Mode-coupling theory for the dynamic heterogeneity in an aging glass: How Do Glassy Domains Grow?

We construct the equations for the growth kinetics of an aging structural glass within mode-coupling theory through a non-stationary variant of the 3-density correlator defined in Phys. Rev. Lett. {\bf 97}, 195701 (2006). We solve a schematic form of the resulting equations to obtain the coarsening of the dynamic heterogeneity, characterized via the 3-point correlator $\chi_3(t,t_w)$, as a function of waiting time $t_w$. For a quench into the glass, we find that $\chi_3$ attains a peak value $\sim t_w^{0.5}$ at $t -t_w \sim t_w^{0.8}$, providing a theoretical basis for the numerical observations of Parisi [J. Phys. Chem. B \textbf{103}, 4128 (1999)] and Kob and Barrat [Phys. Rev. Lett. \textbf{78}, 4581 (1997)]. The aging is not "simple": the $t_w$ dependence cannot be attributed to an evolving effective temperature.