Log-log (top) and log-linear (bottom) plots of the reduced relaxation time and viscosity vs. the reduced cage-rattling amplitude. τ CF and η CF are the conversion factors from the MD time or viscosity units to the actual units, respectively. The two panels emphasize the fast (top) and the slow (bottom) relaxation regimes. The numbers in parenthesis denote the fragility m. The black curve is Eq. (3) . The dashed curves bound the accuracy of the scaling. 32 Experimental details are given elsewhere. 35 For each glassformer is drawn by fitting Eq. (1) to the experimental data τα (or η) vs. 1/⟨u 2⟩ with Eq. (2) as constraint and considering τ CF , or η CF , as adjustable. The conversion factors little depend on the system (with the notable exception of B2O3). 32–34
Self-part of the Van Hove function of the A particles G s (r, t ⋆) (top) and G s (r, ταA ) (bottom) in selected states (ρ, T, p, q) (p and q are the characteristic exponents of the potential Eq. (6) ). Note that states with identical G s (r, t ⋆) have identical G s (r, ταA ), Eq. (5) . The same correlation has been noted in polymers. 44 The clusters of states (ρ, T, p, q) are: C1 cluster: (1.204,1.0,6, 12), (1.204, 0.5,5,8) with ταA ≃ 1.2; C2 cluster: (1.204, 0.6,6, 12), (1.125, 0.425, 6, 12) with ταA ≃ 4.7; C3 cluster: (1.125, 0.525,6, 12), (1.063,0.45,6, 12) with ταA ≃ 15; C4 cluster: (1.204,0.27,5,8), (1.204, 0.481,6,11) with ταA ≃ 62. (Inset): tails of G s (r, t ⋆) (top) and G s (r, ταA ) (bottom). Note the developing exponential decay at large distances on increasing ταA , a signature of dynamic heterogeneity. 47 The exponential tail of G s (r, t) is observed at both short (t = t ⋆, top panel) and long (t = t ⋆, bottom panel) times.
(Top) MSD of the A (left) and B (right) particles in selected states (ρ, T, p, q). (Inset) Plot of Δ(t) ≡ ∂log ⟨r 2(t)⟩/∂log t. The minimum of Δ(t) occurs at the time t ⋆, a measure of the trapping time of the particle (t ⋆ is the same for A and B particles within the errors). (Bottom) Corresponding ISF functions. The q max values for the particles A and the B are and , where is the average distance between B particles. The plotted states have parameters ρ = 1.204, T = 0.45, 0, 48, 0.6, 0.7, 0.8, 1.0, 1.2, 1.5, 2.0, 3.0 (from the rightmost to the leftmost curve in each panel) and Lennard-Jones interacting potential (Eq. (6) with p = 6, q = 12).
Non-gaussian parameter of the states in Fig. 3 . Heterogeneous dynamics develops as time goes by and decays at long times in the diffusive regime, resulting in a maximum value of the parameter . Non-gaussianity increases by decreasing the temperature.
(Top) Correlation plot between the structural relaxation times, the diffusivity (inset), and the picosecond MSD of the components A and B. The dashed lines in the main panel are best-fits with Eq. (1) and parameters as Table I . The dashed lines in the inset are (virtually linear) guides for the eyes. (Bottom) Scaling of the master curves of the A and B components. The black solid line is Eq. (3) . The latter, as Eq. (1) , 32,33 fails for τα ≲ 1. The black solid line in the inset is a (virtually linear) guide for the eyes. The plotted states are: (i) all the states in Fig. 3 ; (ii) states with interacting potential U 5, 8, α, β(r), ρ = 1.204, T = 0.267, 0.270, 0.275, 0.350, 0.400, 0.450, 0.500, 0.600, 0.700, 0.800, 0.900, 1.000, 1.200, 1.500, 2.000, 3.000; (iii) states with LJ interacting potential U 6, 12, α, β(r), ρ = 1.125 with T = 0.350, 0.375, 0.425, 0.450, 0.475, 0.500, 0.525, 0.600, 0.700, 1.000, 2.000, 3.000 and ρ = 1.063 with T = 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 1.0, 2.0, 3.0.
The product D i τα i vs. (top panel) and the reduced ST-MSD (bottom panel), i = A, B for all the states of Fig. 5 . The onset of the SE violation for is indicated with the full vertical lines (uncertainty marked by dashed lines). Notice that the SE breakdown occurs when the second term on the rhs of Eq. (3) becomes larger than the first term. The bottom panel also plots the quantity D P M τα P vs. (empty triangles), referring to a model polymer system with chain length M, and shows that the SE breakdown occurs at the same point. 38,55
Thermodynamic scaling of ST-MSD (top), structural relaxation time (middle), and diffusion coefficient (bottom) of the A component of all the states of Fig. 5 with Lennard-Jones potential. γ = 5.1. The continuous line in the top panel is a polynomial best-fit function. The latter is combined with Eq. (1) with and as in Table I to lead, with no adjustable parameters, to the continuous line in the middle panel. It is known that Eq. (1) fails for τα ≲ 1. 32,33 The dashed line in the bottom panel is a guide for the eyes.
Toy model of the barrier crossing in a liquid. The grey particle overcomes the barrier due to the blue particles by displacing them along the line joining their centers and reaching the transition point (red dot). Initially, the blue particles are at distance 2x apart from each other and the grey particle is at distance a from the transition point. All the particles have unit diameter.
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