Trapping of particles diffusing in a cavity by hidden binding sites analyzed with the Reimann–Schmid–Hanggi steady-state approach

The ordinary narrow escape problem concentrates on finding the mean first-passage time of a particle diffusing in a cavity to one of the small absorbing disks located on the cavity wall, called the narrow escape time. Here, we consider a generalized narrow escape problem in three dimensions by expanding the ordinary narrow escape problem to the case where the absorbing disks are hidden in tunnels. We derive an approximate formula for the generalized narrow escape time, which shows how this time depends on the geometric parameters of the system and the particle diffusivities in the tunnels and cavity. The result is obtained using the Reimann-Schmid-Hanggi steady-state approach. When the tunnel lengths vanish, the generalized narrow escape time reduces to its ordinary counterpart. To check the accuracy of our formula and establish the range of its applicability, we run Brownian dynamics simulations. The comparison shows good agreement between the theoretical predictions and simulation results for not too short tunnels. When the tunnel lengths exceed five tunnel radii, the relative error is smaller than a few per cent.