~\exp\left(-\frac{\epsilon}{kT_g}\right),
  \end{aligned}$$ with $T_g$ being the characteristic temperature. According to the Van’t Hoff-Arrhenius law [@Haenggi1990], at a temperature $T$ these trapping potentials correspond to *mean waiting times* $\tau_k = \tau_0 \exp\left( \frac{E_k}{kT} \right)$, where we put $\tau_0 = 1$ in what follows. The density of the mean waiting times $\tau_k$ is $$\begin{aligned}
    \rho_\tau (t) &= \alpha t^{-1-\alpha}\qquad (t \geq 1),
    \label{equ:rhotau}
  \end{aligned}$$ with $\alpha = T/T_g$. In what follows, we consider lattices in dimensions $D = 1$ and $D = 3$. In 1D the particle’s dynamics on the lattice is described by a continuous-time Markov chain with transition rates $$\begin{aligned}
  w_{k\rightarrow l} &= \begin{cases}
                          -1/\tau_k, &\text{if~} k = l,\\
                          1/(2D\tau_k), &\text{if~} k,l \text{~neighbors},\\
                          0, &\text{else},
                        \end{cases}
                        \label{equ:GeneralRates}
  \end{aligned}$$ with $2D$ being the number of neighbors of site $k$ and $\tau_k$ denoting the mean waiting time associated with the site. If the first moment of the distribution, Eq.(\[equ:rhotau\]), diverges, which is the case for $0 < \alpha < 1$, the dynamics of the system starts to show highly nontrivial behavior (as exemplified by anomalous diffusion in such models). Of course, one could also define a mixture of the trap and barrier model, or generalize the model as in [@BenArous2005].

The model 