The successful prediction of the specific heat of solids is a milestone in the kinetic theory of matter due to Debye. No such success, however, has ever been obtained for the specific heat of liquids, which has remained a mystery for over a century. A theory of specific heat of liquids is derived here using a recently proposed analytical form of the vibrational density of states of liquids, which takes into account saddle points in the liquid energy landscape via the so-called instantaneous normal modes (INMs), corresponding to negative eigenvalues (imaginary frequencies) of the Hessian matrix. The theory is able to explain the typical monotonic decrease in specific heat with temperature observed in liquids in terms of the average INM excitation lifetime decreasing with $T$ (in accordance with the Arrehnius law) and provides an excellent single-parameter fitting to several sets of experimental data for atomic and molecular liquids. It also correlates the height of the liquid energy barrier with the slope of the specific heat in the function of temperature in accordance with the available data. These findings demonstrate that the specific heat of liquids is controlled by the instantaneous normal modes, i.e., by localized unstable (exponentially decaying) vibrational excitations and provide the missing connection among anharmonicity, saddle points in the energy landscape, and the thermodynamics of liquids.

• Received 19 January 2021
• Revised 23 March 2021
• Accepted 9 June 2021

DOI:https://doi.org/10.1103/PhysRevE.104.014103