Narrow-escape-time problem: the imperfect trapping case

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.Comment: 9 page

Narrow escape problem in the presence of the force field

This paper considers the narrow escape problem of a Brownian particle within a three-dimensional Riemannian manifold under the influence of the force field. We compute an asymptotic expansion of mean sojourn time for Brownian particles. As an auxiliary result, we obtain the singular structure for the restricted Neumann Green's function which may be of independent interest.Comment: 27 pages. arXiv admin note: text overlap with arXiv:2101.0795