The low-frequency vibrational and low-temperature thermal properties of amorphous solids are markedly different from those of crystalline solids. This situation is counterintuitive because all solid ...materials are expected to behave as a homogeneous elastic body in the continuum limit, in which vibrational modes are phonons that follow the Debye law. A number of phenomenological explanations for this situation have been proposed, which assume elastic heterogeneities, soft localized vibrations, and so on. Microscopic mean-field theories have recently been developed to predict the universal non-Debye scaling law. Considering these theoretical arguments, it is absolutely necessary to directly observe the nature of the low-frequency vibrations of amorphous solids and determine the laws that such vibrations obey. Herein, we perform an extremely large-scale vibrational mode analysis of a model amorphous solid. We find that the scaling law predicted by the mean-field theory is violated at low frequency, and in the continuum limit, the vibrational modes converge to a mixture of phonon modes that follow the Debye law and soft localized modes that follow another universal non-Debye scaling law.
We numerically study the relaxation dynamics and associated criticality of non-Brownian frictionless soft spheres below jamming in spatial dimensions
d
=
2
, 3, 4, and 8, and in the mean-field ...Mari–Kurchan model. We discover non-trivial finite-size and volume fraction dependences of the relaxation time associated to the relaxation of unjammed packings. In particular, the relaxation time is shown to diverge logarithmically with system size at any density below jamming, and no critical exponent can characterise its behaviour approaching jamming. In mean-field, the relaxation time is instead well-defined: it diverges at jamming with a critical exponent that we determine numerically and differs from an earlier mean-field prediction. We rationalise the finite
d
logarithmic divergence using an extreme-value statistics argument in which the relaxation time is dominated by the most connected region of the system. The same argument shows that the earlier proposition that relaxation dynamics and shear viscosity are directly related breaks down in large systems. The shear viscosity of non-Brownian packings is well-defined in all
d
in the thermodynamic limit, but large finite-size effects plague its measurement close to jamming.
We use computer simulations to study the thermodynamic properties of a glass-former in which a fraction c of the particles has been permanently frozen. By thermodynamic integration, we determine the ...Kauzmann, or ideal glass transition, temperature Formula at which the configurational entropy vanishes. This is done without resorting to any kind of extrapolation, i.e., Formula is indeed an equilibrium property of the system. We also measure the distribution function of the overlap, i.e., the order parameter that signals the glass state. We find that the transition line obtained from the overlap coincides with that obtained from the thermodynamic integration, thus showing that the two approaches give the same transition line. Finally, we determine the geometrical properties of the potential energy landscape, notably the T - and c dependence of the saddle index, and use these properties to obtain the dynamic transition temperature Formula. The two temperatures Formula and Formula cross at a finite value of c and indicate the point at which the glass transition line ends. These findings are qualitatively consistent with the scenario proposed by the random first-order transition theory.
Significance Confirming by experiments or simulations whether or not an ideal glass transition really exists is a daunting task, because at this point the equilibration time becomes astronomically large. Recently it has been proposed that this difficulty can be bypassed by pinning a fraction of the particles in the glass-forming system. Here we study numerically a liquid with such random pinned particles and identify the ideal glass transition point Formula at which the configurational entropy vanishes, thus realizing for the first time, to our knowledge, a glass with zero entropy. We find that as the fraction of pinned particles increases, the Formula line crosses the dynamical transition line, implying the existence of an end point at which theory predicts a new type of criticality.
The low-frequency vibrations of glasses are markedly different from those of crystals. These vibrations have recently been categorized into two types: spatially extended vibrations, whose vibrational ...density of states (vDOS) follows a non-Debye quadratic law, and quasilocalized vibrations (QLVs), whose vDOS follows a quartic law. The former are explained by elasticity theory with quenched disorder and microscopic replica theory as being a consequence of elastic instability, but the origin of the latter is still debated. Here, we show that the latter can also be directly derived from elasticity theory with quenched disorder. We find another elastic instability that the theory encompasses but that has been overlooked so far, namely, the instability of the system against a local dipolar force. This instability gives rise to an additional contribution to the vDOS, and the spatial structure and energetics of the mode originating from this instability are consistent with those of the QLVs. Finally, we construct a model in which the additional contribution to the vDOS follows a quartic law.
The low-frequency vibrations of glasses are markedly different from those of crystals.
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