Dispersion relation for collective modes in classical monatomic liquids and amorphous solids

Existing formulas for the dispersion relation ω(k) of collective modes in liquids and amorphous solids are shown to have the often severe limitation that the velocity of sound is not correctly given. Therefore, a generalization to include wavelength dependence is made of the adiabatic compressibility, via the so-called thermodynamic fluctuation. The velocity of sound is thereby correctly related to the static structure (i.e., pair correlation function) and the ratio of the specific heats γ. This demonstrates that a fully quantitative expression for the dispersion relation ω(k) must involve three- and four-atom correlation functions in liquids and amorphous solids. Theories posed in terms of frequency moments of the dynamical structure factor would have to incorporate up to and including the eighth frequency moment in order to reproduce correctly the velocity of sound in the disordered systems considered here.