Quenched trap model on the extreme landscape: the rise of sub-diffusion and non-Gaussian diffusion

Non-Gaussian diffusion has been intensively studied in recent years, which reflects the dynamic heterogeneity in the disordered media. The recent study on the non-Gaussian diffusion in a static disordered landscape suggests novel phenomena due to the quenched disorder. In this work, we further investigate the random walk in this landscape under various effective temperature $\mu$, which continuously modulates the dynamic heterogeneity. We show in the long time limit, the trap dynamics on the landscape is equivalent to the quenched trap model, in which sub-diffusion appears for $\mu<1$. The non-Gaussian distribution of displacement has been analytically estimated for short $t$, of which the stretched exponential tail is expected for $\mu\neq1$. Due to the localization in the ensemble of trajectory segments, an additional peak arises in $P(x,t)$ around $x=0$ even for $\mu>1$. Evolving in different time scales, the peak and the tail of $P(x,t)$ are well split for a wide range of $t$. This theoretical study reveals the connections among the sub-diffusion, non-Gaussian diffusion, and the dynamic heterogeneity in the static disordered medium. It also offers an insight on how the cell would benefit from the quasi-static disordered structures.