Narrow-escape-time problem: the imperfect trapping case

TL;DR

This paper develops an analytical master equation approach to the narrow escape time problem, incorporating imperfect trapping via a finite transition probability, and validates findings with Monte Carlo simulations.

Contribution

Introduces a novel master equation framework for the NET problem with imperfect trapping, analyzing how transition probabilities affect escape times.

Findings

Analytic expressions for mean escape time with imperfect trapping.

Existence of a global minimum in escape time depends on trapping imperfection.

Monte Carlo simulations confirm theoretical predictions.

Abstract

We present a master equation approach to the \emph{narrow escape time} (NET) problem, i.e. the time needed for a particle contained in a confining domain with a single narrow opening, to exit the domain for the first time. We introduce a finite transition probability, $\nu$, at the narrow escape window allowing the study of the imperfect trapping case. Ranging from 0 to $\infty$, $\nu$ allowed the study of both extremes of the trapping process: that of a highly deficient capture, and situations where escape is certain ("perfect trapping" case). We have obtained analytic results for the basic quantity studied in the NET problem, the \emph{mean escape time} (MET), and we have studied its dependence in terms of the transition (desorption) probability over (from) the surface boundary, the confining domain dimensions, and the finite transition probability at the escape window. Particularly we show that the existence of a global minimum in the NET depends on the `imperfection' of the trapping process. In addition to our analytical approach, we have implemented Monte Carlo simulations, finding excellent agreement between the theoretical results and simulations.