Elastic moduli fluctuations predict wave attenuation rates in glasses

The disorder-induced attenuation of elastic waves is central to the universal low-temperature properties of glasses. Recent literature offers conflicting views on both the scaling of the wave attenuation rate $\Gamma(\omega)$ in the low-frequency limit ($\omega\!\to\!0$), and on its dependence on glass history and properties. A theoretical framework -- termed Fluctuating Elasticity Theory (FET) -- predicts low-frequency Rayleigh scattering scaling in $d$ spatial dimensions, $\Gamma(\omega)\!\sim\!\gamma\,\omega^{d+1}$, where $\gamma\!=\!\gamma(V_{\rm c})$ quantifies the coarse-grained spatial fluctuations of elastic moduli, involving a correlation volume $V_{\rm c}$ that remains debated. Here, using extensive computer simulations, we show that $\Gamma(\omega)\!\sim\!\gamma\,\omega^3$ is asymptotically satisfied in two dimensions ($d\!=\!2$) once $\gamma$ is interpreted in terms of ensemble -- rather than spatial -- averages, where $V_{\rm c}$ is replaced by the system size. In so doing, we also establish that the finite-size ensemble-statistics of elastic moduli is anomalous and related to the universal $\omega^4$ density of states of soft quasilocalized modes. These results not only strongly support FET, but also constitute a strict benchmark for the statistics produced by coarse-graining approaches to the spatial distribution of elastic moduli.