The mathematical 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.
This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.
[10][11] The narrow escape problem was proposed in the context of biology and biophysics by D. Holcman and Z. Schuss,[12] and later on with A.Singer and led to the narrow escape theory in applied mathematics and computational biology.
[13][14][15] The motion of a particle is described by the Smoluchowski limit of the Langevin equation:[16][17]
A common question is to estimate the mean sojourn time of a particle diffusing in a bounded domain
represents the mean sojourn time of particle, conditioned on the initial position
The first order term matters in dimension 2: for a circular disk of radius
, the mean escape time of a particle starting in the center is
The escape time averaged with respect to a uniform initial distribution of the particle is given by
The geometry of the small opening can affect the escape time: if the absorbing window is located at a corner of angle
More surprising, near a cusp in a two dimensional domain, the escape time
grows algebraically, rather than logarithmically: in the domain bounded between two tangent circles, the escape time is:
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
the ratio of the inner to the outer radii, the escape time, averaged with respect to a uniform initial distribution, is:
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.
the gap in the spectrum is not necessarily small between the first and the second eigenvalues, depending on the relative size of the small hole and the force barriers, the particle has to overcome in order to escape.
The first part of the theorem is a classical result, while the average variance was proved in 2011 by Carey Caginalp and Xinfu Chen.
The following closed form result[18][19][20] gives an exact solution that confirms these asymptotic formulae and extends them to gates that are not necessarily small.
Theorem (Carey Caginalp and Xinfu Chen Closed Formula) — In 2-D, with points identified by complex numbers, let
Another set of results concerns the probability density of the location of exit.
[19] Theorem (Carey Caginalp and Xinfu Chen Probability Density) — The probability density of the location of a particle at time of its exit is given by
, the probability that a particle, starting either at the origin or uniformly distributed in
In simulation there is a random error due to the statistical sampling process.
There is also a discretization error due to the finite size approximation of the step size in approximating the Brownian motion.
Using the exact result quoted above for the particular case of the circle, it is possible to make a careful comparison of the exact solution with the numerical solution.
[21][22] This illuminates the distinction between finite steps and continuous diffusion.
A distribution of exit locations was also obtained through simulations for this problem.
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.