# Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior

Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior We consider a model of surface-mediated diffusion with alternating phases of bulk and surface diffusion for two geometries: the disk and rectangles. We develop a spectral approach to derive an exact formula for the mean exit time of a particle through a hole on the boundary. The spectral representation of the mean exit time through the eigenvalues of an appropriate self-adjoint operator is particularly well-suited to investigate the asymptotic behavior in the limit of large desorption rate $$\lambda$$ λ . For a point-like target, we show that the mean exit time diverges as $$\sqrt{\lambda }$$ λ . For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as $$\lambda$$ λ tends to infinity. That implies that the pure bulk diffusion is never an optimal search strategy. We also investigate the influence of rectangle elongation onto the mean exit time, in particular, the dependence of the critical ratio of bulk and surface diffusion coefficients on the rectangle aspect ratio. We show that the intermittent search strategy can significantly outperform pure surface diffusion for elongated rectangles. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior

, Volume 5 (4) – Feb 18, 2015
42 pages

/lp/springer-journals/mean-exit-time-for-surface-mediated-diffusion-spectral-analysis-and-H0y00OxU00
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-015-0098-0
Publisher site
See Article on Publisher Site

### Abstract

We consider a model of surface-mediated diffusion with alternating phases of bulk and surface diffusion for two geometries: the disk and rectangles. We develop a spectral approach to derive an exact formula for the mean exit time of a particle through a hole on the boundary. The spectral representation of the mean exit time through the eigenvalues of an appropriate self-adjoint operator is particularly well-suited to investigate the asymptotic behavior in the limit of large desorption rate $$\lambda$$ λ . For a point-like target, we show that the mean exit time diverges as $$\sqrt{\lambda }$$ λ . For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as $$\lambda$$ λ tends to infinity. That implies that the pure bulk diffusion is never an optimal search strategy. We also investigate the influence of rectangle elongation onto the mean exit time, in particular, the dependence of the critical ratio of bulk and surface diffusion coefficients on the rectangle aspect ratio. We show that the intermittent search strategy can significantly outperform pure surface diffusion for elongated rectangles.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Feb 18, 2015