  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.

Statistical Mechanics;Biological Physics