Statistical Mechanics and Dynamics of a Three-Dimensional Glass-Forming System

In the context of a classical example of glass formation in three dimensions, we exemplify how to construct a statistical-mechanical theory of the glass transition. At the heart of the approach is a simple criterion for verifying a proper choice of upscaled quasispecies that allow the construction of a theory with a finite number of “states.” Once constructed, the theory identifies a typical scale $\xi$ that increases rapidly with lowering the temperature and which determines the $\alpha$-relaxation time ${\tau }_{\alpha }$ as ${\tau }_{\alpha }\sim \exp (\mu \xi /T)$, with $\mu$ a typical chemical potential. The theory can predict relaxation times at temperatures that are inaccessible to numerical simulations.

Figure 1

Time dependence of the correlation functions (2) for a range of temperatures (decreasing from left to right) as shown in the figure. The inset shows the relaxation time ${\tau }_{\alpha }$ in a log-lin plot vs $1/T$, compared to an Arrhenius temperature dependence.

Figure 2

Temperature dependence of the concentrations of the various quasispecies. Symbols are simulation data, and the lines are a guide to the eye.

Figure 3

The approximate linear dependence of the free energies of the chosen quasispecies on the temperature. From the slope we read the degeneracy and from the intercept the enthalpies (up to normalization); cf. Eq. (4). Note that when the free energies are large we do not have data: the concentrations become exponentially small, and in a finite simulation box they disappear completely.

Figure 4

Upper panel: The degeneracies $g_s(n)$ and $g_{\ell }(n)$ read from the slopes of Fig. 3 (in circles) and the degeneracies according to the Gaussian model Eq. (4). Middle panel: The measured enthalpies. Lower panel: Comparison of the measured concentrations of quasispecies to those calculated from Eqs. (5) using the model degeneracies and measured enthalpies. Here symbols are data, and lines are theoretical predictions.

Figure 5

The temperature dependence of $C_{\mathrm{liq}}(T)$ is shown as the upper continuous line. The contributions of the various liquid subspecies are shown with symbols which are identified in the inset.

Figure 6

The relaxation time ${\tau }_{\alpha }(T)$ in terms of the typical scale $\xi (T)$. We show the excellent fit to Eq. (8) with $\mu =0.04$. Note that the intercept at $T\to \infty$ is of the order of unity as it must be.