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Stylised facts of the glass transition. (a) Slowdown with no apparent structural change: structural relaxation time τ for OTP (symbols, data from Refs. 13–15 ); the blue curve is a VFT fit, log τ/τ0 = A/(T − T c ); 15 the red curve is a fit with the parabolic law, log τ/τ0 = (J/T o )2(T o /T − 1)2; the green curve is an MCT fit, τ ∝ |T − T mct|−γ. (See Refs. 14–16 for details on these fits.) (b) Dynamical heterogeneity: projection in space of an equilibrium trajectory of a two-dimensional supercooled mixture, from Ref. 17 ; particles coloured according to overlap with initial positions (displacement by a particle diameter or more is dark red, and no displacement is dark blue); the trajectory length is about a tenth of a relaxation time at these conditions; spatial segregation of dynamics is evident. (c) Anomalous thermodynamic response: temperature variation of the specific heat C p of OTP on cooling (black curve) and heating (green curve), from differential scanning calorimetry; 18 ΔC p is the difference in specific heat between the liquid and the glass. (d) Jamming at zero temperature: for densities below ϕ J particles are not in contact and the system is fluid; at the jamming density ϕ J the system becomes isostatic and mechanically stable; for densities beyond ϕ J particles would overlap, a situation not allowed for hard objects.
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Biasing of ensembles in order to access dynamical or thermodynamical glass transitions. (a) s-ensemble for a Lennard-Jones binary mixture, from Ref. 115 , biased by dynamical activity through the counting field s. The average activity K s displays a crossover from a large value at s = 0 to a small one at s > 0; this becomes sharper for longer observation times (τ is the alpha-relaxation time). At s c , the order parameter distribution displays the bi-modality characteristic of a first-order transition, between a dynamically active equilibrium phase, and a dynamically inactive non-equilibrium one. (b) Suggested “space-time” phase diagram: while dynamics takes place within the active phase, the closeness of the first-order transition to the inactive phase gives rise to fluctuation behavior manifested in dynamic heterogeneity. (c) Result of numerical simulation of pinned harmonic spheres. 122 Large spheres represent pinned particles (rescaled in size by a factor 0.5), small dots are the superposition of the positions of fluid particles obtained from a large number of independent equilibrium configurations in presence of the pinned particles. (d) Phase diagram for pinned particles obtained by renormalization group analysis in Ref. 118 starting from a Ginzburg-Landau action.
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