in surroundings of very slowly moving regions offer a specific example for spatial heterogeneity in glassy systems [@harrowell; @onuki], which was often thought of as the physical mechanism for breakdown of the SE relation.

One simple way to quantify the degree of dynamic heterogeneity directly from the local motions of particles is to calculate the dynamic cooperativity [@doliwa] for one particle dynamic quantities such as [*e.g.*]{}, displacement vectors $X_i \equiv |\vec{r}_i (t+ \Delta t) - \vec{r}_i (t) | $ between the time $t$ and $t+\Delta t$ for some fixed time interval $\Delta t$. We can also choose $X_i$ to be the vector displacement itself $X_i \equiv \vec{r}_i (t+ \Delta t) - \vec{r}_i (t) $. If there are no correlations between the motions of particles, then the variations of the $X_i$’s will satisfy

$$\sigma [ \sum_{i} X_i ] = \sum_{i} \sigma [X_i ],$$

where $\sigma[x]$ denotes the mean square deviations of the random number $x$, $\sigma [x] \equiv \langle (x-<x>)^2 \rangle$. However, some correlations between the particle motions will increase $\sigma [ \sum_{i} X_i ]$ or anti-correlations will decrease it. Following Doliwa and Heuer, we can define the dynamic cooperativity as $$N_{X}^{coop} \equiv {{\sigma [ \sum_{i} X_i ] } \over 
{ 