cooperativity in vector displacement, but only the scalar displacement exhibits appreciable positive cooperativity due to the local regions with high mobilities. Hence, heterogeneity still exists in our lattice Coulomb gas in terms of local mobility distribution, but unlike the case of hard sphere systems, there is no appreciable average local flow.

Also, we may look into the nature of the equilibrium dynamics of the system in wave-vector space. Figure 10 shows the structure factor $S(q) \equiv \langle |\rho_q|^2 \rangle $ at equilibrium where $\rho_q \equiv \sum_j \exp [i 
\vec{q} \cdot \vec{r}_j ]/N $ where $q= {{2 \pi }  \over L} n $, $n=1,2, \cdots 2/L $. We see that the structure factor of our LCG shows some similarity to those of dense liquids with first peak corresponding roughly to the inverse of the average distance between charges. Due to the lattice nature of the LCG, the wave vector has cutoff value at $q_{max}=\pi$ as in the figure.

The diffusive properties of the system can be probed by calculating the incoherent scattering function (ISF) $F_S(q,t)$ which is defined as in our model of LCG

$$F_S (q,t) \equiv \langle \sum_{j=1}^{N_Q} \exp i \vec q 
\cdot [\vec r_j(t)- \vec r_j (0)]
\rangle / 