# Narrow escape problem

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Narrow escape problem

The narrow escape problem is an ubiquitous problem in biology, biophysics and cellular biology.

The formulation is the following: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.

## Formulation

The motion of a particle is described by the Smoluchowski limit of the Langevin equation: $dX_t=\sqrt{2D} \, dB_t+\frac{1}{\gamma}F(x)$

where D is the diffusion coefficient of the particle, γ is the friction coefficient per unit of mass, F(x) the force per unit of mass, and Bt is a Brownian motion.

## Mean first passage time

A common question is to estimate the mean sojourn time of a particle diffusing in a bounded domain Ω before it escapes through a small absorbing window $\partial\Omega_a$ in its boundary $\partial\Omega$. The time is estimated asymptotically in the limit $\varepsilon= \frac{\partial\Omega_a}{\partial\Omega} \ll 1$

The probability density function (pdf) pε(x,t) is the probability of finding the particle at position x at time t.

The pdf satisfies the Fokker–Planck equation $\frac{\partial}{\partial t}p_{\varepsilon}(x,t)=D \Delta p_{\varepsilon}(x,t)-\frac{1}{\gamma}\nabla ( p_\varepsilon (x,t) F(x))$

with initial condition $p_\varepsilon (x,0) = \rho_0(x) \,$

and mixed Dirichlet–Neumann boundary conditions (t > 0) $p_\varepsilon (x,t) = 0\text{ for }x \in \partial\Omega_a$ $D\frac{\partial}{\partial n}p_\varepsilon (x,t) - \frac{p_\varepsilon (x,t)}{\gamma} F(x)\cdot n(x)=0 \text{ for }x \in \partial \Omega - \partial\Omega_a$

The function $u_\varepsilon (y) = \int_\Omega \int_0^\infty p_\varepsilon (x,t y) \, dt \, dx$

represents the mean sojourn time of particle, conditioned on the initial position y. It is the solution of the boundary value problem $D\Delta u_\varepsilon (y) + \frac{1}{\gamma}F(y)\cdot\nabla u_{\varepsilon}(y) = -1$ $u_\varepsilon (y) = 0\text{ for }y \in \partial\Omega_a$ $\frac{\partial u_\varepsilon (y)}{\partial n} = 0\text{ for }y \in \partial\Omega_r$

The solution depends on the dimension of the domain. For a particle diffusing on a disk $u_\varepsilon (y)=\frac{A}{\pi D}\ln\frac{1}{\varepsilon}+O(1)$

where A is the surface of the domain. The function $u_{\epsilon}(y)$does not depend on the initial position y, except for a small boundary layer near due to the asymptotic form. The first order term matters in dimension 2. For a circular disk of radius R, the mean escape time of a particle starting in the center is $E(\tau | x(0)=0 ) = \frac{R^2}{D}\left(\log\left(\frac{1}{\varepsilon}\right) + \log 2 + \frac{1}{4}+O(\varepsilon)\right).$

The escape time averaged with respect to a uniform initial distribution of the particle is given by $E(\tau ) = \frac{R^2}{D}\left(\log\left(\frac{1}{\varepsilon}\right) + \log 2 + \frac{1}{8} + O(\varepsilon)\right).$

The geometry of the small opening can affect the escape time: if the absorbing window is located at a corner of angle α , then $E\tau = \frac{|\Omega|}{\alpha D} \left[\log \frac{1}{\varepsilon} +O(1)\right].$

More surprising, near a cusp in a two dimensional domain, the escape time Eτ grows algebraically, rather than logarithmically: in the domain bounded between two tangent circles, the escape time is $E\tau = \frac{|\Omega|}{(d-1)D} \left(\frac{1}{\varepsilon} + O(1) \right),$

where d > 1 is the ratio of the radii. Finally, when the domain is an annulus, the escape time to a small opening located on the inner circle involves a second parameter which is $\beta = {\frac{R_1}{R_2}} < 1,$ the ratio of the inner to the outer radii, the escape time, averaged with respect to a uniform initial distribution, is $E\tau = \frac{(R_2^2-R_1^2)}D\left[\log \frac{1}{\varepsilon} + \log 2 + 2\beta^2 \right] +\frac{1}{2}\frac{R_2^2}{1-\beta^2}\log\frac{1}{\beta}- \frac{1}{4}R_2^2 + O(\varepsilon,\beta^4)R_2^2.$

This equation contains two terms of the asymptotic expansion of Eτ and is the angle of the absorbing boundary. The case β close to 1 remains open, and for general domains, the asymptotic expansion of the escape time remains an open problem. So does the problem of computing the escape time near a cusp point in three-dimensional domains. For Brownian motion in a field of force $F(x)\neq 0 \,$

the gap in the spectrum is not necessarily between the first and the second eigenvalues, depending on the relative size of the small hole and force barriers the particle has to overcome in order to escape. The escape stream is not necessarily Poissonian.

## Biological Applications

### Stochastic chemical reactions in microdomains 

The forward rate of chemical reactions is the reciprocal of the narrow escape time, which generalizes the classical Smoluchowski formula for Brownian particles located in an infinite medium. A Markov description can be used to estimate the binding and unbinding to a small number of sites.

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