M.-V. VÁZQUEZ ET AL.

287

Figure 2. Narrow escape time for the sphere with two holes

(in Figure 1), in a Log-Log scale to emphasize the differ-

ences between theoretical curves and simulated data. Hill's

formula (11) (dashed line) shows a poor agreement with

simulated data (circles) in the range a > 0.1, while (19)

(solid line) fits better the data for all a in 0.008 ≤ a ≤ 0.9.

time step , so that

6

=10t

3

0

2=210Dt

1.

The actual particle’s position, n, is given by

0ran

, where 0 is the former position, and ran

is a vector of pseudo random numbers generated with a

Gaussian distribution (

r

=

nrrr r r

=0

,0

=2Dt

). Each MFPT

is obtained by averaging the first-passage times of

trajectories whose starting positions are uni-

formly distributed inside the cavity.

4

2.5 10

The system under study is shown in Figure 1, a sphere

of radius R with two round holes of the same size. The

length L between the two absorbing holes is related to

the sizes of the sphere, R, and the hole, a, by the relation,

22

2=LRa

(20)

Equation (20) implies that the volume of the domain is

a function of the length 2L, so the former shrinks as a

increases (and 2L decreases). The height of the sphere

as a function of the axial coordinate x is

22

=rxR x

.

The present methodology can be applied, nonetheless to

any geometry, provided it is radially symmetric.

Figure 2 shows the comparison between Narrow Es-

cape Times computed from Brownian dynamics simula-

tions (circles) and those predicted by Hill’s formula

(dashed line), Equation (11), and our result (solid line),

Equation (19). The simulated data ranges from

to . Equation (11) falls far from the computed

narrow escape times in the range while (19) fits

for all a in a broader range.

=0.007a

=0.900a

>0.1a

4. Acknowledgements

We are grateful to C. Reynaud for her useful comments

on the manuscript.

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