Narrow-escape times for diffusion in microdomains with a particle-surface affinity: Mean-field results

We analyze the mean time t_{app} that a randomly moving particle spends in a bounded domain (sphere) before it escapes through a small window in the domain's boundary. A particle is assumed to diffuse freely in the bulk until it approaches the surface of the domain where it becomes weakly adsorbed, and then wanders diffusively along the boundary for a random time until it desorbs back to the bulk, and etc. Using a mean-field approximation, we define t_{app} analytically as a function of the bulk and surface diffusion coefficients, the mean time it spends in the bulk between two consecutive arrivals to the surface and the mean time it wanders on the surface within a single round of the surface diffusion.