  In heterogeneous environments, the diffusivity is not constant but changes
with time. It is important to detect changes in the diffusivity from
single-particle-tracking trajectories in experiments. Here, we devise a novel
method for detecting the transition times of the diffusivity from trajectory
data. A key idea of this method is the introduction of a characteristic time
scale of the diffusive states, which is obtained by a fluctuation analysis of
the time-averaged mean square displacements. We test our method in silico by
using the Langevin equation with a fluctuating diffusivity. We show that our
method can successfully detect the transition times of diffusive states and
obtain the diffusion coefficient as a function of time. This method will
provide a quantitative description of the fluctuating diffusivity in
heterogeneous environments and can be applied to time series with transitions
of states.
