8 IV. SUMMARY tified and whether the equilibrium curve is included in the time superposition of C$_{q}$ at intermediate and large t$_{w}$ . Using molecular dynamics simulations, we investigated for the strong glass former SiO$_{2}$ the aging dynamics be- low the critical MCT temperature T$_{c}$ , using the BKS po- tential to model the interactions between silicon and oxy- gen atoms. After an instantaneous quench from T$_{i}$ > T$_{c}$ to a temperature T$_{f}$ < T$_{c}$ the dynamics towards equilib- rium was studied as a function of waiting time t$_{w}$ . Note that the temperatures T$_{f}$ were chosen such that equilib- rium was reached on the time span of the simulations (of the order of 30 ns). The central quantities considered in this work are the incoherent intermediate scattering function C$_{q}$ ( t$_{w}$, t$_{w}$ + t ) and the mean square displace- ment Δ r $^{2}$( t$_{w}$, t$_{w}$ + t ). These functions depend on the time origin at t$_{w}$ as long as t$_{w}$ is smaller than the typical relaxation time, τ$_{eq}$ , that is required to equilibrate the system. We find that the decay of C$_{q}$ ( t$_{w}$, t$_{w}$ + t ) (and similarly the rise of Δ r $^{2}$( t$_{w}$, t$_{w}$ + t )) exhibit qualitative changes from short to long waiting times. At short waiting times, relaxation processes are dominant that correspond to the early β relaxation regime at the target temperature T$_{f}$ . In this t$_{w}$ regime, no well-defined plateau is found in C$_{q}$ ( t$_{w}$, t$_{w}$ + t ) (see Fig. 4). Instead, this function first decreases rapidly, followed by a strongly stretched expo- nential decay to zero. At long waiting times, the β re- laxation seems to be very similar to that at equilibrium. The Debye-Waller factor (i.e. the height of the plateau in C$_{q}$ ( t$_{w}$, t$_{w}$ + t )) has reached its equilibrium value, al- though the decay of C$_{q}$ ( t$_{w}$, t$_{w}$ + t ) from the plateau to zero is faster than that at equilibrium. However, the shape of curves describing the long-time decay of C$_{q}$ ( t$_{w}$, t$_{w}$ + t ) is the same as that at equilibrium. Thus, C$_{q}$ follows a sim- ple time superposition for long waiting times, different e.g. from the “activated dynamics scaling” proposed by Wahlen and Rieger for BKS silica. Our results show that the aging dynamics of BKS sil- ica is very similar to that of the KALJ mixture. For both silica and the KALJ mixture three t$_{w}$ -regimes can be identified and C$_{q}$ follows time superposition for suffi- ciently large t$_{w}$ . The only difference between these two systems is that C$_{q}$ scales as C ( q, z ( t$_{w}$, t ) for SiO$_{2}$ but less well for the KALJ mixture. So slightly below its critical temperature T$_{c}$ of MCT, the strong glass-former silica does not seem to be very different from typical frag- ile systems, although one has already reached the low temperature Arrhenius regime (note that activation en- ergies for the self-diffusion, viscosity etc. are of the order of 5 eV, similar to the corresponding activation energies close to the glass transition temperature T$_{g}$ ≈ 1450 K, as measured in various experiments). However, the dynam- ics could be very different at very low temperatures (close to T$_{g}$ ) where the long-time aging regime is not accessi- ble by computer simulations. Thus, more experimental work on the aging dynamics of silica around T$_{g}$ would be very desirable. We also leave for future work to test whether also for other systems three t$_{w}$ -regimes are iden- The Nos´ e Hoover equations of motion are for particles i = 1 , . . . , N at position r$_{i}$ with momentum p$_{i}$ p$_{i}$ KVL thanks A. Zippelius and the Institute of Theoret- ical Physics, University of G¨ ottingen, for hospitality and financial support. Acknowledgments Appendix A: Velocity Verlet Nos´e Hoover where X = 3 N . We integrated with the generalized velocity Verlet form of Fox and Andersen [49] ˙ r$_{i}$ = m$_{i}$ (A1) ˙ p$_{i}$ = F$_{i}$ − ζ p$_{i}$ (A2) ˙ ζ = 1 Q ( N ∑ i =1 p 2 i m$_{i}$ − XkT ) (A3) ¨ r$_{i}$ = F$_{i}$ m$_{i}$ − ζ ˙ r$_{i}$ (A4) d $^{2}$ln s d t 2 = ˙ ζ = 1 Q ( N ∑ i =1 m$_{i}$ ˙ r 2 i − XkT ) (A5) r$_{i}$ ( t + Δ t ) = r$_{i}$ ( t ) + Δ t ˙ r$_{i}$ ( t ) (A6) + (Δ t ) 2 2 ( F$_{i}$ ( t ) m$_{i}$ − ζ ( t ) ˙ r$_{i}$ ( t ) ln s ( t + Δ t ) = ln s ( t ) + Δ t ζ ( t ) (A7) + (Δ t ) 2 2 Q ( N ∑ i =1 m$_{i}$ ˙ r 2 $_{i}$( t ) − XkT ) ζ $^{approx}$( t + Δ t ) = ζ ( t ) (A8) + Δ t Q ( N ∑ i =1 m$_{i}$ ˙ r 2 $_{i}$( t ) − XkT ) ˙ r$_{i}$ ( t + Δ t ) = ˙ r$_{i}$ ( t ) (A9) + Δ t 2 ( F$_{i}$ ( t ) + F$_{i}$ ( t + Δ t ) m$_{i}$ − [ ζ ( t ) + ζ $^{approx}$( t + Δ t ) ] ˙ r$_{i}$ ( t ) ( 1 − Δ t 2 ζ $^{approx}$( t + Δ t ) ζ ( t + Δ t ) = ζ ( t ) + Δ t 2 Q ( N ∑ i =1 m$_{i}$ ˙ r 2 $_{i}$( t ) (A10) + N ∑ i =1 m$_{i}$ ˙ r 2 $_{i}$( t + Δ t ) − 2 XkT ) and thus