  Recently, anomalous subdiffusion, aging, and scatter of the diffusion
coefficient have been reported in many single-particle-tracking experiments,
though origins of these behaviors are still elusive. Here, as a model to
describe such phenomena, we investigate a Langevin equation with diffusivity
fluctuating between a fast and a slow state. We assume that the sojourn time
distributions of these two states are given by power laws. It is shown that,
for a non-equilibrium ensemble, the ensemble-averaged mean square displacement
(MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows
normal diffusion, but an effective diffusion coefficient transiently shows
aging behavior. The propagator is non-Gaussian for short time, and converges to
a Gaussian distribution in a long time limit; this convergence to Gaussian is
extremely slow for some parameter values. For equilibrium ensembles, both
ensemble-averaged and time-averaged MSDs show only normal diffusion, and thus
we cannot detect any traces of the fluctuating diffusivity with these MSDs.
Therefore, as an alternative approach to characterize the fluctuating
diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is
utilized, and it is shown that the RSD exhibits slow relaxation as a signature
of the long-time correlation in the fluctuating diffusivity. Furthermore, it is
shown that the RSD is related to a non-Gaussian parameter of the propagator. To
obtain these theoretical results, we develop a two-state renewal theory as an
analytical tool.
