Narrow escape and leakage of Brownian particles

TL;DR

This paper improves the understanding of narrow escape problems by deriving second-order expansions for escape times and eigenvalues, and analyzes leakage flux of Brownian particles through small perforations.

Contribution

It introduces precise second-term expansions for narrow escape time and eigenvalues, and characterizes leakage flux in domains with small boundary perforations.

Findings

Derived second-order expansion for narrow escape time.

Calculated eigenvalue expansion for mixed boundary conditions.

Quantified leakage flux through small boundary perforations.

Abstract

Questions of flux regulation in biological cells raise renewed interest in the narrow escape problem. The often inadequate expansions of the narrow escape time are due to a not so well known fact that the boundary singularity of Green's function for Poisson's equation with Neumann and mixed Dirichlet-Neumann boundary conditions in three-dimensions contains a logarithmic singularity. Using this fact, we find the second term in the expansion of the narrow escape time and in the expansion of the principal eigenvalue of the Laplace equation with mixed Dirichlet-Neumann boundary conditions, with small Dirichlet and large Neumann parts. We also find the leakage flux of Brownian particles that diffuse from a source to an absorbing target on a reflecting boundary of a domain, if a small perforation is made in the reflecting boundary.