Figure 1

Glassy dynamics for $N=8$. (a) Relaxation of $⟨H_0⟩$, starting from random configurations, at various $T$; $g=0$ except for the dashed line at $T=5$. (b, c) The complex susceptibility ${\tilde{\chi }}_M$ versus $T$, measured by ramping from $T=50$ to $T=0$ and back in steps of $\Delta T=1$, and dwell time per temperature step ${\tau }_{\mathrm{dw}}={10}^4$. The dashed line in (b) indicates the equilibrium ${\chi }_M$ in the dc limit. (d, e) The equilibrium spectrum of ${\tilde{\chi }}_M$ at different $T$. Lines are Havriliak-Negami fits [12]. The inset of (e) plots the inverse peak frequency $1/f_p={\tau }_p$ of ${\chi }_M^{\prime \prime }(f)$ versus $1/T$. Data shown are ensemble-averaged over ${10}^3$ realizations.Reuse & Permissions

Figure 2

Aging, rejuvenation, and memory in thermosort, measuring ${\chi }_M^{\prime \prime }(T)$ at $f=4\times {10}^3$ versus $T$. Here $T$ is ramped in steps of $\Delta T/{\tau }_{\mathrm{dw}}=1/{10}^3$. (a) Dotted curve: reference ${\chi }_M^{\prime \prime }(T)$ without aging. Solid curve: cooling was interrupted at $T_{\mathrm{age}}=150$ for a duration ${\tau }_{\mathrm{age}}=4\times {10}^4$, allowing ${\chi }_M^{\prime \prime }(T)$ to age, forming a dip. Warming back up, the aged ${\chi }_M^{\prime \prime }(T)$ systematically dips below the reference in the vicinity of $T_{\mathrm{age}}$. (b) The difference between the aged and reference ${\chi }_M^{\prime \prime }(T)$, obtained for two different values of $T_{\mathrm{age}}$ (dotted line, cooling; solid line, warming). Data shown are ensemble-averaged over ${10}^5$ realizations for $N=30$ and $g=0$.Reuse & Permissions

Figure 3

(a) The $N=8$ configuration space projected onto the ($M$, $H_0$) plane for $g=0$; filled circles mark locally stable states. (b–d) Thermosort trajectories in the projected view, sampled at various $T$. Each color is an independent trajectory, ${10}^4$ in duration, starting from a random configuration. (e, f) Quantifying trajectory localization: the localization extent in $H_0$ and $M$ versus the observational time scale ${\tau }_{\mathrm{obs}}$. Here, $⟨\Delta H_0({\tau }_{\mathrm{obs}})⟩$ and $⟨\Delta M({\tau }_{\mathrm{obs}})⟩$ are ensemble-averaged over 100 independent trajectories, ${10}^5$ in duration; they are then normalized by their equilibrium ${\tau }_{\mathrm{obs}}\to \infty$ values.Reuse & Permissions

Figure 4

(a) The trajectory of a randomly initiated $N=8$ list ($\mathrm{\text{dark}}\to \mathrm{\text{light}}=1\to 8$), slowly cooled from $T=50$ to $T=0$ with $g=0$. The end state at $T=0$ is a locally stable configuration, consisting of two sorted domains concatenated at a “domain wall” ($k=3$, arrow). (b) The number of locally stable states versus $N$, obtained by enumeration up to $N=9$.Reuse & Permissions