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

Liquids near the glass transition exhibit dynamical heterogeneity, i.e. correlated regions in the liquid relax at either a much faster rate or a much slower rate than the average. This collective phenomenon has been characterized by measurements of a dynamic susceptibility $χ_4(t)$, which are sometimes interpreted in terms of the size of those relaxing regions and the intensity of the fluctuations. We show that the results of those measurements can be affected not only by the collective fluctuations in the relaxation rate, but also by density fluctuations in the initial state and by single-particle fluctuations. 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 time interval $t$ decays as a power law $C(t) \sim t^{-d/2}$. This is much slower than the stretched exponential behavior $C(t) \sim {\rm e}^{-(t/τ)^β}$ previously observed at times within one or two orders of magnitude of the $α$-relaxation time $τ_α$. We find that for times longer than $10-100 τ_α$, the dynamic susceptibility $χ_4(t)$ is dominated by single particle fluctuations, and that $χ_4(t) \approx C(t) \sim t^{-d/2}$. Finally, we introduce a method to extract the collective relaxation contribution to the dynamic susceptibility $χ_4(t)$ by subtracting the effects of single-particle fluctuations and initial state density fluctuations. We apply this method to numerical simulations of two glass forming models: a binary hard sphere system and a Kob-Andersen Lennard-Jones system. This allows us to extend the analysis of numerical data to timescales much longer than previously possible, and opens the door for further future progress in the study of dynamic heterogeneities, including the determination of the exchange time.