Transition from distributional to ergodic behavior in an inhomogeneous diffusion process: Method revealing an unknown surface diffusivity

Diffusion of molecules in cells plays an important role in providing a biological reaction on the surface by finding a target on the membrane surface. The water retardation (slow diffusion) near the target assists the searching molecules to recognize the target. Here, we consider effects of the surface on the diffusivity in three-dimensional diffusion processes, where diffusion on the surface is slower than that in bulk. We show that the ensemble-averaged mean square displacements increase linearly with time when the desorption rate from the surface is finite even when the diffusion on the surface is subdiffusion. Moreover, this slow diffusion on the surface affects the fluctuations of the time-averaged mean square displacements (TAMSDs). We find that fluctuations of the TAMSDs remain large when the measurement time is smaller than a characteristic relaxation time, and decays according to an increase of the measurement time for a relatively large measurement time. Therefore, we find a transition from non-ergodic (distributional) to ergodic diffusivity in a target search process. Moreover, this fluctuation analysis provides a method to estimate an unknown surface diffusivity.