.
Subfields: Applied Mathematics, Molecular Biology
View Paper: https://www.semanticscholar.org/paper/17c60c1f96cbf2417764e02bbbf84d1320625f0a
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In Applied Mathematics
In Molecular Biology
Citations in the corpus
- Time scale of diffusion in molecular and cellular biology (2014, background): “In dimension two, the boundary layer expansion can be computed explicitly (Singer et al 2006a), whereas short-time asymptotics are much harder to estimate (see, for example, the short-time asymptotics given in Schuss and Spivak (2005) for pure diffusion), because they requires the knowledge of all…”
- Stochastic models of intracellular transport (2013, methodology): “…of singular perturbation theory and Green’s function methods for a variety of geometries in two and three dimensions (Benichou and Voituriez, 2008; Chevalier et al., 2011; Cheviakov et al., 2010; Holcman and Schuss, 2004; Pillay et al., 2010; Schuss et al., 2007; Singer et al., 2006a,b).”