Competition between energy- and entropy-driven activation in glasses

In simplified models of glasses we clarify the existence of two different kinds of coexisting activated dynamics, with one of the two dominating over the other. One is the energy barrier hopping that is typically used to understand activation, and the other, which we call entropic activation, is driven by the scarcity of convenient directions in phase space. When entropic activation dominates, the height of the energy barriers is no longer the primary factor governing the system's slowdown. In our analysis, dominance of one mechanism over the other depends on temperature and the shape of the density of states. We also find that at low temperatures a phase transition between the two kinds of activation can occur. Our observations are used to provide a scenario that can harmonize the facilitation and thermodynamic pictures of the slowdown of glasses into a single description.

Figure 1

Left: Phase diagram of the EREM, with the associated characteristic energies and dynamical regimes. For $\beta <{\beta }_{\mathrm{c}}$ we have the equilibrium phase. The solid line is the equilibrium energy $\langle E\rangle$. At $\beta =2{\beta }_{\mathrm{c}}$ we have the transition from the entropically to the energetically activated regime. The dashed line is the attractor energy. The horizontal dashed-dotted line represents $E_{\mathrm{th}}(N=10)$. $E_{\mathrm{th}}$ exists $\forall \beta$ but is only relevant for $\beta >2{\beta }_{\mathrm{c}}$. Right: Two diagrams showing how activated dynamics qualitatively takes place in each of the regimes.

Figure 2

Median $E_{\mathrm{ridge}}(\beta ;N)$ in the EREM (a) and REM (b). Different lines stand for different system sizes. Here and in Fig. 3, results from memory-1 dynamics are shown in solid lines, and results from full-memory dynamics ($N<30$) are shown as markers.

Figure 3

Median $E_{\mathrm{ridge}}$ as a function of $N$ in the EREM and REM (inset). Different lines correspond to different values of $\beta$. The black dashed line represents $E_{\mathrm{th}}\left(N\right)$ for each model, according to Eqs. (4) and (5).

Figure 4

The density of ridge energies in the EREM model, using memory-1 dynamics, for $N=10000$.

Figure 5

Energy as a function of time, in simulations with Gillespie (red) and regular Monte Carlo (blue) dynamics. Each row is a different inverse temperature. On the first column we have the full memory dynamics, while on the second we have the memory-1 dynamics (Appendix pp2).

Figure 6

Energy as a function of time for memory and memory-1 dynamics. The dashed curve is the slope $-Tln\left(t\right)$ that one would expect in the infinite-size limit.