Universal law for the vibrational density of states of liquids

An analytical derivation of the vibrational density of states (DOS) of liquids, and, in particular, of its characteristic linear in frequency low-energy regime, has always been elusive because of the presence of an infinite set of purely imaginary modes-the instantaneous normal modes (INMs). By combining an analytic continuation of the Plemelj identity to the complex plane with the overdamped dynamics of the INMs, we derive a closed-form analytic expression for the low-frequency DOS of liquids. The obtained result explains, from first principles, the widely observed linear in frequency term of the DOS in liquids, whose slope appears to increase with the average lifetime of the INMs. The analytic results are robustly confirmed by fitting simulations data for Lennard-Jones liquids, and they also recover the Arrhenius law for the average relaxation time of the INMs, as expected.

Automated summary

An analytical derivation of the vibrational density of states of liquids, particularly focusing on the low-energy regime, was achieved by considering the presence of infinite purely imaginary modes. The obtained results not only explain the linear frequency term of the DOS in liquids, but also show that the slope increases with the average lifetime of the modes. The analytical results were validated through fitting simulations data for Lennard-Jones liquids and confirmed the Arrhenius law for the average relaxation time of the modes.