Heat capacity and the number of instantaneous-normal modes of a simple fluid

In the last decade, approaches to calculating the thermal properties of liquids have been actively developing. The most developed approaches are those based on the use of the “phonon” theory, which was previously successfully applied to calculate the thermal properties of solids. The “phonon” theory is based on the hypothesis that in addition to longitudinal acoustic waves of a wide range of frequencies, transverse high-frequency acoustic waves can propagate in a liquid. To find the heat capacity of a liquid, the Debye approach is used, which was modified by taking into account only high-frequency transverse waves in the excitation spectrum. Another approach is based on finding the frequency distribution of normal oscillations (instantaneous-normal modes) in a liquid and constructing the thermal and transport properties of the liquid using the found frequencies. In this paper, the equivalence of both approaches is shown in the harmonic approximation, and a simple one-parameter approximation formula is proposed for calculating the isochoric heat capacity of simple liquids.