  Brownian yet non-Gaussian processes have recently been observed in numerous
biological systems and the corresponding theories have been built based on
random diffusivity models. Considering the particularity of random diffusivity,
this paper studies the effect of an external force acting on two kinds of
random diffusivity models whose difference is embodied in whether the
fluctuation-dissipation theorem is valid. Based on the two random diffusivity
models, we derive the Fokker-Planck equations with an arbitrary external force,
and analyse various observables in the case with a constant force, including
the Einstein relation, the moments, the kurtosis, and the asymptotic behaviors
of the probability density function of particle's displacement at different
time scales. Both the theoretical results and numerical simulations of these
observables show significant difference between the two kinds of random
diffusivity models, which implies the important role of the
fluctuation-dissipation theorem in random diffusivity systems.
