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Low-frequency vibrational harmonic modes of glasses are frequently used to rationalize their universal low-temperature properties. One well studied feature is the excess low-frequency density of states over the Debye model prediction. Here, we examine the system size dependence of the density of states for two-dimensional glasses. For systems of fewer than 100 particles, the density of states scales with the system size as if all the modes were plane-wave-like. However, for systems greater than 100 particles, we find a different system-size scaling of the cumulative density of states below the first transverse sound mode frequency, which can be derived from the assumption that these modes are quasi-localized. Moreover, for systems greater than 100 particles, we find that the cumulative density of states scales with the frequency as a power law with the exponent that leads to the exponent β = 3.5 for the density of states. For systems whose sizes were investigated, we do not see a size-dependence of exponent β. 

We propose an alternative theory for the relaxation of density fluctuations in glass-forming fluids. We derive an equation of motion for the density correlation function that is local in time and is similar in spirit to the equation of motion for the average non-uniform density profile derived within the dynamic density functional theory. We identify the Franz–Parisi free energy functional as the non-equilibrium free energy for the evolution of the density correlation function. An appearance of a local minimum of this functional leads to a dynamic arrest. Thus, the ergodicity breaking transition predicted by our theory coincides with the dynamic transition of the static approach based on the same non-equilibrium free energy functional. 

Sound attenuation in low-temperature amorphous solids originates from their disordered structure. However, its detailed mechanism is still being debated. Here, we analyze sound attenuation starting directly from the microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping coefficient. We verify that the sound damping coefficients calculated from our expression agree very well with results from independent simulations of sound attenuation. Small wavevector analysis of our expression shows that sound attenuation is primarily determined by the non-affine displacements’ contribution to the sound wave propagation coefficient coming from the frequency shell of the sound wave. Our expression involves only quantities that pertain to solids’ static configurations. It can be used to evaluate the low-temperature sound damping coefficients without directly simulating sound attenuation. 

The temperature dependence of the thermal conductivity of amorphous solids is markedly different from that of their crystalline counterparts, but exhibits universal behaviour. Sound attenuation is believed to be related to this universal behaviour. Recent computer simulations demonstrated that in the harmonic approximation sound attenuation Γ obeys quartic, Rayleigh scattering scaling for small wavevectors k and quadratic scaling for wavevectors above the Ioffe–Regel limit. However, simulations and experiments do not provide a clear picture of what to expect at finite temperatures where anharmonic effects become relevant. Here we study sound attenuation at finite temperatures for model glasses of various stability, from unstable glasses that exhibit rapid aging to glasses whose stability is equal to those created in laboratory experiments. We find several scaling laws depending on the temperature and stability of the glass. First, we find the large wavevector quadratic scaling to be unchanged at all temperatures. Second, we find that at small wavevectors Γ ∼ k 1.5 for an aging glass, but Γ ∼ k 2 when the glass does not age on the timescale of the calculation. For our most stable glass, we find that Γ ∼ k 2 at small wavevectors, then a crossover to Rayleigh scattering scaling Γ ∼ k 4 , followed by another crossover to the quadratic scaling at large wavevectors. Our computational observation of this quadratic behavior reconciles simulation, theory and experiment, and will advance the understanding of the temperature dependence of thermal conductivity of glasses.