The anomalous density of states and quasi-localized vibration through homogeneous thermalization of an inhomogeneous elastic system.

Amorphous solids exhibit dynamic inhomogeneity due to their lack of translational symmetry, leading to distinct vibrational properties compared to crystalline solids. These include anomalous low-frequency vibrational density of states (VDOS) and associated deviations in low-temperature thermal behavior. While particle-level investigations have advanced rapidly, a fundamental interpretation of these phenomena based on physical laws remains underdeveloped. In this work, we begin with the quasi-equilibrium condition, which mandates a homogeneous distribution of elastic potential energy in an inhomogeneous elastic solid over long timescales. The analytical results suggest that the anomalous low-frequency VDOS scalingD(ω)∝ω4emerges when quasi-equilibrium is imposed on an elastically inhomogeneous system. At higher frequencies, beyond a crossover dictated by the length scale of inhomogeneity, the VDOS transitions toD(ω)∝ω2, consistent with Debye's law for crystals. These findings align with recent particle-level studies and suggests that the universal low-frequency anomaly in amorphous solids arises from homogeneous thermalization.