N_Q,$$

where $\vec r_j(t)$ denotes the position of $j$-th particle on the lattice. Due to the discrete lattice nature of our model sytem, we need to consider the wave-vectors within the first Brillouin zone $q= {{2 \pi }  \over L} n $, $n=0,1,2, \cdots L-1 $. Figure 11 shows the $q$-dependence of $F_S(q,t)$ at temperature $T=0.033$. We find that the long-time behavior of $F_S(q,t)$ also can be fitted to stretched exponential form. For low $q$, the late time $\beta$ exponents were close to one (pure exponential relaxation) but as $q$ increases the exponents decreased down to $\beta \approx 0.73$ for $q = 18 \times 2 \pi /36 $, and $T=0.033$ (Fig. 12). As can be seen from the definition of $F_S(q,t)$, for gaussian distribution for the displacement vector $\Delta \vec{r}_i$, we would get $$F_{G} (q,t) \equiv \langle \exp iq [\Delta r ] \rangle 
= \exp [ -{{q^2} \over 2} \langle (\Delta r )^2 \rangle ].$$ Figure 13 shows that the gaussian approximation is quite good for low $q$. That is, for long distance diffusion, the distribution gets closer to gaussian. However, as $q$ becomes larger, the gaussian approximation gets worse as shown in the figure. Similar features were reported in 