1. New Insights Into the (Old)
Narrow Escape Problem
by Denis Grebenkov
Laboratoire de Physique de la Matière Condensée,
CNRS – Ecole Polytechnique, Palaiseau France
Saint Petersburg, Russia, 22/09/2017

2. All related publications can be found on my webpage:
Outline
Overview
Part I: Role of geometry
Part II: Role of interactions and barrier
All related publications can be found on my webpage:

3. First passage and exit problems
S. Redner, A guide to first-passage processes (2001)
R. Metzler, et al., First-passage phenomena and their applications (2014)
D. Holcman, Z. Schuss, SIAM Rev. 56, (2014)

4. First passage and exit problems
Target size/shape/location/type
Domain shape
Starting point location
Type of diffusive motion
S. Redner, A guide to first-passage processes (2001)
R. Metzler, et al., First-passage phenomena and their applications (2014)
D. Holcman, Z. Schuss, SIAM Rev. 56, (2014)

5. First passage time (FPT)
Survival probability
MFPT:
Mixed boundary value problem
on target
on the rest
Global MFPT:

6. Escape problems
Lord Rayleigh (1877): exit from a sphere through a circular hole of radius a
Exact solution for the disk:
Singer et al. (2006):
Asymptotic behavior:
2D manifolds:
Rayleigh, The theory of sound (1877); Singer et al., J. Stat. Phys. (2006)

7. Various extensions Multiple targets
Partial adsorption on target/remaining part
Gating problem
Interaction with the surface
Anomalous and intermittent diffusion
Probability distribution
Splitting probability
First passage to targets inside the domain
Optimality of the MFPT
R. Metzler, et al., First-passage phenomena and their applications (2014)
D. Holcman, Z. Schuss, SIAM Rev. 56, (2014)

8. Summary: current paradigm
Narrow escape limit:
Global MFPT
is proportional to
and diverges as ln(1/)
(2D)
We will show that this paradigm is incomplete and can be misleading

9. Arbitrary planar domains
Part I
Arbitrary planar domains
DG, PRL 117, (2016)

10. Two basic counter-examples
Common picture: h L
(x,y)  
(x,y)
is independent of y and h
and does not diverge as
T is small while  is small
T is large while  is large
These are not “regular” domains:

11. Conformal mapping is the harmonic measure of seen from x0 D x0 on on
on the rest on
on the rest

12. Integral formula for the MFPT
MFPT from x0 to the whole boundary
eventual reflections

13. Asymptotic behavior of the MFPT
We retrieve the leading logarithmic behavior, with
the perimeter  replaced by the harmonic measure 

14. Two basic counter-examples
New picture: L 
(x,y) 
(x,y) h
Makarov’s theorem:
Makarov, Proc. London Math. Soc. 51, 369 (1985)

15. Illustration for a nontrivial domain
We fix x0 and  and move the region  1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 5 15 s T
MFPT | W |/ p
log(1/ w ) e x0

16. Conclusions Integral formula for the MFPT in planar domains
Dependence on the starting point
Shape of the domain through conformal mapping
Accounting for space-dependent diffusivity
Fundamental relation between the MFPT and the HM,
which substitutes the normalized perimeter 
Simple logarithmic term with the harmonic measure
Scaling with the area is not universal

17. Long-range interactions with boundary and escape barrier
Part II
Long-range interactions with boundary and escape barrier
G. Oshanin
Laboratory of Condensed Matter Theoretical Physics, University Paris-6
DG, G. Oshanin, PCCP 19, 2723 (2017)

18. Circular and spherical domains
Mixed boundary value problem
on target
on the rest
on target
on the rest
Two ingredients: entropic/energetic barrier
long-range radial potential

19. Self-consistent approximation
Mixed Robin-Neumann boundary condition
on target
on the rest
on target
on the rest
Inhomogeneous Neumann boundary condition
Note: solution of the modified problem is not unique!
But one can get the leading (divergent) term

20. Main result MFPT to fully absorbing whole boundary
Contribution due to partial escape region
Contribution due to partial reflections

21. Accuracy of the approximation
0.5 1
1.5 2
2.5 3 10 -1 e
D T /R
SCA
asympt
FEM MC
(no long-range interaction case)

22. Accuracy of the approximation10 1 2 3 50
100
150 k
R/D
D T e /R
SCA (
= 0.1)
FEM (
= 0.2) = p
/4)
(no long-range interaction case)

23. Narrow escape limit
MFPT to fully absorbing whole boundary
Contribution due to partial escape region
Contribution due to partial reflections
Conventional scaling
In presence of entropic/energetic barrier, the escape process is “barrier-limited”, not diffusion-limited

24. Role of long-range interactions
Toy model: linear radial potential with a finite extent
For this model, one can explicitly
find the radial functions satisfying
Repulsive potential (U0>0)
MFPT is increased
Attractive potential (U0<0)
MFPT is decreased

25. Optimal range of the potential60 50 2 40 /R
(3) e 30 DT 20 U
= -1 U
= -2 10 U
= -5
0.2
0.4
0.6
0.8 1 r /R
The optimal range lies near the boundary

26. Conclusions
We proposed an accurate approximation to MFPT for rotation-invariant domains
The escape process is “barrier-limited”,
the asymptotics being , not
The narrow escape behavior is only determined by the potential and its derivative at the boundary
The range of attractive potentials can be optimized to minimize the MFPT

27. Complex Systems in Life Sciences
Theory and Modeling of
Complex Systems in Life Sciences
We thank all the participants!!!
Leonhard Euler ( )
Scientific Organizers
Denis Grebenkov
Sergey Nechaev
Stanislav Smirnov
Local Organizers
Nadia Zalesskaya
Tatiana Vinogradova
Natalia Kirshner
Liza Kruykova
We acknowledge the financial support by