In an article published in the journal PNAS by Matteo Baggioli, and Alessio Zaccone of University of Milano, the law which governs the distribution of vibrational energy states of liquids has been derived mathematically for the first time. The new mathematical theory by Zaccone & Baggioli solved the problem of obtaining the distribution of these complex energy states by combining the so-called Zeldovich regularization for unstable quantum states with the mathematical theory of distributions by L. Schwartz. The final result provides an equation in closed form for the distribution of energy states in liquids, which goes linear in the frequency, as opposed to the Debye law for solids, which goes quadratic in frequency, and also correctly predicts its temperature dependence.

Universal law for the vibrational density of states of liquids | PNAS