  The successful prediction of the specific heat of solids is a milestone in
the kinetic theory of matter, due to Debye (1912). 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 (DOS) 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 of
specific heat with temperature observed in liquids, in terms of the average INM
excitation lifetime decreasing with T (in accordance with 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 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 between anharmonicity, saddle
points in the energy landscape, and the thermodynamics of liquids.

Soft Condensed Matter;Materials Science;Statistical Mechanics