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Liquids near the glass transition exhibit dynamical heterogeneity, i.e.correlated regions in the liquid relax at either a much faster rate or a muchslower rate than the average. This collective phenomenon has been characterizedby measurements of a dynamic susceptibility $\chi_4(t)$, which are sometimesinterpreted in terms of the size of those relaxing regions and the intensity ofthe fluctuations. We show that the results of those measurements can beaffected not only by the collective fluctuations in the relaxation rate, butalso by density fluctuations in the initial state and by single-particlefluctuations. We also show that at very long times the average overlap $C(t)$probing the similarity between an initial and a final state separated by a timeinterval $t$ decays as a power law $C(t) \sim t^{-d/2}$. This is much slowerthan the stretched exponential behavior $C(t) \sim {\rm e}^{-(t/\tau)^{\beta}}$previously observed at times within one or two orders of magnitude of the$\alpha$-relaxation time $\tau_{\alpha}$. We find that for times longer than$10-100 \tau_{\alpha}$, the dynamic susceptibility $\chi_4(t)$ is dominated bysingle particle fluctuations, and that $\chi_4(t) \approx C(t) \sim t^{-d/2}$.Finally, we introduce a method to extract the collective relaxationcontribution to the dynamic susceptibility $\chi_4(t)$ by subtracting theeffects of single-particle fluctuations and initial state density fluctuations.We apply this method to numerical simulations of two glass forming models: abinary hard sphere system and a Kob-Andersen Lennard-Jones system. This allowsus to extend the analysis of numerical data to timescales much longer thanpreviously possible, and opens the door for further future progress in thestudy of dynamic heterogeneities, including the determination of the exchangetime.

Single particle fluctuations dominate the long-time dynamic susceptibility in glass-forming liquids