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 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.
Highlights
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)
Well-known examples of unstable states with overwhelming imaginary part arise in the nuclear α-decay (Gamow states) and, in particle physics—the W and Z 0 bosons [2]— where they are usually called “resonances.” within hydrodynamics and gravitational theories, these excitations are labeled “quasinormal modes” [3], and they are the responsible for the black holes’ ringdown recently observed by LIGO Scientific Collaboration and Virgo Collaboration [4]
The major obstacle resides in the fact that the Plemelj identity, that formally provides the connection between the propagator and the DOS, is defined on the real axis; only states with real frequencies ω, and positive eigenvalues λ = ω2 of the Hessian, are allowed
Summary
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). They play a crucial role in liquids and glasses, where they are often called instantaneous normal modes (INMs) [6,7,8] and correspond to saddle points, with negative eigenvalues, in the energy landscape [9,10,11].
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