Generalized Fluctuation-Dissipation Relation and Effective Temperature Upon Heating a Deeply Supercooled Liquid

We show that a generalized fluctuation-dissipation relation applies upon instantaneously increasing the temperature of a deeply supercooled liquid. This has the same two-step shape of the relation found upon cooling the liquid, but with opposite violation, indicating an effective temperature that is lower than bath temperature. We show that the effective temperature exhibits some sensible time dependence and that it retains its connection with the partitioned phase space visited in aging. We underline the potential relevance of our numerical results for experimental studies of the fluctuation-dissipation relation in glassy systems.

Figure 1

Response versus correlation parametric plots (FD plots) for several temperature up jumps as the final (bath) temperature $T_{\mathrm{bath}}$ is varied. The time $t^{\prime }$ varies between $6\times {10}^2$ and $t=8\times {10}^3$ MC steps. A violation of the FDT (dashed line) characterized by $X>1$ (i.e., $T_{\mathrm{eff}}<T$) is clearly observed at large time lags $t-t^{\prime }$ as evidenced by the straight-line fits of the points departing from the FDT. It can be seen how, increasing the final temperature of the jump, the FDT violation is gradually lost. We show also how two identical FD plots can be generated by instantaneously increasing the volume (at fixed $T$) or the temperature (at fixed $\rho$) in analogy with the findings of Ref. 15. Inset: FD plot in $T$ up jump for two different wave vectors. The fitting line of the $k=5.2$ case fits very well also the $k=7.2$ case indicating that $T_{\mathrm{eff}}$ does not depend on the chosen wave vector as found also in quenches [13].

Figure 2

FD plots for $T$ up jumps starting with different initial temperatures ($6\times {10}^2<t^{\prime }<8\times {10}^3$ MC steps). Notice that as the initial temperature is increased the deviation from the FDT reduces.

Figure 3

FD plots for several $T$ up jumps as the time $t$ increases. Here $t^{\prime }$ varies always from $\sim t/10$ to $t$. The deviation from the FDT (dashed line) gradually decreases as $t$ increases suggesting that the slow degrees of freedom of system slowly thermalize to the bath temperature. The straight lines have slope $-T_{\mathrm{bath}}/T_{\mathrm{int}}(t)$ (only the intercept has been adjusted for fitting), where $T_{\mathrm{int}}$ is the internal temperature obtained by using Eq. (4) [14] (see text).

Figure 4

Left: Average inherent structure energy in equilibrium as a function of $T$. Right: Average inherent structure energy after a $T$ up jump ($T=0.41\to T=0.75$, $\rho =1.20$) as a function of time. By performing the construction indicated by the dashed line, it is possible to relate the aging time $t$ with $T_{\mathrm{eq}}$, i.e., the $T$ at which basins of the same depth as the one visited during aging are explored in equilibrium. After correcting for the vibrational contribution [Eq. (4)], we obtain a value for the internal temperature $T_{\mathrm{int}}$ that compares very well the FDT violation (see Fig. 3).