\sum_{i} \sigma [X_i ]}}.$$ In the case of no correlations between the motions of particles, as in (6), we get $N_X^{coop} =1$. If there exist some positive correlated motions between particles, we would get $N_X^{coop} > 1$, while anti-correlations between the motions of particles would result in $N_X^{coop} < 1$. Doliwa and Heuer investigated the dynamic cooperativity of hard sphere systems in 2D and 3D, where they found finite cooperativity ($N_X^{coop} > 1$) for both vector displacement and the scalar magnitude of the displacement, which is consistent with the snapshots of the particle motions in their work. They argue that the dynamic cooperativity measures the total reduction of degrees of freedom due to the correlations. Here we also studied the dynamic cooperativity of the lattice gas particles by calculating $N_{X}^{coop}$ for both the scalar displacement and the vector displacement itself. Interestingly, we found that the scalar displacement exhibited finite dynamic cooperativity (Fig. 9a), while the vector displacement itself showed weak anti-correlations between particles. as shown in Fig. 9b. In the case of scalar displacement, the cooperativity increases at first as the time interval $\Delta t$ increases and reaches its maximum near the $\alpha$-relaxation time scale $\tau$. Then it decrease back 