New Insights Into the (Old) Narrow Escape Problem by Denis Grebenkov Laboratoire de Physique de la Matière Condensée, CNRS – Ecole Polytechnique, 91128 Palaiseau France Saint Petersburg, Russia, 22/09/2017

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: http://pmc.polytechnique.fr/pagesperso/dg

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, 213-257 (2014)

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, 213-257 (2014)

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

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)

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, 213-257 (2014)

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

Arbitrary planar domains Part I Arbitrary planar domains DG, PRL 117, 260201 (2016)

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:

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

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

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

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

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

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

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)

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

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

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

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)

Accuracy of the approximation 10 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)

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

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

Optimal range of the potential 60 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

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

Complex Systems in Life Sciences Theory and Modeling of Complex Systems in Life Sciences We thank all the participants!!! Leonhard Euler (1707-1783) Scientific Organizers Denis Grebenkov Sergey Nechaev Stanislav Smirnov Local Organizers Nadia Zalesskaya Tatiana Vinogradova Natalia Kirshner Liza Kruykova http://inadilic.fr/conference-2017/ We acknowledge the financial support by