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

TL;DR

This paper analyzes the mean exit time for surface-mediated diffusion using spectral methods, revealing its asymptotic behavior as the desorption rate increases, with implications for optimizing diffusion processes.

Contribution

It introduces a spectral approach to compute and analyze the mean exit time in surface-mediated diffusion, especially its asymptotic behavior for large desorption rates.

Findings

Mean exit time diverges as √λ for point targets

For extended targets, the mean exit time approaches a finite limit

Mean exit time increases asymptotically with λ

Abstract

We consider a model of surface-mediated diffusion with alternating phases of pure bulk and surface diffusion. For this process, we compute the mean exit time from a disk through a hole on the circle. We develop a spectral approach to this escape problem in which the mean exit time is explicitly expressed through the eigenvalues of the related self-adjoint operator. This representation is particularly well suited to investigate the asymptotic behavior of the mean exit time in the limit of large desorption rate $\lambda$. For a point-like target, we show that the mean exit time diverges as $\sqrt{\lambda}$. 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 $\lambda$ tends to infinity. We also revise the optimality regime of surface-mediated diffusion. Although the presentation is limited to the unit disk, the spectral approach can be extended to other domains such as rectangles or spheres.