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

Universal law for the vibrational density of states of liquids