Universal law for the vibrational density of states of liquids

TL;DR

This paper derives a universal analytical expression for the vibrational density of states in liquids, explaining the linear low-frequency behavior and its dependence on the lifetime of instantaneous normal modes, confirmed by simulations.

Contribution

It provides the first analytical derivation of the low-frequency DOS in liquids, linking it to the dynamics of overdamped instantaneous normal modes.

Findings

The derived DOS explains the linear frequency dependence observed in liquids.

The slope of the DOS correlates with the lifetime of INMs.

Simulation data for Lennard-Jones liquids confirms the analytical results.

Abstract

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 purely 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.