# Narrow escape and leakage of Brownian particles

###### Abstract

Questions of flux regulation in biological cells raise renewed interest in the narrow escape problem. The often inadequate expansions of the narrow escape time are due to a not so well known fact that the boundary singularity of Green’s function for Poisson’s equation with Neumann and mixed Dirichlet-Neumann boundary conditions in three-dimensions contains a logarithmic singularity. Using this fact, we find the second term in the expansion of the narrow escape time and in the expansion of the principal eigenvalue of the Laplace equation with mixed Dirichlet-Neumann boundary conditions, with small Dirichlet and large Neumann parts. We also find the leakage flux of Brownian particles that diffuse from a source to an absorbing target on a reflecting boundary of a domain, if a small perforation is made in the reflecting boundary.

###### pacs:

05.40.-Jc., 87.10.-e^{†}

^{†}preprint: APS/123-PRE

## I Introduction

The narrow escape problem in diffusion theory, which goes back to Lord Rayleigh Rayleigh , is to calculate the mean first passage time, also called the narrow escape time (NET), of a Brownian particle to a small absorbing window on the otherwise reflecting boundary of a bounded domain. The renewed interest in the small hole problem is due to its relevance in molecular biology and biophysics. The small hole often represents a small target on a cellular membrane, such as a protein channel, which is a target for ions Hille , a receptor for neurotransmitter molecules in a neuronal synapse Nicoll , a narrow neck in the neuronal spine, which is a target for calcium ions EJN , and so on. The physiological role of the small hole is often to regulate flux, which carries a physiological signal. For example, the NMDA channels in the post synaptic membrane in the neuronal cleft are small targets for diffusing glutamate molecules released from a vesicle at the pre synaptic membrane. The leakage problem here is to find the fraction of the released molecules that reach the channels before being irreversibly absorbed by the surrounding medium (e.g., glia transporters) Patton , Gallo (see also http://en.wikipedia.org/wiki/Chemical_synapse). The position and the number of the NMDA and AMPA receptors regulate synaptic transmission and is believed to be a part of coding memory Nicoll , Malinow .

The narrow escape problem is connected to that of calculating the principal eigenvalue of the mixed Dirichlet-Neumann problem for the Laplace equation in a bounded domain, whose Dirichlet boundary is only a small patch on the otherwise Neumann boundary. Specifically, the principal eigenvalue is asymptotically the reciprocal of the narrow escape time in the limit of shrinking patch.

The recent history of the problem begins with the work of Ward, Keller, Henshaw, Van De Velde, Kolokolnikov, and Titworth Ward1 ; Ward2 ; Ward3 ; Ward4 on the principal eigenvalue and is based on boundary layer theory and matched asymptotics, in which the boundary layer equation is the classical electrified disk problem, solved explicitly by Weber in 1873 Weber ; Jackson . The work of Holcman, Singer, Schuss, and Eisenberg HS ; SSH1 ; SSH2 ; SSH3 ; PNAS ; PLA-holes ; JPA-holes ; NarrowEscape4 on the NET for diffusion with and without a force field and for several small windows and its applications in biology, is based on the known structure of the singularity of Neumann’s function at the boundary Jackson ; CourantHilbert ; Kellog ; Garabedian and on the Helmholtz integral equation Helmholtz (see Lurie ). The most recent work of Bénichou and Voituriez Benichou on the NET in diffusion and anomalous diffusion finds the dependence of the NET on the initial point inside the boundary layer and finds the scaling laws for sub-diffusions. In these papers the leading term in the asymptotic expansion was calculated in the shrinking window limit.

Neither the second term, nor its order of magnitude were calculated for the three dimensional problem, except in the case of a spherical domain with a small circular absorbing window, where an explicit solution was constructed by a generalization of Collins’ method (an error in the coefficient of the second term, given in SSH1 , is corrected here). The difficulty in finding, or even estimating, the second term can be attributed to the practically unknown (to mathematicians and physicists) structure of the singularity of Neumann’s function on the boundary. While classical texts in partial differential equations and in classical mathematical physics Jackson ; CourantHilbert ; Kellog ; Garabedian mention only the leading order singularity of the Newtonian potential and a regular correction, Kellog shows (in an exercise) that Neumann’s function for a sphere has a logarithmic singularity at the boundary. The logarithmic boundary singularity of Neumann’s function for the Laplace equation in a general regular domain seems to have been discovered by Popov Popov and elaborated by Silbergleit, Mandel, and Nemenman Silbergleit (which cites neither Kellog nor Popov ).

Another small window problem is that of a leaky conductor of Brownian particles, which is a bounded domain with a source of particles on the boundary or in the interior, and a (big) target, which is an absorbing part of the boundary. The remaining boundary is reflecting. If the boundary has a small absorbing patch (a hole), some of the Brownian particles may leak out and never make it to the big absorbing target. The calculation of the leakage flux is not the same as that in the narrow escape problem, because the total flux on the boundary remains bounded as the small hole shrinks. The calculation of the leakage flux was attempted in Savtchenko for diffusion in a flat cylinder with a source at the reflecting top and a small absorbing window at the reflecting bottom, and absorbing lateral envelop. The three-dimensional diffusion in the cylinder was assumed to be well approximated by radial diffusion in a circular disk.

In this paper, we find the structure of the boundary singularity of the Neumann function for the Poisson equation and of the Green-Neumann function for the mixed problem (with Dirichlet and Neumann boundary conditions) in a general bounded domain , whose boundary is sufficiently smooth. Our calculations use the method of Popov ; Silbergleit . We find that for , the structure of the Neumann function (in dimensionless variables) is

(1) |

where and are the principal curvatures of at and is a bounded function of in . If is a ball of radius , the above mentioned result of Kellog Kellog is recovered, because .

We find that the NET through a circular disk of (dimensionless) radius , centered at on the boundary, is

(2) |

where is the diffusion coefficient. If is a ball of radius , then

(3) |

The result (3) corrects that given in SSH1 . The case of an elliptic window is handled in a straightforward manner, as in SSH1 .

The principal eigenvalue of the Laplace equation in with Dirichlet conditions given on a circular disk of dimensionless radius and Neumann boundary conditions elsewhere has the asymptotic expansion

(4) |

The result (4) provides the missing second term and estimate of the remainder, which was not given in Ward1 ; Ward2 ; Ward3 ; Ward4 ; SSH1 .

For a leaky conductor, we find that the leakage flux through a circular hole of small (dimensionless) radius , centered at , is

(5) |

where is the solution of the reduced problem (without the leak) at the hole.

Equation (5) can be viewed as a generalization of (4) in the sense that the factor in (4) can be interpreted as the uniform concentration of the Brownian particle in . The uniform concentration is the solution of the stationary diffusion equation problem with Neumann conditions on the entire boundary, which is the reduced problem for narrow escape. Thus the concentration is a generalization of the fixed concentration in (4).

## Ii The singularity of Neumann’s function

Consider a bounded domain , given by , where . Our purpose is to determine the singularity of Green’s function for the Laplace equation in with Neumann boundary conditions (called Neumann’s function) and of Green’s function for the mixed Dirichlet and Neumann boundary conditions.

The Neumann function for this domain is the solution of the boundary value problem

(7) |

where is the outer unit normal to the boundary . If or (or both) are in , then only a half of any sufficiently small ball about a boundary point is contained in , which means that the singularity of Neumann’s function is . Therefore Neumann’s function for is written as

(8) |

where satisfies

(9) |

and the boundary condition

(10) |

Green’s identity requires the evaluation of two integrals. The first is the volume integral, which by (II) is

and the second is the surface integral, which by (7) is

Thus, for Green’s identity gives

(11) | |||||

To determine the singularity of this integral when approaches , we use the method of successive approximations to expand as

(12) |

where is more regular than (see Silbergleit ). For or (or both) in , the first term is the most singular part

(13) |

To extract its dominant part, we reproduce here, for completeness, the analysis of Popov with only minor modifications. We consider and assume that the boundary near is sufficiently smooth. Moving the origin to , we set . Taking a sufficiently small patch about , we assume that it can be projected orthogonally onto a circular disk of radius in the tangent plane to at . We can assume, therefore, that can be represented as

(14) |

If is sufficiently small, then . This canonical representation (14) assumes that has at least one non-zero curvature and that the quadratic part in Taylor’s expansion of about the origin is represented in principal axes.

The asymptotically dominant part as is determined by the integral over the patch , which we write as

(15) |

In the representation (14)

so that

(16) |

The patch is represented in polar coordinates in as

(17) |

so transforming into spherical coordinates

we can write (16) as

(18) |

where

(19) | |||||

Integration with respect to gives

for . It follows from (18) that for the leading order singularity is

(20) |

For further analysis of the term, see Silbergleit .

## Iii Application to the narrow escape problem

### iii.1 Escape through a small circular hole

As mentioned in the Introduction, the narrow escape problem Ward1 ; Ward2 ; Ward3 ; Ward4 ; HS ; SSH1 ; SSH2 ; SSH3 ; PNAS is to calculate the mean escape time of a Brownian particle from a bounded domain , whose boundary is reflecting, except for a small absorbing patch (or patches PLA-holes ; JPA-holes ) . We assume here that is a circular disk of radius and that a ball of radius can be rolled on inside . This means that there are no narrow passages in . We denote and and investigate the limit . We assume that all coordinates have been scaled with , so that all variables and parameters are dimensionless.

The MFPT from a point to is the solution of the mixed boundary value problem

(24) |

where is the diffusion coefficient. The compatibility condition,

(25) |

is obtained by integrating (III.1) over and using (III.1) and (24).

Green’s identity and the boundary conditions (7), (III.1), and (24) give

(26) |

where

(27) |

Following the argument in SSH1 , we note that is an integrable function independent of , whose integral is uniformly bounded, whereas as . Setting for and using the boundary condition (III.1), we obtain from (26) the integral equation for the flux density in ,

(28) |

which, in view of (8), (20) now becomes the generalized Helmholtz equation Helmholtz , SSH1

(29) | |||

where are the principal curvatures at the center of . To solve (29), we expand , where for and choose

(30) |

It was shown in Rayleigh , Lurie , SSH1 that if is a circular disk of radius , then

(31) |

It follows that satisfies the integral equation

(32) |

Setting , and changing to polar coordinates in the integral on the right hand side of (32), we obtain

(33) |

which gives in the limit (e.g., keeping fixed and ) that

(34) |

As in the pair (30), (31), we obtain that

(35) |

Finally, to determine the asymptotic value of the constant , we recall that and use in (25) the approximation

(36) |

We obtain the narrow escape time (in dimensionless variables) as

(37) |

The principal eigenvalue of the Laplace equation in with the mixed Dirichlet-Neumann boundary conditions (III.1), (24) has the asymptotic expansion for

(38) |

The result (38) provides the missing second term and estimate of the remainder, which was not given in Ward1 ; Ward2 ; Ward3 ; Ward4 ; SSH1 .

If is a ball of radius , then and the narrow escape time is given (in dimensional variables) by

(39) |

The result (39) corrects that given in SSH1 . Specifically, equation (3.52) in SSH1 is missing the factor of equation (39), which should have been carried from eq.(3.51) in SSH1 . The case of an elliptic window is handled in a straightforward manner, as in SSH1 .

### iii.2 Leakage in a conductor of Brownian particles

A conductor of Brownian particles is a bounded domain , with a source of particles on the boundary or in the interior and a target, which is an absorbing part of . The remaining boundary is reflecting. Some of the Brownian particles may leak out of if contains a small absorbing hole . The calculation of the leakage flux is not the same as that in the narrow escape problem, because the total flux on the boundary remains bounded as the small hole shrinks. Our purpose is to find the portion that leaks through the small hole out of the total flux.

The (dimensionless) stationary density of the Brownian particles satisfies the mixed boundary value problem

(40) | |||||

where is the flux density of the source on the boundary. Next, we derive an asymptotic expression for the flux through ,

(41) |

in terms of the solution of the reduced problem (without ), thus avoiding the need to construct boundary layers. First, we find the flux of each eigenfunction and then, using eigenfunction expansion, we calculate . Every eigenfunction of the homogeneous problem (III.2) satisfies

(44) |

The matched asymptotics method of Ward1 -Ward4 gives the expansion of the eigenvalues

(45) |

where is the eigenvalue of the reduced problem (for without any small holes).

We define the reduced Green function (without the small hole) as the solution of the mixed boundary value problem with ,

(48) |

Multiplying (III.2) by and integrating over , we get

(49) |

In view of the boundary condition (44), we get from (49) for all

(50) |

The integral on the left hand side of (50) can be expanded about the center of in the form

(51) |

where the origin is assumed to be in the center of and the plane is that of .

As in Section III.1, Green’s function for the mixed boundary value problem has the form

(52) |

for , where depends locally on the curvatures of the boundary and is a continuous function of and on . We assume that is bounded. Using (52) and the expansion (51) in (50), we obtain the Helmholtz equation

(53) |

The leading order singularity of and (31) suggest the expansion

(54) |

where is yet an undetermined coefficient, that is,

(55) | |||||

which reduces at to

It follows that

so that

(56) |

Now, (54) gives the flux through as