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We examine the dynamics of hard spheres and disks at high packing fractions in two and three dimensions, modeling the simplest systems exhibiting a glass transition. As it is well known, cooperativity and dynamic heterogeneity arise as central features when approaching the glass transition from the liquid phase, so an understanding of their underlying physics is of great interest. Cooperativity implies a reduction of the effective degrees of freedom, and we demonstrate a simple way of quantification in terms of the strength and the length scale of dynamic correlations among different particles. These correlations are obtained for different dynamical quantities X(i)(t) that are constructed from single-particle displacements during some observation time t. Of particular interest is the dependence on t. Interestingly, for appropriately chosen X(i)(t) we obtain finite cooperativity in the limit t-->infinity.
Doliwa et al. (Thu,) studied this question.