Ergodic property of Langevin systems with superstatistical, uncorrelated or correlated diffusivity

Brownian yet non-Gaussian diffusion has recently been observed in numerous biological and active matter system. The cause of the non-Gaussian distribution have been elaborately studied in the idea of a superstatistical dynamics or a diffusing diffusivity. Based on a random diffusivity model, we here focus on the ergodic property and the scatter of the amplitude of time-averaged mean-squared displacement (TAMSD). Further, we individually investigate this model with three categories of diffusivities, including diffusivity being a random variable $D$, a time-dependent but uncorrelated diffusivity $D(t)$, and a correlated stochastic process $D(t)$. We find that ensemble-averaged TAMSDs are always normal while ensemble-averaged mean-squared displacement can be anomalous. Further, the scatter of dimensionless amplitude is determined by the time average of diffusivity $D(t)$. Our results are valid for arbitrary diffusivities.