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We study the mean first exit time $T_{\ve}$ of a particle diffusing in acircular or a spherical micro-domain with an impenetrable confining boundarycontaining a small escape window (EW) of an angular size $\ve$. Focusing on theeffects of an energy/entropy barrier at the EW, and of the long-rangeinteractions (LRI) with the boundary on the diffusive search for the EW, wedevelop a self-consistent approximation to derive for $T_{\ve}$ a generalexpression, akin to the celebrated Collins-Kimball relation in chemicalkinetics and accounting for both rate-controlling factors in an explicit way.Our analysis reveals that the barrier-induced contribution to $T_{\ve}$ is thedominant one in the limit $\ve \to 0$, implying that the narrow escape problemis not "diffusion-limited" but rather "barrier-limited". We present thesmall-$\ve$ expansion for $T_{\ve}$, in which the coefficients in front of theleading terms are expressed via some integrals and derivatives of the LRIpotential. On example of a triangular-well potential, we show that $T_{\ve}$ isnon-monotonic with respect to the extent of the attractive LRI, being minimalfor the ones having an intermediate extent, neither too concentrated on theboundary nor penetrating too deeply into the bulk. Our analytical predictionsare in a good agreement with the numerical simulations.

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