Diffusive escape through a narrow opening: new insights into a classic problem

TL;DR

This paper develops a self-consistent approximation to analyze the mean first exit time for diffusing particles in micro-domains with small escape windows, highlighting the dominant role of energy barriers over diffusion in narrow escape problems.

Contribution

It introduces a general expression for escape time considering energy barriers and long-range interactions, revealing the barrier-limited nature of the problem and providing analytical formulas validated by simulations.

Findings

Barrier effects dominate as escape window size decreases.

Escape time is minimized at intermediate interaction ranges.

Analytical results agree well with numerical simulations.

Abstract

We study the mean first exit time $T_{\ve}$ of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size $\ve$. Focusing on the effects of an energy/entropy barrier at the EW, and of the long-range interactions (LRI) with the boundary on the diffusive search for the EW, we develop a self-consistent approximation to derive for $T_{\ve}$ a general expression, akin to the celebrated Collins-Kimball relation in chemical kinetics and accounting for both rate-controlling factors in an explicit way. Our analysis reveals that the barrier-induced contribution to $T_{\ve}$ is the dominant one in the limit $\ve \to 0$, implying that the narrow escape problem is not "diffusion-limited" but rather "barrier-limited". We present the small-$\ve$ expansion for $T_{\ve}$, in which the coefficients in front of the leading terms are expressed via some integrals and derivatives of the LRI potential. On example of a triangular-well potential, we show that $T_{\ve}$ is non-monotonic with respect to the extent of the attractive LRI, being minimal for the ones having an intermediate extent, neither too concentrated on the boundary nor penetrating too deeply into the bulk. Our analytical predictions are in a good agreement with the numerical simulations.