Origin of end-of-aging and subaging scaling behavior in glassy dynamics

Linear response functions of aging systems are routinely interpreted using the scaling variable $t_{\text{obs}}/t_w^{\mu }$, where $t_w$ is the time at which the field conjugated to the response is turned on or off, and where $t_{\text{obs}}$ is the “observation” time elapsed from the field change. The response curve obtained for different values of $t_w$ are usually collapsed using values of $\mu$ slightly below one, a scaling behavior generally known as subaging. Recent spin glass thermoremanent magnetization experiments have shown that the value of $\mu$ is strongly affected by the form of the initial cooling protocol [G. F. Rodriguez et al., Phys. Rev. Lett. 91, 037203 (2003)], and even more importantly [G. G. Kenning et al., Phys. Rev. Lett. 97, 057201 (2006)], that the $t_w$ dependence of the response curves vanishes altogether in the limit $t_{\text{obs}}⪢t_w$. The latter result shows that $t_{\text{obs}}/t_w^{\mu }$ scaling of linear response data cannot be generally valid, thereby casting some doubt on the theoretical significance of the exponent $\mu$. In this work, a common mechanism is proposed for the origin of both subaging and end of aging behavior in glassy dynamics. The mechanism combines real and configuration space properties of the state produced by the initial thermal quench which initiates the aging process.

Figure 1

(Color online) The Thermoremanent Magnetization, calculated according to Eq. (8), is plotted for $t_w=50$, 100, 1000, and $10000$ versus $t_{\text{obs}}/t_w$, (left panel), and versus $t_{\text{obs}}/t_w^{\mu }$, (middle panel). The left panel shows that a $t/t_w$ scaling does not work satisfactorily. The middle panel shows that the standard $t/t_w^{\mu }$ scaling procedure visibly improves the quality of the data collapse. All data are for $T=0.60T_g$. Similar scaling plots were also obtained for other temperatures (not shown). In each case the exponent $\mu$ was estimated as the value providing the best data collapse. In the rightmost panel, the $\mu$ values, thus, obtained are plotted versus $T/T_g$ (squares). The dotted line is only a guide to the eye. A clear maximum is visible at $T/T_g=0.83$.

Figure 2

(Color online) The TRM decay rate, multiplied by the age $t$ is plotted versus $t/t_w$ for $T/T_g=0.83$ and $t_w=50$ (right pointing triangles), 100 (circles), 300 (squares), 630 (diamonds), 1000 (pentagrams), 3600 (hexagrams), 6310 (asterisks) and $\left(10000\right)$ (left pointing triangles). The line is given by Eq. (4). The insert compares the TRM decay measured at $t_w=100\text{s}$ (red line), with the theoretical estimates (blue circles) obtained by Eq. (3).