  The mean-square displacement (MSD) is widely utilized to study the dynamical
properties of stochastic processes. The time-averaged MSD (TAMSD) provides some
information on the dynamics which cannot be extracted from the
ensemble-averaged MSD. In particular, the relative standard deviation (RSD) of
the TAMSD can be utilized to study the long time relaxation behavior. In this
work, we consider a class of Langevin equations which are multiplicatively
coupled to time-dependent and fluctuating diffusivities. Various interesting
dynamics models such as entangled polymers and supercooled liquids can be
interpreted as the Langevin equations with time-dependent and fluctuating
diffusivities. We derive a general formula for the RSD of the TAMSD for the
Langevin equation with the time-dependent and fluctuating diffusivity. We show
that the RSD can be expressed in terms of the correlation function of the
diffusivity. The RSD exhibits the crossover at the long time region. The
crossover time is related to a weighted average relaxation time for the
diffusivity. Thus the crossover time gives some information on the relaxation
time of fluctuating diffusivity which cannot be extracted from the
ensemble-averaged MSD. We discuss the universality and possible applications of
the formula via some simple examples.
