Two types of Brownian yet non-Gaussian diffusion in the quenched disordered environment

Anomalous diffusion has been widely observed in quenched disordered environments such as living cells, and has attracted wide attention. Within the framework of the space–time correlated continuous-time random walk (CTRW), this manuscript studies two types of Brownian yet non-Gaussian diffusion. One is that when the waiting time density satisfies the weak asymptotic form $\omega \left(\tau \right)\sim {\tau }^{-(1+\sigma )}$ with $1<\sigma <2$, its mean-squared displacement (MSD) exhibits a crossover phenomenon during the evolution process and eventually increases linearly with time at large timescales, while its probability density function (PDF) shows a sharp shape. The other is that when the time density follows an exponential distribution $\omega \left(\tau \right)\sim exp(-\tau )$, its MSD always increases linearly with time, yet its PDF exhibits a generalized exponential shape and crosses over to a standard Gaussian distribution at large timescales. The diffusive behaviors are analyzed and discussed by calculating the MSD and PDF. The results reveal that the anomalous diffusion of the former is due to the weak asymptotic property of $\omega \left(\tau \right)$ while the latter is due to the nonergodicity of particles in the quenched disordered environment at the initial moment. The model explains the random motions in complex inhomogeneous environments and reproduces different observational results in biological and physical systems.