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.

由于存在无限数量的纯虚模，瞬时状态，液体的振动态密度（DOS）的分析推导，尤其是其在低频率能量状态下的线性特征，一直是难以捉摸的。正常模式（INM）。通过将对Plemelj身份的解析连续性与复杂的INM动力学相结合，我们得出了液体低频DOS的闭式解析表达式。所获得的结果从第一原理出发，解释了在液体中DOS的频度线性观察到的线性关系，其斜率似乎随着INM的平均寿命而增加。分析结果通过Lennard-Jones液体的拟合模拟数据得到了有力的证实，