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Yet another look at narrow escape through a tube

Victorya Richardson, Yick Hin Ling, Sean D Lawley

Abstract

The narrow escape problem concerns the time needed for a diffusing particle to exit a confining domain through a small hole in the boundary. While this problem is now well-understood, determining the escape time for a particle that must exit through a narrow tube has proven challenging. Indeed, relying on analogies with electrodynamics, parameter fits to simulations, and heuristics, a variety of conflicting estimates for this escape time have been offered over the last three decades, some of which are counterintuitive and arguably non-physical. In this paper, we combine matched asymptotic analysis and probabilistic methods to determine the exact asymptotics of the narrow escape time through a tube. We obtain a new escape time formula which reduces to the various prior estimates in certain special cases. If the diffusivity in the tube differs from the diffusivity in the rest of the domain, our results reveal the importance of the form of the multiplicative noise inherent to any diffusivity that varies in space. We discuss our results in the context of asymmetric cell division.

Yet another look at narrow escape through a tube

Abstract

The narrow escape problem concerns the time needed for a diffusing particle to exit a confining domain through a small hole in the boundary. While this problem is now well-understood, determining the escape time for a particle that must exit through a narrow tube has proven challenging. Indeed, relying on analogies with electrodynamics, parameter fits to simulations, and heuristics, a variety of conflicting estimates for this escape time have been offered over the last three decades, some of which are counterintuitive and arguably non-physical. In this paper, we combine matched asymptotic analysis and probabilistic methods to determine the exact asymptotics of the narrow escape time through a tube. We obtain a new escape time formula which reduces to the various prior estimates in certain special cases. If the diffusivity in the tube differs from the diffusivity in the rest of the domain, our results reveal the importance of the form of the multiplicative noise inherent to any diffusivity that varies in space. We discuss our results in the context of asymmetric cell division.
Paper Structure (18 sections, 94 equations, 2 figures)

This paper contains 18 sections, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inner problem
  3. PDE
  4. Probabilistic interpretation
  5. Far-field behavior
  6. Inner problem asymptotics
  7. Taking $\rho:=(D_0/D_1)^{\alpha}\delta\to0$
  8. Taking $\rho:=(D_0/D_1)^{\alpha}\delta\to\infty$
  9. Sigmoid approximation
  10. KMC simulations
  11. Narrow escape through a tube
  12. Constructing the domain
  13. Diffusion process
  14. Tube residence time
  15. Matched asymptotic analysis
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Diffusive path of a particle in a three-dimensional domain consisting of a "bulk" volume connected to a tube. The particle starts at the green ball and is eventually absorbed at the red region at the end of the tube. The diffusivity is $D_0$ in the bulk and $D_1$ in the tube
  • Figure 2: The markers show estimates of $C$ in \ref{['eq:corefar']} computed from KMC simulations for the case that $\Gamma$ is the unit disk. The solid curve is the sigmoid approximation in \ref{['eq:sigmoiddisk']}.