Hopping in the glass configuration space: Subaging and generalized scaling laws

Aging dynamics in glassy systems is investigated by considering the hopping motion in a rugged energy landscape whose deep minima are characterized by an exponential density of states $\rho {\mathrm{}\left(E\right)=T}_g^{-1}\exp {(E/T}_g),$ $-\infty <E<~0.$ In particular we explore the behavior of a generic two-time correlation function $\Pi {(t}_w{+t,t}_w)$ below the glass transition temperature $T_g$ when both the observation time t and the waiting time $t_w$ become large. We show the occurrence of ordinary scaling behavior, $\Pi {(t}_w{+t,t}_w\left)\sim F_1{(t/t}_w^{{\mu }_1}\right),$ where ${\mu }_1=1\mathrm{}$ (normal aging) or ${\mu }_1<1\mathrm{}$ (subaging), and the possible simultaneous occurrence of generalized scaling behavior, $t_w^{\gamma }\left[1-\Pi {(t}_w{+t,t}_w\right)\left]\sim F_2{(t/t}_w^{{\mu }_2}\right)$ with ${\mu }_2<{\mu }_1$ (subaging). Which situation occurs depends on the form of the effective transition rates between the low-lying states. Employing a “partial equilibrium concept,” the exponents ${\mu }_{1,2}$ and the asymptotic form of the scaling functions are obtained both by simple scaling arguments and by analytical calculations. The predicted scaling properties compare well with Monte Carlo simulations in dimensions $d=1\mathrm{}-1000$ and it is argued that a mean-field-type treatment of the hopping motion fails to describe the aging dynamics in any dimension. Implications for more general situations involving different forms of transition rates and the occurrence of many scaling regimes in the $t-t_w$ plane are pointed out.