  Brownian yet non-Gaussian phenomenon has recently been observed in many
biological and active matter systems. The main idea of explaining this
phenomenon is to introduce a random diffusivity for particles moving in
inhomogeneous environment. This paper considers a Langevin system containing a
random diffusivity and an $\alpha$-stable subordinator with $\alpha<1$. This
model describes the particle's motion in complex media where both the long
trapping events and random diffusivity exist. We derive the general expressions
of ensemble- and time-averaged mean-squared displacements which only contain
the values of the inverse subordinator and diffusivity. Further taking specific
time-dependent diffusivity, we obtain the analytic expressions of ergodicity
breaking parameter and probability density function of the time-averaged
mean-squared displacement. The results imply the nonergodicity of the random
diffusivity model for any kind of diffusivity, including the critical case
where the model presenting normal diffusion.

Statistical Mechanics