Narrow Escape, Part II: The Circular Disk

Abstract

We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ω a , which is absorbing. As ε = |∂Ω a |/|∂Ω| decreases to zero the mean time to absorption in ∂Ω a , 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 smooth 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] = R 2 D log 1 ε + log 2 + 1 4 + O(ε) . 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 ∂Ω a at the endpoints of ∂Ω a , and find the value of the flux near the center of the window.

Activation through a narrow opening

“…Summary and discussion. The narrow escape problem of a Brownian particle through a small absorbing window in an otherwise reflecting boundary was discussed in [8], [17], [18], and [19]. Here we solve the narrow escape problem for a Brownian particle in a force field.…”

“…The first is due to the potential, while the second is due to geometry of the absorbing window alone. Unlike the free diffusion case [17], [18], [19], geometrical properties of the domain, such as its volume, are not included in the leading order asymptotics of the reaction rate.…”

“…In [18], [19] we have calculated the O(1) term for diffusion in a circular disk, in a circular annulus, and on a sphere. These results extend in a straightforward way to domains that can be mapped conformally onto these shapes (e.g., all simply connected planar domains).…”

“…For the special case of the mixed Neumann problem with a small Dirichlet window in the boundary, the leading term obtained in [17], [18], [19] can be obtained by the application of the matched asymptotics expansion to this problem. In this paper we generalize the method of [17], [18], [19] to obtain the leading term for the corresponding boundary value problem for the Fokker-Planck operator, though matched asymptotics can be applied to this problem as well. The advantage of our method, as demonstrated in [17], is that it reveals the order of magnitude of the second term in three dimensions, while the matched asymptotics method does not indicate this in a simple way.…”

“…The same setup also describes the process of trafficking receptors on biological membranes [8]. The escape of a free Brownian motion (without drift) through a small window was discussed in [17], [18], [19]. Here we consider the narrow escape problem for a Brownian motion in a field of force.…”