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We analyze the correlated back and forth dynamics and dynamic heterogeneities, i.e., the presence of fast and slow ions, for a lithium metasilicate system via computer simulations. For this purpose we define, in analogy to previous work in the field of glass transition, appropriate three-time correlation functions. They contain information about the dynamics during two successive time intervals. First we apply them to simple model systems in order to clarify their information content. Afterwards we use this formalism to analyze the lithium trajectories. A strong back-dragging effect is observed, which also fulfills the time-temperature superposition principle. Furthermore, it turns out that the back-dragging effect is long ranged and exceeds the nearest-neighbor position. In contrast, the strength of the dynamic heterogeneities does not fulfill the time-temperature superposition principle. The lower the temperature, the stronger the mobility difference between fast and slow ions. The results are then compared with the simple model systems considered here as well as with some lattice models of ion dynamics.
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